ANGLE TRISECTOR, AS VALIDATED TO PERFORM ACCURATELY OVER A WIDE RANGE OF DEVICE SETTINGS BY A NOVEL GEOMETRIC FORMING PROCESS; ALSO CAPABLE OF PORTRAYING FINITE LENGTHS THAT ONLY COULD BE APPROXIMATED BY MEANS OF OTHERWISE APPLYING A COMPASS AND STRAIGHTEDGE TO A GIVEN LENGTH OF UNITY; THAT FURTHERMORE FUNCTIONS AS A LEVEL WHOSE INHERENT GEOMETRY COULD BE ADAPTED FOR MANY OTHER USES SUCH AS BEING INCORPORATED INTO THE DESIGN OF A HYDRAULIC CAR LIFT
20180201050 ยท 2018-07-19
Inventors
Cpc classification
G01B3/56
PHYSICS
G01B1/00
PHYSICS
International classification
Abstract
A newly proposed articulating invention, each of whose four constituent embodiments is designed to trisect any of a multitude of suitably described angles by means of becoming properly set to its designated magnitude; thus automatically portraying a motion related solution for the trisection of an angle that discloses complete routing details of a pathway that leads from such designated magnitude all the way back to its trisector; thereby discerning the whereabouts of certain intersection points which evade detection when attempting to otherwise locate them by means of applying only a straightedge and compass to an angle of such designated magnitude; furthermore projecting finite lengths of any trisector that bears cubic irrational trigonometric properties, being those that cannot be duplicated, but only approximated, when applying a straightedge and compass to a given length of unity; and being of a unique design that could be adapted to function as a level.
Claims
1. A flowchart, developed for the express purpose of unlocking important secrets concerning trisection which have remained shrouded in mystery for millennia, thereby plaguing mankind due to their persistence; as based upon the premise that a quadratic irrational number is the magnitude of any length which can be geometrically constructed from a given length of unity other than that which is of a rational value, and a cubic irrational number is any real number other than that which is rational or quadratic irrational; and organized into series of interrelated tasks that are comprised of: identifying some fundamental mathematics limitation which can be held responsible for difficulties normally experienced when attempting to perform trisection; uncovering some relevant, but as yet unknown geometric property; delineating the degree of imposition that such fundamental mathematics limitation causes; revealing a primary function which could be incorporated into a device in order to successfully overcome the degree of imposition that such fundamental mathematics limitation prefers; assuring that such deficiency could be effectively mitigated by means of designing some device which performs such newly revealed primary function; providing inputs which consist of: a trisection rationale, to be introduced in order to account for how a newly proposed geometric forming process can account for limitations otherwise encountered when attempting to perform certain unattainable Euclidean operations such as trying to solve the classical problem of the trisection of an angle; an improved drawing pretext, to be developed as a extension to conventional Euclidean practice, whose novel format could dramatically aid in simplifying what otherwise would be considered to be a rather convoluted procedure for substantiating that such newly proposed geometric forming process actually can compensate for some unknown difficulty which is suspected to prevent the classical problem of the trisection of an angle from being solved; a mathematic demarcation, to be established in order to expose an overall order which relates major fields of mathematics to one another; a set of rules, to be applied to mechanisms comprised largely of linkages and interconnecting pivot pins that undergo articulation, thereby supplementing rules which instead govern conventional Euclidean practice and thus apply only to stationary geometric construction patterns; and a probabilistic proof of mathematic limitation, to be specified as an entirely independent mathematical method for verifying that the classical problem of the trisection of an angle cannot be solved solely by conventional Euclidean means; devising a geometric forming process that is based upon, as well as accounts for, such above described inputs; discrediting claims falsely alleging that the classical problem of the trisection of an angle actually can be solved; dispelling the notion that a singular drawing solution can substantiate trisection over a multitude of trisecting emulation mechanism settings; specifying supplemental device capabilities that accompany trisection events; categorizing trisection inventions according to specific design features that they share in common; preparing a requirements chart which lists provisions that must be satisfied in order to acceptably substantiate that an articulating mechanism can perform trisection accurately throughout its entire range of device settings: refining a proposed invention design to the point where it complies with all of the provisions listed in such requirements chart; checking to assure that such proposed invention design, once having been properly refined, truly adheres to all of the provisions listed in such requirements chart; and substantiating that a trisecting emulation mechanism can automatically portray a motion related solution for the problem of the trisection of an angle whenever it becomes properly set.
2. A comprehensive methodology, consisting of findings which represent an amalgamation of pertinent results obtained after conducting all of the tasks described in claim 1, and hereby contending that: the classical problem of the trisection of an angle actually cannot be solved because conventional Euclidean practice has limited capabilities; an unknown geometric property, now to be referred to as an overlapment point and defined as any intersection point that cannot be located solely by conventional Euclidean means exclusively with respect to any rendered information that is contained within the very geometric construction pattern which it appears in, as considered to reside within any geometric construction pattern whose rendered angle is of a magnitude that amounts to exactly three times the size of its given angle; the degree of imposition caused by an availability of overlapment points is that within any geometric construction pattern which they might reside in at least one pathway which leads from rendered information all the way back to a given set of previously defined geometric data cannot be fully backtracked upon; chiefly because such overlapment points elude detection solely by conventional Euclidean means in such circumstances, thereby characterizing any of such patterns as being completely irreversible; the primary function of a trisecting emulation mechanism is to regenerate a static image for virtually any designated angle which such device could be properly set to, thereby automatically portraying a motion related solution for the problem of the trisection of an angle; the Euclidean deficiency of being unable to fully backtrack upon any irreversible geometric construction pattern whose rendered angle is of a magnitude which amounts to exactly three times the size of its given angle, being a detriment which prevents the classical problem of the trisection of an angle from being solved solely by conventional Euclidean means, actually can be mitigated by means of setting a trisecting emulation mechanism to some designated magnitude in order to automatically portray a motion related solution for the problem of the trisection of an angle; hence locating overlapment points that complete the pathway leading from such designated magnitude all the way back to a trisector that otherwise would remain indistinguishable; the very manner in which a newly proposed geometric forming process is intended to perform and the capabilities it is expected to afford need to be fully accounted for, thereby requiring a method to be introduced which explains in detail exactly what becomes accomplished by each major segment that contributes to its overall structure, as comprising: a trisection rationale which presents: the logic behind the existence of overlapment points and the vital role they play in characterizing irreversible geometric construction patterns, and the basis for fully describing certain motions by means of replication, being a process whereby a virtually infinite number of singular geometric construction patterns, all related to one another in some specified Euclidean manner, become arranged in some particular fashion and thereafter animated in order to project such motion on a screen, with the full understanding that each of such representations not only distinguishes the overall shape of such motion at a singular point in time, but furthermore can be depicted with respect to any other drawing appearing therein; an improved drawing pretext which, with respect to the substantiation of any motion related solution for the problem of the trisection of an angle, assumes the form of a newly discovered Euclidean formulation, being defined as an entire family of geometric construction patterns, or more specifically, an infinite number of geometric construction patterns whose overall shapes vary imperceptibly from one to the next, thereby making each of such drawings entirely unique due to a slight variation appearing in the magnitude of its given angle, as accounted for in the very first step of a specific sequence of Euclidean operations from which all of such drawings can be exclusively derived; a mathematic demarcation which, presented in the form of a chart, indicates that: only a geometric forming process, as represented by some particular Euclidean formulation, can account for all real number types, including those which qualify as being cubic irrational numbers, and thereby can be defined to consist of all real numbers except for those which are either rational or quadratic irrational; whereas conventional Euclidean practice, as depicted by any singular geometric construction pattern, can account for only rational and quadratic irrational number types; and only a geometric forming process, as represented by some particular Euclidean formulation, can account for all algebraic equation and associated function format types which express singular unknown variables, including cubic equation and cubic function format types; whereas conventional Euclidean practice, as depicted by any singular geometric construction pattern, can account for only linear and quadratic equations, as well as their linear and quadratic function counterparts; a set of rules that address geometric properties which remain constant, or unalterable as mechanisms comprised largely of linkages and interconnecting pivot pins become articulated in a specific manner, thereby enabling certain conclusions to be drawn as to the very nature which some of their members might assume with respect to each other during certain motions; a probabilistic proof of mathematic limitation which indicates that when applying only a straightedge and compass to a given angle of arbitrarily selected size in order to thereby geometrically construct a rendered angle that amounts to exactly three times its magnitude, the likelihood that such rendered angle turns out to be of a size which equals the designated magnitude of an angle that has been slated for trisection approaches zero percent, thereby validating that since the classical problem of the trisection of an angle cannot be solved in such manner, the only remaining recourse is to implement the alternative approach of attempting to backtrack upon some particularly selected irreversible geometric construction pattern, being a mathematic limitation that cannot be accomplished by conventional Euclidean means; any claim alleging that the classical problem of the trisection of an angle has been solved can be discredited on the grounds that it can be shown either to contain faulty logic, or else introduce relevant information, thereby contributing to some geometric solution that applies to an entirely different trisection problem; supplemental device capabilities that accompany trisection events include: automatically portraying angles whose trigonometric properties are of cubic irrational value, as represented by the trisector portions of certain motion related solutions for the problem of the trisection of an angle, thereby amounting to angular magnitudes that cannot be duplicated, but only approximated, when applying a straightedge and compass to a given length of unity and not otherwise introducing relevant information into such attempts; and affording leveling provisions by means of designing a trisecting emulation mechanism in a manner such that one of its members remains parallel to a stationary x-axis reference line at all times during device actuation; trisection inventions should be categorized by features that they share in common in order that they be properly grouped together in accordance with the convention that: CATEGORY I sub-classification A mechanisms consist of articulating devices that trisect by means of featuring fan arrays that consist of four linkages that all emanate from a central hub for the express purpose of portraying designated angles that have been divided into three equal portions; CATEGORY I sub-classification B mechanisms instead trisect by means of featuring fan arrays that consist of only three linkages that all radiate from a central hub for the express purpose of portraying designated angles in relation to angles that amount to exactly one-third their respective sizes; and CATEGORY II mechanisms instead trisect by means of portraying static images during flexure which furthermore can be fully described by geometric construction patterns belonging to an Archimedes formulation, being a Euclidean formulation that contains a virtually unlimited number of geometric construction patterns comply with such famous Archimedes proposition; and CATEGORY III mechanisms instead trisect by means of portraying specific contours that represent a composite of trisecting angles, or aggregate of previously established trisection points, with respect to angles which amount to three times their respective sizes; the provisions to be included in a requirements chart are to consist of: RQMT 1furnishing an indication as to exactly which settings, or particular range thereof, a proposed invention can trisect, thereby disclosing whether it can account for acute, as well as obtuse angles; RQMT 2stating the reason why the classical problem of the trisection of an angle cannot be solved, thereby greatly contributing to unmasking Euclidean limitation whose mitigation would prove useful in the performance of trisecting emulation mechanisms; RQMT 3indicating how such proposed invention is to be operated, whereby if it needs to be specifically arranged before being set, it avoid confusion to furthermore indicate that all configurations which such device could be arranged to do not automatically portray a motion related solution for the problem of the trisection of an angle; RQMT 4revealing the primary function such proposed invention is expected to perform, and thereby making it clear that if it can overcome the Euclidean deficiency of being unable to fully backtrack upon any irreversible geometric construction pattern whose rendered angle is of a magnitude which amounts to exactly three times the size of its given angle in some manner other than regenerating a static image for virtually any designated angle which such device properly could be set to in order to automatically portray a motion related solution for the problem of the trisection of an angle, then such alternate approach would become burdened with the task of substantiating that each and every one of its settings could produce a valid solution on its own right; and RQMT 5explaining why each device setting automatically portrays a unique motion related solution for the problem of the trisection of an angle, thereby neither mistaking one unique solution for another, nor incorrectly claiming that such unique trisection solution applies to an entire range of device settings; a proposed invention design might need to be refined in the event that certain provisions listed in such requirements chart have not been satisfactorily addressed, whereby if not complied with, an explanation instead becomes afforded as to why an exception can been taken and how such device can be adequately substantiated without being equipped with such capability; and a thorough substantiation of a trisecting emulation mechanism needs to identify a singular geometric construction pattern which fully describes the overall shape of virtually any independent motion related solution for the problem of the trisection of an angle that such device could possibly automatically portray by means of becoming properly set; in effect, a Euclidean formulation is a composite of the configurations that its fundamental architecture could assume whenever a trisecting emulation mechanism becomes properly set.
3. A newly proposed invention whose linkages preferably are fabricated either out of a light weight metal such as aluminum, or a durable plastic such as polycarbonate throughout whose four constituent embodiments consist of: a first embodiment, also referred to as a basic configuration, comprising: a compass assembly which could be specifically arranged merely by means of being laid upon its side so that a reference linkage could be rotated relative to the shorter member of an L-shaped rigid right angled positioning linkage that is of equal span in order to form, and thereafter suitably maintain by means of being secured in some fashion, an acute angle which algebraically is to be denoted as being of 903 magnitude, as extending from the longitudinal centerline of such reference linkage to that of the shorter member of such positioning linkage about an axis where such two longitudinal centerlines meet, as furthermore being where the vertex of such inverted L-shaped rigid right angled positioning linkage is located and where such reference linkage is hinged to it so as to be situated within the ninety degree arc which is described by such rigid framework; such that a complementary angle, as algebraically expressed to be of size 3, would represent the designated magnitude of an acute angle that is intended to be trisected, as projected about such vertex between the longitudinal centerline of the longer member of such L-shaped rigid right angled positioning linkage, serving as an appendage to such produced compass, and the longitudinal centerline of such reference linkage; a counterbalance compass assembly which also could be specifically arranged merely by means of being laid upon its side in order to duplicate, and thereby secure in similar fashion, an acute angle of magnitude of 903, as subtended between the longitudinal centerline of its opposing reference linkage and that of its opposing positioning linkage; whose axis where such longitudinal centerlines meet furthermore constitutes the very vertex of such counterbalance compass assembly; such that the spans of such two described linkages are designed to be of equal length to the spans of the reference linkage and shorter member of the L-shaped rigid right angled positioning linkage belonging to such compass assembly; an interconnecting linkage which is hinged along its longitudinal centerline about respective endpoints of its span to the vertex of the L-shaped rigid right angled positioning linkage belonging to such compass assembly on one side, and to the vertex of such counterbalance compass assembly on the other; whose span is of equal length to the span of the shorter member of the L-shaped rigid right angled positioning linkage and span of the reference linkage belonging to such compass assembly, as well as to the span of the opposing reference linkage and span of the opposing positioning linkage belonging to such counterbalance compass assembly; a second embodiment, also referred to as a modified configuration that is of identical design to such first embodiment excepting that: the reference linkage belonging to its compass assembly is to be replaced by an elongated linkage whose construction is to be extended beyond that of the vertex of a positioning linkage that is to be of identical design to that belonging to the positioning linkage belonging to the compass assembly of such first embodiment, as now belonging to a modified compass assembly of such second embodiment, such that the length of its overall span becomes doubled; whereby upon being specifically arranged in the same manner as described for such first embodiment, an angle algebraically expressed to be of size 1803 would represent the designated magnitude of an obtuse angle that is intended to be trisected, as projected about the vertex of the L-shaped rigid right angled positioning linkage belonging to such modified compass assembly between the longitudinal centerline of its longer member, otherwise referred to as its appendage, and the longitudinal centerline of such elongated linkage; a third embodiment, also referred to as a rhombus configuration, comprising: a rhombus shaped mechanism which bears the overall geometry of a rhombus whose four sides all have been widened, such that each is free to rotate with respect to its two adjacent sides by means of being hinged at each end of its span by a dowel or other suitable interconnecting pivot pin; as more particularly consisting of: a left linkage; a right linkage; a lower linkage; an upper linkage; and four interconnecting pivot pins located at its four corners; a middle linkage which is of the same overall span as any of the four linkages belonging to such rhombus shaped mechanism; two additional interconnecting pivot pins which pass through the longitudinal centerline of such middle linkage very close to each of its ends, such that the distance between their radial centerlines amounts to the length of its overall span; whereby one of such interconnecting pivot pins furthermore passes through a slot made in the left slotted linkage of such rhombus shaped mechanism, with the other passing through a slot cut into its right slotted linkage; a protractor board which features a protractor upon its face which includes angular readings appearing about a circle whose: radius is of a length which equals the span of any of the linkages contained in such rhombus shaped mechanism; origin becomes secured in some manner to one of the four interconnecting pins contained in such rhombus shaped mechanism, so as to be situated directly underneath it; and ninety degree reading becomes secured in some manner to an adjacent interconnecting pin contained in such rhombus shaped mechanism, so as to be situated directly underneath it, whereby yet another interconnecting pin then would reside atop such circle somewhere in between its zero to ninety degree readings; and supporting members which could be introduced as necessary in order to maintain an accurate parallelism between the longitudinal centerline of such middle linkage with that of the lower linkage of such rhombus shaped mechanism during circumstances when it becomes translated within the slots cut through its left linkage and right linkage; with such added components including, but not limited to: a cross linkage; a stabilizer linkage; a slide; a cross dowel; and a retaining ring; a fourth embodiment, also referred to as a car jack configuration because its design closely resembles the ratcheting portion of a device of such design, as more specifically could be described by two linkages which are fitted together in order to form an inverted T-shaped rigid framework in which the longitudinal centerline of one of such linkages perpendicularly bisects the longitudinal centerline of the other, the latter of which would then be considered to be its base; whereby, instead of either raising or slowly lowering heavy objects resting upon an adjoining coupler, such geometry could form the basis of a design that could trisect angles of acute and obtuse designated magnitudes, merely by means of removing such coupler, as well as the ratcheting capability of such perpendicularly bisecting member, and thereafter fitting two addition side members such inverted T-shaped rigid framework, each of which is to be hinged about one of its span terminations, as located along its longitudinal centerline, to a span termination of such base, as located along its longitudinal centerline, in a manner so that their respective longitudinal centerlines are free to converge somewhere along the longitudinal centerline of such perpendicularly bisecting member during conditions in which the angle subtended between the longitudinal centerline of one of such side members and that of such base becomes varied in size from zero degrees to some acute angle design limit which becomes dictated by the shortest span of such two side members; thereby featuring a geometry which serves as the basis for a slotted linkage arrangement, as well as a miniaturized slider arrangement which more easily can be transported, such that its: slotted linkage arrangement is comprised of: a trisector solid linkage which serves the function of such aforementioned base; a bisector slotted linkage that is rigidly attached to it in a manner in which its longitudinal centerline perpendicularly bisects that of such trisector solid linkage; a given acute angle slotted linkage of the same span as such trisector solid linkage; and a transverse slotted linkage of sizably longer span than that of such trisector solid linkage which contains a hole bored about its longitudinal centerline such that its radial centerline is set a distance away from that of one its overall span terminations which is equal to the span of such trisector solid linkage; a control slotted linkage of the same span as such trisector solid linkage; a given obtuse angle solid linkage of the same span as such trisector solid linkage; a slide linkage of sizably shorter span; a protractor strip that features angular readings appearing about a circle whose radius is of a length which is equal to the span of such trisector solid linkage; a dowel whose radial centerline is to be centered at either span termination of such given acute angle slotted linkage which, after becoming inserted through it, furthermore is to pass through a spacer, a span termination of such trisector solid linkage, a span termination of such control slotted linkage, a span termination of such given obtuse angle solid linkage, and then through the origin of such protractor strip, without extending beyond it, such that it can become permanently secured to it in some fashion in order to effectively hinge together those aforementioned components which become sandwiched in between the head of such dowel and such protractor strip; a dowel whose radial centerline is to be centered at the span termination of such transverse slotted linkage which resides away from the radial centerline of the hole already bored through it a distance which is equal to the span of such trisector solid linkage which, after becoming inserted through it, furthermore is to pass through the remaining span termination of such trisector solid linkage, without extending beyond it, such that it can become permanently secured to it; a dowel which is to be inserted into the slot cut through such given acute angle slotted linkage, then into the slot cut through such transverse slotted linkage which resides in between the hole bored through it and its span termination which resides away from the radial centerline of such hole a distance which is equal to the span of such trisector solid linkage, whereby such slotted portion is not to encroach upon any hole which might become bored through such transverse slotted linkage, thereafter to be passed through a spacer, and finally through the slot afforded in such the bisector slotted linkage where it thereafter can be permanently secured to a retaining ring which lies underneath them all; a dowel whose radial centerline is to be centered at the span termination of such slide linkage, which, after becoming inserted through it, furthermore is to pass through the hole bored through such transverse slotted linkage, then through a spacer, and finally through the slot cut into such control slotted linkage in order that it lastly can be permanently secured to a retaining ring which resides underneath them; a dowel which is to be inserted into the remaining slot cut through such transverse slotted linkage, then through the remaining span termination of such given obtuse angle solid linkage such that it finally can be permanently secured to it; and a shoulder screw which can be inserted either: through the unused span termination of such control slotted linkage, and then screwed into a threaded hole located at the ninety degree reading of such protractor strip, thereby specifically arranging such device so that it can trisect any angle of acute designated magnitude that it could be set to, as algebraically expressed to be of 3 size; or through the unused span termination of such given acute angle slotted, and then screwed into a threaded hole located at the ninety degree reading of such protractor strip, thereby specifically arranging such device so that such device can trisect any angle of obtuse designated magnitude that it could be set to, as algebraically expressed to be of 2706 size; and slider arrangement is comprised of: a trisector linkage with the term TRISECTOR imprinted upon it; a bisector linkage that is rigidly attached to it in a manner so that its longitudinal centerline perpendicularly bisects that of such and trisector linkage; a given acute angle linkage with the term GIVEN ACUTE ANGLE imprinted upon it that is of the same span as such trisector linkage; and a transverse linkage of sizably longer span than that of such trisector linkage which contains a hole bored about its longitudinal centerline such that its radial centerline is set a distance away from that of one its overall span terminations which is equal to the span of such trisector linkage; a control linkage of the same span as such trisector linkage; a given obtuse angle linkage with the term GIVEN OBTUSE ANGLE imprinted upon it that is of the same span as such trisector linkage; an adjustment linkage with the term ADJUSTMENT LINKAGE imprinted upon it; a protractor/instructions piece of paper that features angular readings appearing about a circle whose radius is of a length which is equal to the span of such trisector linkage; a toploader which such device can be transported within, but operated from outside of; a pair of easels which could be glued to the back of a toploader in order to vertically mount it for easy viewing; standoffs which provide the necessary clearances to enables such device to articulate as needed within the confines of such toploader; a rivet whose radial centerline is to pass through one of two sleeves of such toploader, a washer, the span termination the its given obtuse angle linkage which appears after the term GIVEN OBTUSE ANGLE which is imprinted upon it; the span termination of such given acute angle linkage which precedes the term GIVEN ACUTE ANGLE which is imprinted upon it, the span termination of its trisector linkage which precedes the term TRISECTOR imprinted upon it, one of the span terminations of its control linkage, through another washer, through its protractor/instructions piece of paper, and then through the remaining sleeve of such toploader, whereby such rivet, after being pulled up, would thereby sandwich components housed at such location in between the two sleeves of such toploader, but allows for their free rotation relative to one another; a rivet whose radial centerline first is to pass through the span termination of such transverse linkage which resides away from the radial centerline of the hole bored through it a distance which is equal to the span of such trisector linkage, and then through a shim, next through the unused, or remaining span termination of such trisector linkage, thereby appearing after the term TRISECTOR imprinted upon it, and finally through a washer which it becomes pulled up inside of in order secure such components and also enable such transverse linkage to rotate freely with respect to such trisector linkage, each of which come into contact with opposing faces of such shim; a rivet which first is passed through an overlapping portion of a slider in order to sandwich its head in between such overlapping portion and another portion of such slider which was wrapped around such bisector linkage, then through a shim, then through another overlapping portion of a slider which was wrapped around such given acute angle linkage, and lastly though yet another overlapping portion of a slider which was wrapped around a portion of such transverse linkage at some location in between its hole and span termination which resides a distance away from the radial centerline of such hole which is equal to the span of such trisector linkage, thereby becoming pulled up in a direction which is opposite to the manner in such previously described rivets were pulled up, and causing the longitudinal centerlines of such transverse linkage and given acute angle linkage to always converge somewhere along the longitudinal centerline of such bisector linkage whenever such device becomes articulated; a rivet which first is passed through an overlapping portion of a slider in order to sandwich its head in between such overlapping portion and another portion of such slider which was wrapped around such control linkage, then through a shim, then through a span termination of such adjustment linkage, as located along its longitudinal centerline, then through another shim, and lastly through the hole bored through such transverse, thereby becoming pulled in the same direction as the rivet which was described directly above; a rivet which first is passed through an overlapping portion of a slider in order to sandwich its head in between such overlapping portion and another portion of such slider which was wrapped around such transverse linkage, then though the unused, or remaining span termination of such given obtuse angle linkage, thereby preceding the term GIVEN OBTUSE ANGLE imprinted upon it, and lastly through a washer where it then becomes pulled up; a rivet which is passed through such control linkage, situated so that its radial centerline aligns upon the unused, or remaining span termination of such control linkage, along its longitudinal centerline, which furthermore had its rivet center pin removed after pull-up operations, thereby leaving a small hole which can be seen to pass entirely through such rivet; a rivet which is passed through such given acute angle linkage, situated so that its radial centerline aligns upon the unused, or remaining span termination of such given acute angle linkage, along its longitudinal centerline, hence being located at a position which appears after the term GIVEN ACUTE ANGLE which is imprinted upon it, which furthermore had its rivet center pin removed after pull-up operations, thereby leaving a small hole which can be seen to pass entirely through such rivet; a pin which can be inserted through one sleeve of such toploader where it aligns with the ninety degree reading of a protractor/instructions piece of paper that was inserted into it, and then is passed through either: the hole of the rivet that is retained by such control linkage, then through the ninety degree reading of such inserted protractor/instructions piece of paper, then out the other sleeve of such toploader where it becomes housed by a clutch which surrounds its sharp end, thereby specifically arranging such device so that it can trisect any angle of acute designated magnitude that it could be set to, as algebraically expressed to be of 3 size; or the hole of the rivet that is retained by such given acute angle linkage, then through the ninety degree reading of such inserted protractor/instructions piece of paper, then out the other sleeve of such toploader where it becomes housed by a clutch which surrounds its sharp end, thereby specifically arranging such device so that it can trisect any angle of obtuse designated magnitude that it could be set to, as algebraically expressed to be of 2706 size;
4. The newly proposed invention described in claim 3, wherein: such first embodiment, once becoming specifically arranged to an acute angle of 3 designated magnitude, thereafter could become properly set, merely by means of translating the distal span termination of the opposing reference linkage of such counterbalance compass assembly, or some suitable slide mechanism which additionally might be featured at such location in order to facilitate such operation, along the longitudinal centerline of the shorter member of the inverted L-shaped positioning linkage belonging to its compass assembly until such time that the distal span termination of the reference linkage belonging to such compass assembly aligns somewhere upon the longitudinal centerline of the opposing positioning linkage of such counterbalance compass assembly; thereby automatically portraying a trisector that would become algebraically expressed as being of size , as measured about the vertex of the inverted L-shaped positioning linkage of such compass assembly and extending from the longitudinal centerline of the appendage portion of its positioning linkage to the longitudinal centerline of the interconnecting linkage of such first embodiment; as based upon the understanding that the: distal span termination of the opposing reference linkage of such counterbalance compass assembly represents the radial centerline of its span termination, running along its longitudinal centerline, that resides opposite to, or farthest away from its other span termination, being located at the vertex of such counterbalance compass assembly; and distal span termination of the reference linkage of such compass assembly represents the radial centerline of its span termination, running along its longitudinal centerline, that resides opposite to, or farthest away from its other span termination, being located at the vertex of such the inverted L-shaped positioning linkage belonging to its compass assembly; such second embodiment, once becoming specifically arranged to an acute angle of 3 designated magnitude, thereafter could become properly set, merely by means of translating the distal span termination of the opposing reference linkage of such counterbalance compass assembly, or some suitable slide mechanism which additionally might be featured at such location in order to facilitate such operation, along the longitudinal centerline of the shorter member of the duplicate inverted L-shaped positioning linkage belonging to its modified compass assembly until such time that the distal span termination of the elongated linkage portion that replaced the reference linkage of such compass assembly aligns somewhere upon the longitudinal centerline of the opposing positioning linkage of such counterbalance compass assembly; thereby automatically portraying a trisector that would become algebraically expressed as being of size , as measured about the vertex of the duplicate inverted L-shaped positioning linkage of belonging to such compass assembly and extending from the longitudinal centerline of the appendage portion of its duplicate positioning linkage to the longitudinal centerline of that portion of the equilateral template which replaced the interconnecting linkage of such first embodiment; as well as automatically portraying a trisector for an additional specifically arranged obtuse angle of 1803 designated magnitude, thereby becoming algebraically expressed as being of size 60, as measured about the vertex of the duplicate inverted L-shaped positioning linkage belonging to such modified compass assembly and extending from the longitudinal centerline of the appendage portion of its duplicate positioning linkage to the longitudinal centerline of another portion of the equilateral template whose span also terminates at the vertex of such compass assembly; such third embodiment could be properly set, without first having to be specifically arranged, merely by means of translating its slide until such time that the longitudinal centerline of its middle linkage intersects a point on the circular imprint of its protractor board whose reading amounts to the designated magnitude of an acute angle that is intended to be trisected, algebraically denoted to be of 3 magnitude, and forming an angle about the origin of its protractor board that extends from the zero degree reading upon its circular imprint to a location where it intersects the longitudinal centerline of its middle linkage; such that its trisector automatically would be portrayed about such origin, extending from the zero degree reading upon such circular imprint to a location where it intersects the longitudinal centerline of the lower linkage of its rhombus shaped mechanism, algebraically expressed as being of magnitude, and furthermore distinguished by the very reading which the longitudinal centerline of the lower linkage of its rhombus shaped mechanism points to upon such protractor board; wherein a trisector for an obtuse supplementary angle, algebraically expressed to be of 1803 size and subtended from such zero degree reading to the extension of a straight line which could be drawn from such 3 reading upon the circular imprint of its protractor board to its origin, furthermore could be located, merely by means of drawing another straight line that passes through the origin of such protractor board and makes a sixty degree angle with the longitudinal centerline of the lower linkage of its rhombus shaped mechanism, thereby terminating along its circular imprint at a reading that amounts to 60 magnitude with respect to such zero degree reading; the slotted linkage arrangement of such fourth embodiment, once becoming specifically arranged so that it could trisect acute angles of 3 designated magnitudes, thereafter could become properly set, merely by means of translating its slide linkage so that the longitudinal centerline of its given acute angle slotted linkage appears in line with, or points to a reading which appears upon the face of its protractor strip that matches the designated magnitude of an acute angle that is intended to be trisected; thereby automatically portraying two angles; one being of specific 3 designated magnitude that is intended to be trisected, as measured about the origin of its protractor strip and subtended between its zero degree reading and the longitudinal centerline of such given acute angle slotted linkage; and the other being its trisector, algebraically denoted to be of magnitude, as measured about the origin of its protractor strip and subtended between its zero degree reading and the longitudinal centerline of such trisector solid linkage or specifically arranged so that it could trisect obtuse angles of 2706 designated magnitudes, thereafter could become properly set, merely by means of translating its slide linkage so that the longitudinal centerline of its given obtuse angle solid linkage appears in line with, or points to a reading which appears upon the face of its protractor strip that matches the designated magnitude of an obtuse angle that is intended to be trisected; thereby automatically portraying two angles; one being of specific 2706 designated magnitude that is intended to be trisected, as measured about the origin of its protractor strip and subtended between its zero degree reading and the longitudinal centerline of its given obtuse angle solid linkage; and the other being its trisector, algebraically denoted to be of 902 magnitude, as measured about the origin of its protractor strip and subtended between its zero degree reading and the longitudinal centerline of such trisector solid linkage; and the slider arrangement of such fourth embodiment, once becoming specifically arranged so that it could trisect acute angles of 3 designated magnitudes, thereafter could become properly set, merely by means of translating its adjustment linkage from outside of its toploader so that the longitudinal centerline of such given acute angle linkage appears in line with, or points to a reading which appears upon the face of its protractor/instructions piece of paper that matches the designated magnitude of an acute angle that is intended to be trisected; thereby automatically portraying two angles; one being of specific 3 designated magnitude that is intended to be trisected, as measured about the origin of its protractor/instructions piece of paper and subtended between its zero degree reading and the longitudinal centerline of such given acute angle linkage; and the other being its trisector, algebraically denoted to be of magnitude, as measured about the origin of its protractor/instructions piece of paper and subtended between its zero degree reading and the longitudinal centerline of such trisector linkage; or specifically arranged so that it could trisect obtuse angles of 2706 designated magnitudes, thereafter could become properly set, merely by means of translating its adjustment linkage from outside of its toploader so that the longitudinal centerline of its given obtuse angle solid linkage appears in line with, or points to a reading which appears upon the face of its protractor/instructions piece of paper that matches the designated magnitude of an obtuse angle that is intended to be trisected; thereby automatically portraying two angles; one being of specific 2706 designated magnitude that is intended to be trisected, as measured about the origin of its protractor/instructions piece of paper and subtended between its zero degree reading and the longitudinal centerline of its given obtuse angle linkage; and the other being its trisector, algebraically denoted to be of 902 magnitude, as measured about the origin of its protractor/instructions piece of paper and subtended between its zero degree reading and the longitudinal centerline of such trisector linkage.
5. The newly proposed invention described in claim 3, wherein any feet which either have been fitted onto, or perhaps otherwise form an integral part of various linkages belonging to any the first three of its four constituent embodiments and/or the slotted linkage arrangement of such fourth embodiment, as well as any washers and/or shims which might be featured in the slider arrangement of such fourth embodiment have been positioned at strategic locations for the express purpose of maintaining essential linkages parallel to one another at all times during device articulation in order to enable accurate trisection to become performed.
6. The newly proposed invention described in claim 3, wherein its four constituent embodiments, by belonging to CATEGORY I sub-classification B, automatically portray each and every motion related solution for the problem of the trisection of an angle as a distinct trisector; thereby operating in a completely unique manner than could any mechanism which would be considered to belong CATEGORY I sub-classification A because it would feature an additional linkage into its design for the express purpose of dividing an angle into three equal parts in order to perform trisection.
7. The newly proposed invention described in claim 3, wherein the overall shape of virtually any motion related solution for the problem of the trisection of an angle that each of its four constituent embodiments possibly could automatically portray furthermore could be fully described by a geometric construction pattern that is not based upon the famous Archimedes proposition which is specified on page 309 in The Works of Archimedes, as first published in the English language in 1897; thereby substantiating that each of its four embodiments is entirely unique from any CATEGORY II trisecting articulating device which otherwise would need to substantiate each automatically portrayed motion related solution for the problem of the trisection of an angle by demonstrating that such generated overall shape could be fully described only by a geometric construction pattern that could be derived from such stated Archimedes proposition.
8. The newly proposed invention described in claim 3, wherein each of its four constituent embodiments performs trisection without charting specific contours that represent a composite of trisecting angles, or aggregate of previously established trisection points, with respect to angles which amount to three times their respective sizes; thereby being unique designs from any CATEGORY III qualifying design.
9. The newly proposed invention described in claim 3, wherein: its modified configuration is capable of trisecting obtuse angles that its basic configuration clearly is not equipped to handle; thereby differing from it by means of featuring enhanced capabilities; its rhombus configuration is capable of performing trisection by means of precisely regulating the movement of a mechanism which resembles the overall shape of a rhombus, being a process which differs substantially from the method of control exercised by such basic and modified configurations which instead each regulate the positioning of a counterbalance compass assembly with respect to a compass assembly; thereby differing substantially in design; and its car jack configuration is capable of performing trisection by means of regulating the movement of a control mechanism which resembles the overall shape of a car jack, being an action that clearly cannot be duplicated by such rhombus configuration, and thereby differing substantially in its design.
10. The newly proposed invention described in claim 3, wherein each of its four constituent embodiments is designed to perform the primary function of regenerating static images, or projecting overall shapes that are indicative of the various configurations which its fundamental architecture might assume during articulation; thereby disclosing the relative positioning of the longitudinal centerlines of its constituent linkages and radial centerlines of its interconnecting pivot pins.
11. The newly proposed invention described in claim 3, wherein each of its four constituent embodiments is designed to regenerate a static image that automatically portrays a motion related solution for the problem of the trisection of angle any time such device becomes properly set.
12. The newly proposed invention described in claim 3, wherein each of its four constituent embodiments is designed so that any a motion related solution for the problem of the trisection of angle which it possibly could automatically portray would project an overall outline that furthermore fully can be described by a geometric pattern, as belonging to a particular Euclidean formulation, whose rendered angle is of a magnitude that amounts to exactly three times the size of its given angle; thereby substantiating that such device would perform trisection accurately over a wide range of device settings and, in so doing, qualify as a bona fide trisecting emulation mechanism.
13. The newly proposed invention described in claim 3, wherein each of its four constituent embodiments is designed so that any motion related solution for the problem of the trisection of an angle which it possibly might automatically portray, as the result of becoming properly set to some arbitrarily selected designated magnitude, furthermore would constitute a unique trisection solution, in its own right, because the overall shape which thereby would become projected could not be matched, at least down to its true proportion, by otherwise attempting to reset such device.
14. The newly proposed invention described in claim 3, wherein each of its four constituent embodiments is designed to automatically portray angles of cubic irrational trigonometric proportion which thereby cannot be duplicated, but only approximated, when otherwise applying a straightedge and compass to a given length of unity.
15. The newly proposed invention described in claim 3, wherein any static image that possibly could be regenerated by means of properly setting one of its four constituent embodiments would map out a complete pathway that leads from an angle whose designated magnitude is intended to be trisected all the way back to its trisector; in effect divulging locations of overlapment points within such layout that otherwise would remain entirely undetectable if instead attempting to backtrack, solely by conventional Euclidean means, upon an irreversible geometric construction pattern that fully describes the overall shape of such static image; thereby enabling a motion related solution for the problem of the trisection of an angle to become automatically portrayed that can overcome such major difficulty, but cannot overturn the realization that such fundamental Euclidean limitation nevertheless always will prevent the classical problem of the trisection of an angle from being solved.
16. The newly proposed invention described in claim 3, wherein the slotted linkage arrangement of its fourth embodiment furthermore can function as a level, once being specifically arranged to trisect angles of acute designated magnitudes, by means of maintaining the radial centerlines of the span ends of its given acute angle slotted linkage and given obtuse angle solid linkage, as distally disposed away from those span ends which instead congregate about the origin of its protractor strip, parallel to the straight line which stretches in between such origin and the zero degree reading upon such protractor strip at all times during device actuation; whereby such device furthermore could be made more robust in order to accurately control the actuation of a hydraulic car lift.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
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=tan(3)={square root over (5)}/7=(3z.sub.Rz.sub.R.sup.3)/(13 z.sub.R.sup.2); and
z.sub.R.sup.3+z.sub.R.sup.2+z.sub.R+=0 when =({square root over (5)}+7),=7{circle around (5)}+12, and =12{square root over (5)}.
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DETAILED DESCRIPTION
[0657] Certainly by now it should have been made quite clear that in order to unlock vital secrets, highly suspected to be hidden deep within the very recesses of a perplexing trisection mystery, a paradigm shift most definitely is warranted; one that expressly should recommend some fundamental change in overall approach concerning how to properly account for difficulties encountered when trying to solve the classical problem of the trisection of an angle.
[0658] Only by means of exposing such closely held secrets could the basic objective of a comprehensive trisection methodology become realized, as presented in the flowchart appearing in
[0659] Accordingly, a detailed discussion of such flowchart should precede the introduction of such newly proposed invention. In this way, any requirements posed relating to the design of its four constituent embodiments would be presented well before explaining exactly how they are to complied with. Such accounting begins with a process box entitled MATHEMATIC LIMITATION IDENTIFIED 1 therein, representing the task within such flowchart where some unknown mathematical limitation is identified that supposedly prevents the classical problem of the trisection of an angle from being solved. Obviously, since such solution must depend solely upon the communication of a straightedge and compass with respect to an angle of designated magnitude, any mathematic limitation alluded to therein must be some pronounced difficulty having to do with conventional Euclidean practice!
[0660] The process box referred to as UNKNOWN GEOMETRIC PROPERTY UNCOVERED 2 is where, in the course of such
[0661] The third process box, entitled DEGREE OF IMPOSITION DELINEATED 3 is reserved for describing the extent of difficulty that such newly uncovered geometric property is anticipated to impose upon conventional Euclidean practice.
[0662] The process box referred to as DEVICE PRIMARY FUNCTION REVEALED 4 is where an as yet unknown capability thereby becomes revealed which assumes the form of some specially added equipment that articulating mechanisms can be fitted with that enables them to overcome, correct, or compensate for such undermining influence, as now suspected to be a mathematic limitation.
[0663] Next, the decision box entitled DEFICIENCY MITIGATED 5 within such
[0664] The input box entitled TRISECTION RATIONALE 6, as shown in
[0665] Such trisection rationale discussion specifically directs attention to the first four processes listed in such
[0666] The very fact that overlapment points remain entirely inconspicuous in this manner furthermore evidences that it is impossible to specify a distinct set of Euclidean commands which can identify their whereabouts solely with respect to such rendered information.
[0667] Without such vital input, a specific sequence of Euclidean operations furthermore could not be developed that instructs how to apply a straightedge and compass in order to trace out a pathway which begins at such rendered information and leads all the way back to a given set of previously defined geometric data; whereby the very presence of overlapment points serves to circumvent reversibility!
[0668] Since the very concept of reversibility is entirely new with regards to conventional Euclidean practice, a validation that isosceles triangle MNP, as posed in
[0673] In order to demonstrate the actual difficulty which an intrusion of overlapment points causes, notice in
[0674] Taking any of the specific geometric construction patterns which collectively constitute such Archimedes formulation into account, this becomes evident upon realizing that overlapment points M and N, as represented in such
[0677] For the particular hypothetical case when QPS amounts to exactly ninety degrees, such thirty degree trisector very easily could be geometrically constructed, simply by bisecting any angle or side of an equilateral triangle. However, the computation of dividing such ninety degree angle by a factor of three in order to arrive at the magnitude of such thirty degree trisector unfortunately cannot be duplicated solely by conventional Euclidean means. Hence, to do so only would create a corrupted version of the classical problem of the trisection of an angle; thereby solving an entirely different problem!
[0678] Hence, in such capacity, overlapment points function as obstructions serving to confound attempts to redefine an entire geometric construction pattern solely with respect to its rendered information.
[0679] Consequently, any pathway consisting of previously distinguished intersection points which originally led from given angle RMP all the way to rendered angle QPS, as depicted in
[0680] In that such discussion particularly should account for difficulties experienced when attempting to solve the classical problem of the trisection of an angle, it thereby becomes formally stipulated that it is impossible to fully backtrack upon any geometric construction pattern whose rendered angle is of a magnitude that amounts to exactly three times the size of its given angle; simply because such drawing would harbor overlapment points!
[0681] As such, a presence of overlapment points within such specific types of geometric construction patterns entirely thwarts attempts to generate such overall pathways in complete reverse order, solely by conventional Euclidean means; thereby preventing the classical problem of the trisection of an angle from being solved!
[0682] In summary, overlapment points have an affinity to impede the completion of geometric construction patterns that are replete with them for the mere reason that they cannot be entirely reconstituted solely via straightedge and compass in complete reverse order.
[0683] For the benefit of any remaining skeptics, it furthermore should be added that only when the magnitude of a trisected angle becomes furnished beforehand can a geometric construction pattern which specifies such trisector, in the very the form of its given angle, become fully reversible; thereby enabling some corrupted version of the classical problem of the trisection of an angle to be solved.
[0684] During such condition, overlapment points, by definition, then would become distinguishable intersection points with respect to such given trisecting angle; thereby making such geometric construction pattern fully reversible. However, to attempt such activity would defeat the purpose of trying to trisect an angle solely by conventional Euclidean means in the very first place; simply because the very information being sought after already has been furnished. In other words, it would be entirely senseless to generate geometric quantities such as straight lines, circles, and angles aforehand exclusively for purposes of then determining them solely via straightedge and compass. Nevertheless, a notable history of this exists which mostly has been directed towards improper attempts to trisect angles solely via straightedge and compass.
[0685] Such foolish endeavors stand is sharp contrast to most, if not all, other standard Euclidean procedures, such as bisection; whereby a bisector remains totally unknown until such time that it actually becomes geometrically constructed from an angle of given magnitude.
[0686] When only the magnitude of an angle that is intended to be trisected becomes designated, its associated geometric construction pattern remains completely unspecified. This presents a heightened problem because there virtually are a countless number of other geometric construction patterns, besides those represented in
[0687] Even when a specific geometric construction pattern becomes selected as a vehicle for attempting to perform trisection, such as in the case of the rendition of the Archimedes formulation, as posed in
[0688] Such pronounced geometric construction limitation of not being able to encroach upon overlapment points when being launched from a particular direction can, in fact, be rectified rather simply; merely by affording a means for discerning overlapment points that reside within irreversible geometric construction patterns, and thereby making them entirely distinguishable with respect to rendered angles which otherwise cannot be backtracked upon!
[0689] Such elementary recommendation, despite its rather unsuspecting and seemingly outlandish nature, nevertheless describes exactly how a trisecting emulation mechanism can trisect virtually any designated angle which it can be set to; thereby portraying a of motion related solution for the problem of the trisection of an angle.
[0690] Such strange phenomena perhaps most easily can be described with respect to the motion of any CATEGORY I sub-classification A articulating trisection device because such types of devices do not first have to be specifically arranged before displaying their settings. As any of such devices becomes cycled, eventually reaching all of the settings within its entire operating range, its fundamental architecture sweeps out, or regenerates, a multitude of static images, each representing a still shot cameo of two angles, the larger of which not only amounts to exactly three times the size of the other, but furthermore is calibrated to a specific device setting.
[0691] The beauty of such design concept is that once any of such types of devices becomes set to a preselected designated angle, the portion of the smaller angle contained within the static image which becomes regenerated thereby automatically portrays its associated trisector.
[0692] In other words, by means of properly setting any trisecting emulation mechanism, its fundamental architecture becomes rearranged to a particular position such that the static image which becomes regenerated automatically portrays a motion related solution for the problem of the trisection of an angle!
[0693] In effect, such motion related solution distinguishes overlapment points whose availability otherwise would prevent the classical problem of the trisection of an angle from being solved!
[0694] Accordingly, instead of attempting to perform that which is impossible; essentially consisting of retracing a distinguishable pathway within an irreversible geometric construction pattern in complete reverse order solely by conventional Euclidean means, a trisecting emulation mechanism otherwise functions like the Dewey decimal system in a library wherein the exact name of a document that is being searched for becomes either input into a computer, or otherwise looked up in some card deck, whereby an alpha-numeric code that provides an indication of its whereabouts, thereby allows such information to forthwith become retrieved. The only glaring difference in the case of a trisecting emulation mechanism is that the magnitude of a designated angle which is slated for trisection becomes set into such device, thereby causing the regeneration of a particular static image that automatically portrays its associated trisector!
[0695] Accordingly, a fundamental architecture might be thought of as a mechanical means for conveniently storing a multitude of static images within the very memory of some particularly designed trisecting emulation mechanism; thereby enabling a motion related solution for the problem of the trisection of an angle of designated magnitude to be automatically portrayed at will.
[0696] To conclude, a unique pathway which leads from one angle all the way to another that amounts to exactly three times its size automatically becomes portrayed each and every time a static image become regenerated by means of configuring a trisecting emulation mechanism to any of its discrete device settings; thereby disclosing the actual whereabouts of nuisance overlapment points which reside along the way; simply by means of exposing them to be nothing more than commonly known intersection points. In so doing, any obstructions that otherwise normally would be encountered when attempting to solve the classical problem of the trisection of an angle, would be overcome merely by means of properly setting a trisecting emulating mechanism.
[0697] A basic tenet of conventional Euclidean practice is that all activity must proceed exclusively from a given set of previously defined geometric data, or else from intersection points which become located with respect to it.
[0698] It may well be that a purposeful adherence to such rule might explain why any serious attempt to completely retrace a geometric construction pattern exclusively from its rendered information all the way back to its given set of previously defined geometric data, solely by application of a straightedge and compass, entirely might have been overlooked in the past.
[0699] Moreover, only on very rare occasions, such as in the particular case of attempting to solve the classical problem of the trisection of an angle, could the prospect of possibly even engaging upon such activity arise, thereafter culminating in an avid interest to solve such classical problem without considering that a pathway leading from a rendered angle within any geometric construction pattern all the way back to a given angle whose magnitude amounts to exactly one-third of its size lies at the very heart of such difficulty!
[0700] Remarkably, only by means of analyzing conventional Euclidean practice from this other seldom viewed perspective could irreversibility be indentified as being caused by an intrusion of overlapment points.
[0701] By otherwise neglecting such critical information, it would become virtually impossible to substantiate that any qualifying CATEGORY I sub-classification A or CATEGORY II articulating trisection mechanism could perform trisection accurately throughout a wide range of device settings.
[0702] The input box entitled IMPROVED DRAWING PRETEXT 7, as posed
[0703] Whereas such
[0704] Accordingly, the rather seemingly antiquated idea of generating singular, but unrelated geometric construction patterns thereby very easily could become dwarfed simply by means of considering the prospect that they furthermore might become linked to one another in some particular fashion through the use of an improved drawing pretext for the express purpose of geometrically describing motion!
[0705] The wording above is intended to infer that improved drawing pretexts, other than that of the Euclidean formulation could be devised, thereby associating their constituent drawing patterns in some distinct manner other than through specified sequences of Euclidean operations; and, upon becoming replicated might thereby describe important motions which are known to be of service to mankind!
[0706] Such discussion is building to the proposition that by means of properly partitioning all observed phenomena which can be described geometrically, including that of certain motions, it thereby becomes possible to envision a certain order that becomes evident within a farther reaching mathematics.
[0707] Such is the very purpose of the input box entitled MATHEMATICS DEMARCATION 8, as posed in
[0708] As it pertains to trisection matters, the drawing pretext entry appearing in the third column of such
[0709] Headings appearing in
[0710] Moreover, inasmuch as the field of geometry concerns itself with mathematically quantified depictions, algebra, on the other hand, by representing the overall language of mathematics, instead bears the biggest brunt of responsibility in validating that such alleged order truly exists; doing so by associating algebraic format types through some newly proposed equation sub-element theory!
[0711] One principal reference, standing as a harbinger of a newly proposed equation sub-element theory, is a relatively unknown treatise that was published in 1684; as written by one Thomas Baker and entitled, The Geometrical Key or the Gate of Equations Unlocked. After a close affiliation with Oxford University, Mr. Baker successfully provided a solution set pertaining to biquadratic equations, perhaps more commonly referred to today as either quartic, or fourth order equations. However, it seems quite plausible that because of a serious competition among rival institutions going all the way back to that time period, Gerolamo Cardano's preceding work of 1545, as it appeared in Ars Magna, nevertheless, still managed to eclipse his later contributions. In brief, Cardano applied a transform to remove the second, or squared, term from cubic equations in order to modify them into an overall format that very easily could be resolved. However, because of such gross simplification, the all important fact that each algebraic equation is unique, in its own right, was largely ignored; hence, failing to attribute deliberate meaning to the various equation types that actually govern third order algebraic equation formats. The very stigma which such abbreviated process instilled unfortunately served to direct attention away from developing an all purpose solution that applies to all cubic equation formats, as posed in a single variable; one which obviously would lie at the very heart of any newly proposed sub-element theory; thereby not requiring that cubic equations which express second terms first become transformed in order to solve them! In retrospect, it now appears very likely, indeed, that a hit-and-miss mathematics approach of such nature most probably delayed the actual debut of a newly proposed equation sub-element theory by some four hundred years!
[0712] To conclude, by means of now introducing an all-purpose cubic equation solution, as presently has remained absent for all these years, the very relevancy of each format type can remain preserved so that further comparisons could be made in order to avail a more comprehensive understanding of an overall order that actually prevails within all of mathematics.
[0713] In such
[0714] That is to say, whenever the angular portion within a regenerated static image that has been calibrated to a particular device setting bears cubic irrational trigonometric properties, so must the angular portion therein which serves as its trisector. Accordingly, there is no way to relate either rational or quadratic irrational trigonometric properties of a trisector to an angle which amounts to exactly three times its size that bears cubic irrational trigonometric properties.
[0715] In other words, it requires, not one, but three angles that all exhibit cubic irrational trigonometric properties in order to geometrically construct an angle which exhibits either rational or quadratic irrational trigonometric properties. Such angle very well could be geometrically constructed in a manner which is analogous, or consistent with virtually any of the nine the arrangements of such products, sums, and sums of paired products, as posed in the algebraic equations previously expressed in such definition of a cubic irrational number.
[0716] Accordingly, any geometric construction pattern that belongs to a Euclidean formulation which furthermore is known to replicate the articulated motion of the fundamental architecture of any CATEGORY I sub-classification A trisecting emulation mechanism which thereby becomes reset every time it becomes articulated only can be approximated in size if it is meant to depict a static image either of whose two included angular portions portrays cubic irrational trigonometric properties!
[0717] An elementary, but nonetheless very revealing example of this concerns attempts to trisect a sixty degree angle solely by conventional Euclidean means!
[0718] Although such sixty angle can be distinguished merely by geometrically constructing an equilateral triangle, its associated twenty degree trisector, on the other hand, is known to exhibit transcendental trigonometric properties that cannot be geometrically constructed, when proceeding either exclusively from a given length of unity, or solely from any angle whose trigonometric properties exhibit either rational or quadratic irrational values.
[0719] Such explicitly stated impossibility is what actually distinguishes the realm between where angles can be portrayed which bear cubic irrational trigonometric property values, and other angles that do not which thereby can be expressed solely by conventional Euclidean means!
[0720] Further note in such
[0721] The fact that cubic equations appear only under the heading referred to as geometric forming process therein is a little more difficult to explain; having to do with the fact that by depicting actual motions, Euclidean formulations moreover can be expressed algebraically as continuums.
[0722] The most commonly known algebraic continuum is an infinite series whose terms become summed over some specific predetermined range of performance.
[0723] It naturally follows then that their integral counterparts, as realized within the field of calculus, also could apply, as well, to certain relative motions which furthermore can be geometrically described by Euclidean formulations. Quite obviously, this presumption moreover assumes that such motions actually do appear as complete continuums to any would be observer, wherein the time interval pertaining to such integral sign would approach zero; thereby confirming the very validity of yet another rather intrusive mathematical involvement.
[0724] Furthermore, other types of algebraic equations are considered to be continuous, beginning with that of a straight line whose linear equation of y=mx+b validates that for each and every real number x which becomes specified, a corresponding value of y truly exists.
[0725] With particular regard to a motion related solution for the problem of the trisection of an angle, algebraically expressed continuums relate to Euclidean formulations by well known cubic equations of a single variable in which trigonometric values of an angle of size 3 become associated to those of an angle of size .
[0726] The key factor pertaining to such relationships is that no matter what values might be applied to either of such angles, a three-to-one correspondence nevertheless would hold between their respective angular amplitudes!
[0727] As an example of this, consider various motion related solutions for the problem of the trisection of an angle which could be portrayed when cycling such famous Kempe prior art from a 20 degree setting to one of 120 degrees.
[0728] In such case, not only would an entire Euclidean formulation with representative geometric construction pattern as fully described by
[0729] That is to say, within such Euclidean formulation, angle ABC, when amounting to virtually any designated magnitude 3 within the limits of 20ABC120, furthermore would algebraically relate to an angle ABD therein, of resulting size , by such aforementioned famous algebraic cubic function.
[0730] Algebraically, such relationship could be confirmed for virtually any angle within such postulated range. For example, below such functional relationship is confirmed algebraically for the particular condition when angle ABC amounts to exactly 60:
ABC=3=60
=60/3=20=ABD;
cos(ABC)=cos(3)=cos 60=0.5;
cos(ABD)=cos =cos 20=0.93969262 . . . ;
4 cos.sup.33 cos =4(0.93969262 . . . ).sup.33(0.93969262 . . . )
=3.3190778622.819077862
=0.5.
[0731] Additionally, a specific nature that is found to be evident within algebraic continuums furthermore shall become addressed, wherein: [0732] a Euclidean formulation, each of whose constituent geometric construction patterns exhibits a rendered angle whose magnitude amounts to exactly three times the size of its given angle, is to become obtained by means of having the value of its sine described by a length of 3 sin 4 sin.sup.3 ; thereby conforming to the famous cubic function 3 sin 4 sin.sup.3 =sin (3); and [0733] a graph is to become developed that distinguishes between the continuity of such well known cubic function 4 cos.sup.3 3 cos =cos (3) and the discontinuity that clearly is evident within a function that otherwise assumes the form (4 cos.sup.3 6)/(20 cos )=cos (3).
[0734] Note that in this presentation such issue is addressed even before a more important detailed discussion that shall describe the very designs of such four newly proposed embodiments.
[0735] One method of algebraically relating a quadratic equation to two independent cubic functions that share a common root, wherein each function is limited only to a singular variable, is to link their respective coefficients together by means of what commonly is referred to as a simultaneous reduction process.
[0736] Since such common root, as denoted as z.sub.R below, occurs only when the value y in such functions equals zero, the following second order parabolic equation, thereby assuming the well known form ax.sup.2+bx+c=0, can be derived from the following two given cubic equations:
y.sub.1=0=z.sub.R.sup.3+.sub.1z.sub.R.sup.2+.sub.1z.sub.R+.sub.1;
y.sub.2=0=z.sub.R.sup.3+.sub.2z.sub.R.sup.2+.sub.2z.sub.R+.sub.2;
z.sub.R.sup.3+.sub.1z.sub.R.sup.2+.sub.1z.sub.R+.sub.1=0=z.sub.R.sup.3+.sub.2z.sub.R.sup.2+.sub.2z.sub.R+.sub.2;
.sub.1z.sub.R.sup.2+.sub.1z.sub.R+.sub.1=0=.sub.2z.sub.R.sup.2+.sub.2z.sub.R+.sub.2;
0=(.sub.2.sub.1)z.sub.R.sup.2+(.sub.2.sub.1)z.sub.R+(.sub.2.sub.1); and
0=az.sub.R+bz.sub.R+c.
[0737] Therein, whenever coefficients a, b, and c become specified, a straight line of length equal to such common root z.sub.R can be determined solely by conventional Euclidean means, simply by developing a geometric construction pattern that is representative of the famous Quadratic Formula z.sub.R=(b{square root over (b.sup.24ac)})/2a. Since such approach is not germane just to trisection, but nevertheless is relevant to a proper understanding of the dichotomy which exists between cubic functions of a single variable and an algebraically related famous parabolic equation, such geometric construction approach is to be described later on; after the four embodiments of such newly proposed invention first become formally introduced. Moreover, such particular resolution shall pertain to the specific circumstance when the coefficients in such well known parabolic equation, assuming the particular form az.sub.R.sup.2+bz.sub.R+c=0=ax.sup.2+bx+c become assigned the respective values of a=2, b=0.4, and c=0.75, thereby later being described by the second order equation of a single variable of the particular form 0.2x.sup.2+0.4x+0.75=0.
[0738] In such
z.sup.3+z.sup.2+z+=y; and
z.sup.3+3z.sup.2+z+(y)=0
[0739] In such first case, the variable z can change in value, thereby promoting a new corresponding value for y.
[0740] However, in such second case, generally a specific value of z is being sought after based upon the particular values which are assigned to its second order coefficient , its linear coefficient , and its scalar coefficient y. Notice that in such particular later reformatting, no attention whatsoever is directed to the fact that such value y also signifies a particular height above an x-axis within an orthogonal coordinate system at which a horizontal line passes through the curve that can be algebraically expressed as z.sup.3+z.sup.2+z+=y at three specific locations whose corresponding values away from the y-axis amount to the respective magnitudes of z. Such perceived distinctions also suitably should be accounted for; in order to serve as yet another rudimentary elements, as contained within an all-encompassing newly proposed equation sub-element theory.
[0741] In such
[0742] Such relationships are further addressed in section 9.3, as entitled Cubic Equation Uniqueness Theorem, also appearing within such above cited treatise; wherein it is stated that with respect to equation formats of singular variable, Only cubic equations allow solely rational and quadratic irrational numerical coefficients to co-exist with root sets comprised of cubic irrational numbers.
[0743] Such technical position doesn't address higher order equations merely because they represent byproducts of cubic relationships which are fashioned in a singular variable.
[0744] Neither does such contention dispute, nor contradict the fact that cubic irrational root pairs can, and do exist within quadratic equations of singular unknown quantity.
[0745] An example of this follows with respect to the parabolic equation presented below, followed by an associated abbreviated form of the Quadratic Formula:
[0746] After examining such abbreviated Quadratic Formula, it becomes obvious that the only way in which such roots can be of cubic irrational value is when either coefficient b and/or c also turns out to be cubic irrational.
[0747] As such, a corollary furthermore states, Cubic irrational root pairs which appear in parabolic equations or their associated functions require supporting cubic irrational coefficients.
[0748] Just as in the general case of conventional Euclidean practice where stringent rules apply, so to should they be specified in support of a geometric forming process. With respect to such flowchart, as posed in
[0749] A few of the very simple rules which apply to geometric forming are elicited directly below. Their intent is to simplify the overall administration of such process by means of requiring fewer lines in any attendant substantiation. As duly furnished below, some of them might appear to be rather straightforward, even to the point where they may be considered as being somewhat obvious such that: [0750] one principal rule is that the overall length of a linkage which belongs to any trisecting emulation mechanism is considered to remain constant throughout device flexure. Naturally, such rule applies so long as the linkage under consideration remains totally inelastic and intact during device flexure. From such rule, a wide variety of relationships thereby can be obtained, a small portion of which are listed as follows: [0751] when two straight solid linkages of equal length become attached along their longitudinal centerlines at a common end by an interconnecting pivot pin which situated orthogonal to it, such three piece assembly thereby shall function as an integral hinged unit, even during conditions when one of such linkages becomes rotated respect to the other about the radial centerline of such interconnecting pivot pin; and [0752] whenever one free end of such integral three piece unit thereby becomes attached along its longitudinal centerline to the solid end of another straight slotted linkage along its longitudinal centerline by means of inserting an second interconnecting pivot pin through a common axis which is orthogonal to such longitudinal centerlines, and thereafter the remaining unattached end of such initial integral three piece hinged unit has a third interconnecting pivot pin inserted orthogonally through its longitudinal axis whose radial centerline lies equidistant away from the radial centerline of its hinge as does the radial centerline of such added second interconnecting pivot pin, whereby such third interconnecting pivot pin furthermore passes through the slot of such slotted linkage, the longitudinal centerlines of such three linkages, together with the radial centerlines of such three interconnecting pivot pins collectively shall describe an isosceles triangle shape in space, even during device flexure. For example, when viewing prior art, as posed in
and [0760] lastly, an example is afforded for the particular case when a quadratic irrational number is to be further characterized, such that when:
and [0761] conversely, whenever trigonometric values of triads .sub.1, .sub.2, and .sub.3 become afforded as given quantities, geometric construction patterns can be approximated which are analogous to the above equations. For example, a unit circle can be drawn which exhibits three radii that emanate from its origin describing angles of , (+120), and (+240) with respect to its x-axis and terminate upon its circumference. Accordingly, from the equation below, the sum of their three ordinate values always must be equal to zero, verified algebraically as follows:
[0762] Before even trying to solve the classical problem of the trisection of an angle, either the designated magnitude of an angle which is intended to be trisected or some geometric construction pattern which fully describes it first needs to be furnished!
[0763] To the contrary, if such information instead were to be withheld, then the exact size of an angle which is intended to be trisected would not be known; thereby making it virtually impossible to fulfill the task of dividing into three equal parts.
[0764] In effect, such provision of an a priori condition performs the very important role of identifying exactly which classical problem of the trisection of an angle is to be solved out of a virtually infinite number of possible forms it otherwise could assume depending upon which designated magnitude comes under scrutiny!
[0765] For example, attempting to trisect a sixty degree angle solely by conventional Euclidean means poses an entirely different problem than trying to trisect a seventeen degree angle by means of applying the very same process.
[0766] From an entirely different point of view, whenever a motion related solution for the problem of the trisection of an angle becomes portrayed, it signifies that an actual event has taken place. Such is the case because some period of time must elapse in order to reposition a trisecting emulation mechanism to a designated setting.
[0767] If this were not the case, specifically meaning that an element of time would not be needed in order to effect trisection, then a motion related solution for the problem of the trisection of an angle thereby could not occur; simply because without time, there can be no motion!
[0768] In support of such straightforward line of reasoning, however, it surprisingly turns out that a trisecting emulation mechanism furthermore can portray a stationary solution for the problem of the trisection of an angle, as well; not as an event, but by sheer coincidence; meaning that such portrayed solution materializes before time can expire!
[0769] The only way this could occur is by having such solution be portrayed before an a priori condition becomes specified; thereby suggesting that such solution becomes posed even before defining the full extent problem which it already has solved.
[0770] Essentially, such stationary solution for the problem of the trisection of an angle consists of a condition in which the designated magnitude of an angle which is intended to be trisected just so happens to match the particular reading that a trisecting emulation mechanism turns out to be prematurely set to before such activity even commences.
[0771] The only problem with such stationary solution scenario is that its probability of occurrence approaches zero; thereby negating its practical application. Such determination is computed as such singular reading selection divided by the number all possible readings which such device could be set to, generally comprised of a virtually unlimited number of distinct possibilities, and thereby amounting to a ratio which equates to 1/.fwdarw.0.
[0772] The input box appearing in such
[0773] Unfortunately although posing a legitimate solution for the classical problem of the trisection of an angle, such rather elementary approach also proves to be entirely impractical; simply because there is no way of assuring that such generated rendered angle matches the designated magnitude of an angle which is intended to be trisected; as had to be specified as an a priori condition even before attempting to generate such solution!
[0774] Since such a priori condition might have specified any of an infinite number of possible designated magnitudes, the probability of such geometric construction activity proving successful approaches zero, as again calculated by the ratio 1/.fwdarw.0.
[0775] Therefore, the practicality of actually attempting to solve such classical problem of the trisection of an angle solely by conventional Euclidean means now easily can be evaluated; whereby any singular geometric construction pattern which could be generated in such manner that the magnitude of its rendered angle amounts to exactly three times the size of a given angle, as well as turns out to be equal to a designated magnitude which previously was identified, because it bears a probability that approaches zero percent of posing a legitimate solution for such classical trisection problem, pretty much should be considered to be an impossible avenue for obtaining such solution!
[0776] Another interpretation is that an angle could be divided into three equal parts by means of applying only a straightedge and compass to it, but only under the highly unusual condition that an unlimited number of opportunities become extended, thereby assuring success. Unfortunately, such alternate approach also should be viewed to be quite unacceptable because it would take forever to complete.
[0777] To follow through with such discussion, it should be mentioned, however, that an approach to solve such classical problem of the trisection of an angle in this very manner already was discovered. As copyrighted in chapter six of my never before published 1976 treatise entitled, Trisection, an Exact Solution, as filed under copyright registration number TXu 636-519, such infinite point solution can trisect in a precise manner by means of performing a multitude of consecutive angular bisections, all geometrically constructed upon just a single piece of paper. Since such solution was authored more than forty years ago, it is included herein for purposes of being shared with the general public for the very first time, but only after formally introducing the four embodiments of such newly proposed articulating trisection invention first.
[0778] In
[0779] Within such flowchart, although such process box is limited basically to trisection matters, a geometric forming process nevertheless is indicative of a whole gamut of improved drawing pretexts, besides that Euclidean formulations, which could be developed in order to chart certain other distinct motions which lie outside of its presently discussed purview, or very narrow scope which hereinafter is to be addressed in this presentation. Accordingly, it is important to note that such overall process, at some future date, furthermore could prove to be the source of countless other discoveries which would require either a motion related geometric substantiation, and/or an analogous higher order algebraic solution; thereby evidencing the enormity of a geometric forming capability with regard to its profound influence upon other forms of mathematics.
[0780] In 1893, Thomas Alva Edison at long last showcased his kinetoscope. Obviously, such discovery spurred on the development of a cinematic projector by the Lumire brothers shortly afterwards. Unfortunately, many instances can be cited in human history in which follow-up inventions of far larger practical importance succeed earlier landmark cases. Ironically, such type of mishap befell Edison on another occasion, as well; being when he developed a direct current capability which thereafter became improved upon by Tesla during such time that he introduced alternating current. Accordingly, one fitting way to suitably address such above described disparity would be to unequivocally state that due to a series of ongoing technical developments, an entire motion picture industry eventually became ushered in; whereby a great fanfare finally arose, as caused by a rather unsuspecting audience who became more and more accustomed to witnessing the actual footages of world events at the cinema, as opposed to just reading about them in the newspapers. Over time, the general public began to welcome viewing news in a more fashionable setting. In retrospect, Kempe's attempts to disclose how to articulate an anti-parallelogram linkage assembly for the express purpose of performing trisection most certainly appeared to receive far less critical attention. Whether or not there existed a large interest in such subject matter is hard to fathom, for just consider: A full fifteen years prior to Lumire's actual cinematic projector debut, dating back all the way to the late 1880's, it obviously would have been very difficult, if not impossible, to reveal in sufficient detail to any awaiting crowd, and that much less to one that might have been gathered some distance way, just how to articulate an anti-parallelogram linkage device in order to satisfactorily perform trisection. Moreover, consider: Had a presentation to this effect successfully been pulled off at that very time period, it more fittingly might have been mistaken for some sort of magic act! Be that as it may, had there also been a considerable demand levied beforehand, for example by some predisposed mathematics party who might have expressed an interest in viewing such purported trisection capability, it evidently would have had very little effect in the overall scheme of things. As it were, way back in the 1880's, with such industrial community seriously lagging behind in development, as least in comparison to what actually had become accomplished just ten to fifteen years later, fewer news organizations would have been available to disseminate important technical information of that kind. In sharp contrast, only rather recently has it truly become possible to pictorially describe just how a Kempe anti-parallelogram trisection device actually functions. In today's technology, a presentation very easily could be made, merely by means of simulating the relative movement of such Kempe anti-parallelogram device within a modern day computer. However, without being predisposed to such type of information, or even to a lesser extent, thoroughly apprised of such professed trisection capabilities, it most certainly would be very difficult, indeed, to foresee that the overall technique used to create the very illusion of motion all those prior years, merely by means of animating some ragtag assortment of pictures, or possibly even some collection of photographs whose overall shapes would have been known to differ imperceptibly from one to the next, furthermore could have been applied to replicate an observed motion by means of instead animating an entire family of related geometric construction patterns! Hover, had such association truly been made those many years before, it well might have contributed to substantiating that some articulating prior art mechanism actually could perform trisection effectively throughout its wide range of device settings.
[0781] Another possible reason for such noticeable omission could be a reticence, or complacency stemming from the fact that, not only had conventional Euclidean practice proved entirely satisfactory for use on most prior occasions, but moreover that, up until now, generating a singular drawing pattern was the preferred way to pictorially display various aspects of mathematics.
[0782] Unfortunately, as it just so happens to turn out, one of the very few instances in which a singular conventional Euclidean practice approach should not be applied, just for the very reasons expressed above, is when attempting to provide the solution for the classical problem of the trisection of an angle!
[0783] As such, it might well be that a recommendation never before was raised, thereby proposing to extend conventional Euclidean practice into a geometric forming process that is fully capable of describing certain motions, simply because such aforementioned complacency very well by now actually might have escalated into a full blown reluctance on the part of a seemingly silent majority of mathematical authoritarians to overcome the crippling Euclidean limitation of not being able to backtrack upon irreversible geometric construction patterns!
[0784] With regard to the particular damage levied upon trisection matters over the years by not otherwise adopting a formal geometric forming process, consider the very first English language trisection involvement, tracing all the way back to a particular drawing which appears on page 309 of such 1897 The Works of Archimedes. Inasmuch as such drawing is accompanied by a complete accounting of such previously referred to Archimedes proposition, as well as a suitable algebraic proof needed to substantiate it, the apparent problem is that such drawing only is a singular geometric construction pattern, thereby applying only to the specific chord length which appears within its depicted circle. In order for such drawing depiction to be fully consistent with such Archimedes proposition and supporting algebraic proof, it should be represented by an entire Euclidean formulation, replete with an infinite number of other chord lengths which furthermore could be described within such circle, and which such Archimedes proposition and supporting algebraic proof also apply to. Without such incorporation, such drawing remains quite adequate for substantiating the arbitrarily selected chord pattern which is illustrated therein, but nonetheless remains grossly impractical because it cannot represent such infinite number of other chord shapes and attendant sizes with its circle, and thereby also remain subject to the very requirements posed by such included proposition. Whereas such drawing evidently was presented as a convention of the time, it must be presumed that it was provided merely as an example of all of the other possible geometric construction patterns which also could have been drawn while still satisfying all of the requirements of such proposition. Unfortunately, the key element that never was stated therein is that all of such other possible geometric construction patterns furthermore must stem from the very same sequence of Euclidean operations that governs such singular drawing, as is represented therein.
[0785] Based upon such prior trisection rationale discussion, it becomes apparent that a singular geometric construction pattern can depict only one event which takes place during an entire articulation process, thereby representing only a momentary viewing which neither can provide an indication of where a particular motion might have originated from, nor where it might have ended up.
[0786] Accordingly, such singular drawing format remains somewhat deficient from the standpoint that it cannot even define all of the various geometries needed to characterize an entire articulated motion!
[0787] As such, a singular geometric construction pattern can be likened to a still photograph. Whereas the latter gave birth to the motion picture industry, it seems only appropriate that the former should serve as the basis for an improved geometric approach that becomes capable of characterizing motion!
[0788] Such newly proposed geometric forming process capitalizes upon the novel prospect that it requires an entire family of geometric construction patterns to adequately represent all of the unique shapes needed to represent a complete articulation event.
[0789] Accordingly, Euclidean formulations can be of service in motion related problems which cannot be fully interpreted by a singular geometric construction pattern.
[0790] With particular regard to trisection matters, the magnitude of at least one rendered angle exhibited within any constituent geometric construction pattern that belongs to a substantiating Euclidean formulation, quite obviously would need to amount to exactly three times the size of its given angle.
[0791] Hence, by means of verifying that its outline matches the overall shape of a corresponding regenerated static image that becomes automatically portrayed once a trisecting emulation mechanism becomes properly set, its smaller static image portion thereby could be substantiated to qualify as an associated trisector for such device setting.
[0792] As such, a Euclidean formulation, recognizable by its double arrow notation, could dramatically simplify the overall process needed to substantiate that some proposed invention has been designed so that it can perform trisection accurately over a wide range of device settings and, in so doing, thereby become referred to as a bona fide trisecting emulation mechanism; as duly is depicted in the lower right hand portion of such
[0793] Hence, applying such novel geometric forming process in this respect thereby validates that overlapment points, normally considered to be detrimental because they remain inconspicuous, can be supplanted with intersection points that become fully distinguishable as regenerated static images become automatically portrayed by means of properly setting trisecting emulating mechanisms
[0794] In closing, it should be mentioned that when imposing a controlled motion, it becomes possible to discern overlapment points; whereby such Euclidean limitation of otherwise not being able to distinguish them by means of backtracking exclusively from a rendered angle within an irreversible geometric construction becomes rectified!
[0795] Recapping, an overall explanation just has been afforded for the very first time which maintains that a discernment of overlapment points leads to trisection. Hence, it couldn't possibly have been referred to in any prior art.
[0796] Moreover, since such explanation alone accounts for how a motion related solution for the problem of the trisection of an angle can be portrayed, prior art couldn't possibly have rendered a differing substantiation that actually accounts for such professed capabilities.
[0797] Any further discussion concerning specific amounts of time which may be needed to arrange trisecting emulation mechanisms to particular device settings are omitted herein because such input is irrelevant when attempting to substantiate a motion related solution for the problem of the trisection of an angle; especially when considering that such times obviously would vary depending upon a user's dexterity, as well as the varying distances encountered when going from where such device might be temporarily positioned to a particular device setting.
[0798] In conclusion, if the logic proposed in such
[0799] Moreover, when considering that it is necessary to exert a motion in order to properly set any trisecting emulation mechanism, such warranted flexure could not, in any way, be fully described solely by a singular geometric construction pattern!
[0800] The process box entitled CLASSICAL PROBLEM OF THE TRISECTION OF AN ANGLE SOLUTION DISCREDITED 12 is to serve as the principal focal point within such flowchart, as represented in
[0803] The fact that a duration of time is needed in order to effect a motion related solution for the problem of the trisection of an angle eliminates the possibility that such form of solution potentially might double as a solution for the classical problem of the trisection of an angle. This is because any geometric construction pattern, once drawn, cannot be modified just by applying time to it; thereby affording a probability that still approaches zero that its overall outline just might happen to superimpose upon that which otherwise could be automatically portrayed whenever a static image becomes regenerated by means of properly setting any trisecting emulation mechanism.
[0804] Moreover, when investigating whether a geometric solution furthermore might qualify as a solution for the classical problem of the trisection of an angle, it should be remembered that if extraneous information were to become introduced into such problem that turns out to be relevant to determining its solution, then only a solution for some corrupted version of the classical problem of the trisection of an angle could be obtained; thereby solving an entirely different problem and, in so doing, discrediting any potential claims that might incorrectly allege that the classical problem of the trisection of an angle has been solved.
[0805] Lastly, for those remaining skeptics who otherwise would prefer to believe that a solution for the classical problem of the trisection of an angle might yet be specified, all they need to do is disprove that an availability of overlapment points actually prevents backtracking upon a rendered angle within any geometric construction pattern all the way back to a given angle whose magnitude amounts to exactly one-third of its size!
[0806] In other words, to dispute the new theory that is presented herein, it is now up to them to identify some as yet unidentified geometric construction pattern which would enable an angle of virtually any designated magnitude they might decide upon to be trisected; when neither violating the rules which pertain to conventional Euclidean practice, not introducing any extraneous information which could be considered to be relevant to its solution!
[0807] Over time, as such ascribed overlapment attribution finally becomes acknowledged to be the real cause for being unable to solve the classical problem of the trisection of an angle, ongoing analysis thereby could be performed in order to confirm, beyond any shadow of doubt, that trisection of an angle of any magnitude cannot be performed solely by means of applying only a straightedge and compass to it!
[0808] The process box entitled SINGULAR DRAWING SOLUTION DISPELLED 13 is included in such
[0809] Conversely, any proposed articulating trisection invention that only specifies a singular motion related solution for the trisection of angle couldn't possibly substantiate a trisection capability for its remaining wide range of settings!
[0810] The process box described as SUPPLEMENTAL DEVICE CAPABILITIES SPECIFIED 14 is the principal location in such
[0811] Such fact is duly reflected in such
[0812] For the particular case of the fourth embodiment of such newly proposed invention, a supplemental device leveling capability also is to be thoroughly described.
[0813] Within a right triangle, if the ratio between the length of one of its sides to that of its hypotenuse is cubic irrational, so must be the other. In other words, if one trigonometric property of a right triangle is cubic irrational, so must be all of its trigonometric properties!
[0814] It then logically would follow that for any right triangle that exhibits cubic irrational trigonometric properties whose hypotenuse amounts to one unit in length, the lengths of its constituent sides each would have to be of a cubic irrational value.
[0815] Such association enables the lengths of the sides of such right triangle to compensate for each other. With regard to the Pythagorean Theorem, this means that only the sum of the squares of two cubic irrational values can equal a value of one; thereby avoiding the common pitfall of otherwise attempting to equate such rational unitary value to the square of a cubic irrational value added to the square of either a rational or quadratic irrational value!
[0816] The reason that a right triangle which exhibits cubic irrational trigonometric properties truly can be geometrically constructed is because of the large number of geometric construction patterns which exist, all meeting such criteria; whereby the probability of drawing just one of them out of sheer coincidence increases dramatically.
[0817] Attempting to reproduce any one of them just be conventional Euclidean means, however, nevertheless would prove fruitless, resulting only in a mere approximation thereof; one which might prove suitable when being considered as a duplicate rendering, but not when taking into account differences between them which possibly only would become discernable well beyond what the capabilities of the human eye could detect.
[0818] By finally acknowledging that angles which exhibit cubic irrational trigonometric properties actually can be portrayed, their exact measurements would become revealed for the very first time, despite the fact that their real values can be described only by decimal patterns that are never-ending. Perhaps such new found capability very well might become perceived as an uncharted gateway that unfortunately was overlooked time and time again in the past!
[0819]
[0820] Its associated trisector NMP=QMP=RMP must be equal to exactly one-third of its size, amounting to a value which computes to 60/3=20.
[0821] Upon interpreting
[0822] Moreover consider that the notch appearing in its ruler resides away from its endpoint, M, one unit of measurement.
[0823] In isosceles triangle NMP, since length MN=length NP=1, it logically follows that twice the cosine of angle NMP would amount to the ratio between length MP length MN, whereby the following relationship thereby could be obtained:
[0824] Hence, a cubic irrational value 1.879385242 . . . must be the exact length of base MP of isosceles triangle NMP; whereby the three dots notated after such number indicates that such decimal pattern extends on indefinitely.
[0825] Since the cosine of twenty degrees furthermore is a transcendental, number, the above procedure also could distinguish such number types, thereby constituting a subset of cubic irrational numbers.
[0826] Once having devised a suitable geometric forming process, it thereby becomes possible to verify that device candidates which wish to qualify as trisecting emulation mechanisms conform to the various elements which funnel into such process box. For example, all devices must be shown to be fully capable of performing the primary function of regenerating static images, or be bound by the same set of rules. Devices which meet such criteria, but thereafter are found to share common design traits, should be categorized as such in order to assure that each item appearing within any particular group features some fundamental performance difference which qualifies it as being individually unique. The TRISECTION INVENTIONS CLASSIFIED 15 process box represents the location within such
[0827] The process box therein entitled REQUIREMENTS CHART PREPARED 16 is intended to distinguish that, although CATEGORY I and CATEGORY II prior art devices actually can perform trisection over a wide range of device settings, certain aspects of such capability never before were completely substantiated. The remainder of such
[0828] In closing, a novel geometric forming process just has been proposed which suitably explains how to rectify a major Euclidean limitation, essentially consisting of an incapability to distinguish overlapment points; as achieved simply by means of imposing a controlled motion which makes it possible to discern them!
[0829] Although trisection today can be performed because of such identified motion related compensation, were such deleterious behavior otherwise to remain unchecked, then trisection, as sought after by countless futile attempts to solve the famous classical problem of the trisection of an angle still would remain a very illusive problem!
[0830] Accordingly it is concluded that a geometric forming process thereby eclipses a rather limited conventional Euclidean practice that has been in vogue for millennia!
[0831] Having just concluded the prerequisite discussion pertaining to such flowchart, as posed in
[0832] Its first embodiment, as represented in
[0849] In order to properly reconcile the above listed components with the various elements which constitute the representative geometric construction pattern of such Euclidean formulation, as posed in
[0852] A more detailed description of the various components which comprise the first embodiment of such newly proposed invention is provided below: [0853] main dowel 203, whose shank is chamfered about its lower end, and also bears a very small white colored circle inscribed upon its upper face whose center point coincides with its vertically positioned radial centerline; [0854] intermediate dowel 204, whose envelope dimensions (including that of its chamfer) are identical to those respectively projected by main dowel 203, except that its shank only amounts to two-thirds of such length; additionally having a small sight hole of circular shape bored completely through it about its vertical centerline which is circumscribed by a large four pointed star imprinted around its upper face whose center point also resides upon such vertical centerline; [0855] shoulder screw 205, whose shoulder is of the same diameter, but only one-third as long as the overall shank length of main dowel 203; with its remaining threaded portion being of a slightly smaller maximum outer diameter and double the length of its shoulder; [0856] reference linkage 201, whose uniform cross section consists of two convexly opposed half circles of equal size with a rectangle interposed in between them two of whose opposing sides each superimpose directly upon the outside diameters of half circles; thereby collectively projecting the envelope of a rectangular bar that is rounded in an outwardly fashion about two ends whose semicircular shaped contours radiate about respective vertical centerlines that, by construction, must pass through each of the center points of the two half circles resident in each uniform cross section within such component; whose thickness throughout measures exactly one-third the shank length of main dowel 203; which has two circular shaped holes bored through it located so that their respective vertical centerlines coincide with those which its two circular shaped ends radiate about, thereby placing the outer half sections of each of its two holes into concentricity with the respective semicircular shaped contours of its two ends; with its circular shaped hole of slightly larger proportion, being suitably sized to fit comfortably inside of reference linkage 201 without breeching its outer wall in any manner, or considerably degrading its overall structural integrity, while furthermore being capable of fitting snugly around the shank of main dowel 203 without presenting a noticeable gap, but not to the degree where any clamping becomes apparent that otherwise could cause an appreciable buildup in frictional resistance to occur, thereby otherwise possibly thwarting relative rotation attempts; with such hole also being positioned to precede the appearance of the words GIVEN ACUTE ANGLE inscribed along the longitudinal centerline of the upper face of its midsection; with its circular shaped hole of slightly smaller proportion positioned beyond the imprinted slogan GIVEN ACUTE ANGLE located such that its vertical centerline coincides with that which its neighboring circular end radiates about, appropriately sized to form an interference fit with the shank of intermediate dowel 204, but not to the degree where reference linkage 201 could become seriously overstressed over time, consequently limiting its useful service life; [0857] adjustment linkage 202, whose envelope dimensions are the same sizes as those respectively projected by reference linkage 201, including its identically sized, opposite facing semicircular shaped extremities, excepting that its midsection is sized to be about one and one-half times longer; furthermore which exhibits a circular hole that is of identical size to the larger circular hole bored through reference linkage 201 whose vertical centerline is positioned to coincide with that which either of its semicircular shaped extremities radiates about; which additionally features a slot that extends along the longitudinal centerline of its entire remaining inner portion, whose cutout width is equal to the diameter of the circular hole bored through it and whose two internal extremities, being of identically shaped, but convexly opposed semicircular contours, represent mere extensions of its slot, carefully located so that they neither communicate with, nor encroach upon, either the circular hole bored through it or its neighboring outer extremity; [0858] positioning linkage 200, which contains two members radiating from a central hub at ninety degrees to one another, thereby constituting a rigid right angled framework; with its member which resides in a direction that is ninety degrees counterclockwise from its other member, when looking down upon such device, consisting of a midsection onto whose free end is attached an integral end piece that consists of an extremity that is sandwiched in between two feet, all three portions being of identical uniform cross section and precisely aligned with respect to one another, whereby only the extremity portion of such integral end piece is precisely fitted to such midsection, thereby forming a natural extension to it which is its same width and thickness where they join; with its remaining clockwise member consisting of another midsection whose envelope is identical in size to the midsection featured by its counterclockwise member, onto whose free end is attached an extended integral end piece that consists of a singular extremity which has a lower foot affixed onto it, both portions of which are of identical uniform cross section and precisely aligned with respect to each other, whereby only the extremity portion of such extended integral end piece is precisely fitted to the midsection of such clockwise member, thereby forming a natural extension to it which is its same width and thickness where they join; such that its two midsections, hub, integral end piece extremities and adjoining three feet all are of the same thickness as the constant overall depth featured by reference linkage 201, with the widths of its two midsections being equal to the constant width of the midsection of reference linkage 201; with its two integral end pieces bearing the same overall cross section as either of the two ends of reference linkage 201, with the only exception being that the constant width portion of the singular extremity of such extended integral end piece is longitudinally extended by an additional length that is either equal to or larger than the radius of the head of shoulder screw 205; with the vertex of said rigid right angled framework represented as a vertical centerline that runs directly through the intersection point of two imaginary longitudinal centerlines which respectively run across the upper faces of its two midsections; whose hub features an inner ninety degree circular contour and a diametrically opposed larger outer ninety degree circular contour, oriented in the same direction, the later of which is formed a common distance away from its vertex which amounts to one-half the width of its midsections in order to transition seamlessly into their respective neighboring portions; with its counterclockwise member being sized to a length such that the distance from the very tip of its integral end piece to the outer periphery of its hub, when measured directly along the extended imaginary longitudinal centerline which runs across the upper face of its midsection, is equal to the length of the imaginary longitudinal centerline which extends across the upper face of reference linkage 201 beginning at the very tip of one of its ends and terminating at that of its other end; with positioning linkage 200 further featuring a hole of circular shape bored through it whose size is identical to that of the circular hole of slightly larger proportion bored through reference linkage 201 whose vertical centerline is positioned to coincide with that of its vertex; whereby another circular hole that thereafter is tapped with threads is made through the upper foot and adjoining singular extremity of its integral end piece, without penetrating into its neighboring lower foot, such that their common vertical centerline is positioned to coincide directly with that which its semicircular shaped contour radiates about, sized so that the threads tapped into such bored hole mate effortlessly with the threads of shoulder screw 205; additionally featuring a slot that runs along the longitudinal centerline of the entire midsection of its counterclockwise member whose cutout width over its length is equal to the diameter of the circular hole drilled through its vertex, and whose internal extremities also assume the shape of semicircular shaped contours, carefully located so that they neither communicate with, nor encroach upon, either the tapped circular hole machined into it or the circular hole drilled through its vertex; which furthermore bears a small diamond shape inscribed upon its clockwise member whose center point is located along the imaginary longitudinal centerline that runs across the upper face of its midsection at a position which resides the same distance away from its vertex as does the common vertical centerline of the tapped circular hole machined into its counterclockwise member; wherefore each of the two spans that extend from its vertex and terminate respectively at the center point of the small diamond shape inscribed upon the upper face of its clockwise member, as well as at the vertical centerline of the tapped circular hole machined into its clockwise member, are of equal length to the span which extends between the respective vertical centerlines of the two circular holes of slightly different sizes bored through reference linkage 201; [0859] opposing main dowel 213, whose envelope dimensions, including that of its chamfer, are the same as those respectively projected by main dowel 203, except that its shank is one-third longer, and it additionally features a small four pointed star imprinted upon its upper face whose center point coincides with its vertical oriented, radial centerline; [0860] opposing intermediate dowel 214, which is identical in every respect to said opposing main dowel 213, except that instead of a small four pointed star, it bears a small dark grey circle inscribed upon its upper face whose center point coincides with its vertically positioned radial centerline; [0861] opposing shoulder screw 215, which is an exact duplicate of shoulder screw 205 in every respect; [0862] opposing reference linkage 211, whose envelope dimensions are the same sizes as those respectively projected by reference linkage 201, including its identically sized, opposite facing semicircular shaped ends; which has two circular holes of identical size to the larger circular hole bored through reference linkage 201 bored through it, positioned so that their vertical centerlines coincide with those which its respective two ends are generated about; wherefore the span between the respective vertical centerlines of the two circular holes of identical size bored through opposing reference linkage 211 must be equal to that which bridges between the respective vertical centerlines of the two circular holes of slightly different sizes bored through reference linkage 201; [0863] opposing adjustment linkage 212; which is an exact duplicate of adjustment linkage 202 in every respect; [0864] opposing positioning linkage 210, whose midsection, along with both extremity portions of two integral end pieces which it is precisely fitted to, thereby forming natural extensions to such midsection, in tandem exhibit envelope dimensions which are the very same sizes as those projected by reference linkage 201, including its semicircular shaped ends; whose rightmost endowed integral end piece, as located about axis O, is fitted with a foot above its extremity portion, and whose leftmost endowed integral end piece, as located about axis U, is fitted with a foot below its extremity portion, with each of such two feet furthermore being of the same shape as the three identically shaped feet incorporated into positioning linkage 200, whose bounding semicircular shaped foot contours become directly aligned with the corresponding shapes of their respective adjoining extremity portions; whereby a circular hole, whose diameter is equal to that of the larger circular hole bored through reference linkage 201, also is bored entirely through its rightmost endowed integral end piece about a common vertical centerline which is positioned to coincide with the vertical centerline about which its circular contour is generated; whereupon yet another circular hole which thereafter is tapped with threads, whose respective dimensions are identical to that of the tapped circular hole machined through the counterclockwise member of positioning linkage 200 in every respect, is machined entirely through such leftmost endowed integral end piece about a common vertical centerline which is positioned to coincide with the vertical centerline about which its circular contour radiates; such that the span which lies between the respective vertical centerlines of the circular hole and the tapped circular hole which engage opposing positioning linkage 210 is equal to the span which exists between the respective vertical centerlines of the two circular holes of slightly different sizes bored through reference linkage 201; furthermore which features the term LONGITUDINAL AXIS imprinted somewhere along its upper face, accompanied by a straight line marking that is inscribed along its imaginary upper longitudinal centerline; [0865] slide mechanism 216, whose envelope dimensions are identical to those projected by such reference linkage 201, except that its midsection is of an abbreviated, or lesser, overall length; which contains the term SLIDE imprinted longitudinally along the left side of its upper face, after which a circular hole is bored through it that is identical in all respects to the circular hole of smaller size bored through such reference linkage 201, and whose vertical centerline coincides with that which its contoured surrounding end has been fashioned about; and [0866] interconnecting linkage 230, whose respective envelope dimensions are the very same sizes as those projected by reference linkage 201, including its semicircular shaped ends; whereupon the term TRISECTOR is imprinted longitudinally along the midsection of its upper face; in which two circular holes, identical in all respects to the circular hole of smaller proportion bored through reference linkage 201, also are bored through it whose respective vertical centerlines align with those which its respective semicircular ends were designed about; such that the span between the respective vertical centerlines of the two circular holes bored through such interconnecting linkage 230 is equal to that which lies between the respective vertical centerlines of the two circular holes of slightly different sizes bored through reference linkage 201.
[0867] The description of reference linkage 201 should not be construed to mean that such design is frozen to just that specific configuration, whereby it furthermore could exhibit an assorted variety of alternate configurations so long as any modifications do not degrade its overall fit and function. Acceptable alterations could involve realigning its inscription, or even a possible abridging its words, such as just GIVEN ANGLE. Also, the overall envelope of reference linkage 201 could be changed so long as no additional material becomes introduced which would obstruct its uninhibited overall motion pattern; whereby such possible changes could include adjusting the overall shape of its uniform cross section, or possibly even introducing a shape which is not of uniform cross section.
[0868] Naturally, such refinements furthermore could apply to other device linkages, as well; whereby their envelope patterns, as previously distinguished with respect to that of reference linkage 201, also would be permitted to change accordingly.
[0869] Moreover, modifications which do not adversely affect device form, fit, or function always could be unilaterally applied without reservation.
[0870] For example, without degrading the overall performance of such first embodiment in any manner, straight line markings very easily could be added to indicate the exact positioning of the longitudinal centerlines which run along the upper surfaces of reference linkage 201 and opposing reference linkage 211. Such refinements might better be used to demarcate the fundamental architectures of such compass and counterbalance compass assemblies, especially with respect to the appendage OV of positioning linkage 200; thereby serving to abet the currently pictured way of distinguishing them which consists of: [0871] a white colored circle, as inscribed upon the upper face of main dowel 203; [0872] a small dark grey circle, as inscribed upon the upper face of opposing intermediate dowel 214; [0873] a large four pointed star, as imprinted around the upper face of intermediate dowel 204; [0874] a small four pointed star, as imprinted upon the upper face of opposing main dowel 213; [0875] the straight line which runs along the upper face of opposing positioning linkage 210 which is indicative of its longitudinal centerline; and [0876] a small diamond shape inscribed upon the clockwise member of positioning linkage 200.
[0877] Whereas both angle TOU and angle TOU, as represented in
90=VOU+(903)
0=VOU3
3=VOU.
[0881] Trisection is achieved merely by specifically arranging and thereafter setting such device in the following manner: [0882] laying the basic configuration of this invention down upon a table top or flat surface; [0883] loosening shoulder screw 205; [0884] applying slight finger pressure upon such clockwise member of positioning linkage 200; [0885] rotating reference linkage 201 about axis O until such time that its longitudinal centerline resides at an angle of 3 with respect to the longitudinal centerline of clockwise member of positioning linkage 200. Once configured in this fashion, an imaginary straight line which runs from the white colored circle inscribed upon the upper face of main dowel 203 to the large four pointed star imprinted around the upper face of intermediate dowel 204 would reside exactly 3 counterclockwise of another imaginary straight line which runs between such white colored circle inscribed upon the upper face of main dowel 203 and the small diamond shape inscribed upon the clockwise member of positioning linkage 200 with respect to axis O; thereby placing into position the GIVEN ACUTE ANGLE magnitude of such compass assembly, as duly notated upon its reference linkage 201; [0886] tightening shoulder screw 205 such that the magnitude of vertex angle TOU of such compass assembly, as viewed about axis O and depicted in
[0893] Whenever a designated angle VOU of such first embodiment, as posed in
[0894] During such types of events, both designated angle VOU, as well as trisected angle VOO, as measured about axis O, become identified in the following manner: [0895] angle VOU of magnitude 3 becomes subtended between two imaginary lines, both of which emanate from the white colored circle inscribed upon the upper face of main dowel 203 and run, respectively, to the small diamond shape which is inscribed upon the clockwise member of positioning linkage 200 and to the large four pointed star imprinted around the upper face of intermediate dowel 204; and [0896] angle VOO of size becomes subtended between two imaginary lines, both of which emanate from the white colored circle inscribed upon the upper face of main dowel 203 and run, respectively, to the small diamond shape inscribed upon the clockwise member of positioning linkage 200 and to the small four pointed star which is imprinted upon the upper face of opposing main dowel 213.
[0897] Accordingly, arc VO, as it extends from such small diamond shape to such small four pointed star, amounts to exactly one-third the size of arc VU, as it extends from such small diamond shape to such large four pointed star. In other words, such small star trisects the very angle which is established by such large star.
[0898] During such time that such first embodiment becomes properly set in this manner, such counterbalance compass assembly translates relative to such compass assembly in a practically frictionless manner. This enables such compass assembly to remain stationary with respect to the table top which it has been placed upon.
[0899]
[0900] By means of implementing the above cited trisection procedure, the basic configuration of such newly proposed invention can trisect angles which cannot otherwise be determined when otherwise unsuccessfully attempting to solve such classical problem of the trisection of an angle; thereby surpassing Euclidean capabilities.
[0901] Such trisection approach enables both the compass assembly and counterbalance compass assembly to meet the previous stipulated rule that their respective fundamental architectures perform as isosceles triangles for all possible configurations which they might assume. Accordingly, once specifically arranged in accordance with such
[0904] In such above scenario such magnitude is calculated by subtracting any acute designated magnitude of 3 value from ninety degrees.
[0905] Furthermore, by means of thereafter properly setting such device, as described above, such chosen geometric construction pattern then furthermore would fully describe the overall outline of a static image which the first embodiment would regenerate at such time; thereby substantiating that a motion related solution for the trisection of an angle was automatically portrayed.
[0906] The major difference between the fundamental architecture of such basic configuration, as represented in
[0907] Because of this outstanding difference, the overall height of such first embodiment additionally needs to be assessed from the standpoint that the concentricity and tolerance stack-ups at device interconnections permits constituent linkages to move only within particularly designated elevations, whereby they can be assured to perform parallel to one another at all times. Without invoking such design requirement, each particular trisection could not be validated as being precise!
[0908] In order to provide such substantiation, a First Embodiment Stacking Chart, as presented in
[0909] Later, it also will be validated that the components of such first embodiment actually can be assembled to perform properly in such prescribed manner.
[0910] More specifically stated, the design of such first embodiment, as presented in
[0911] Such perspective drawing, as posed in
[0912]
[0913] The second major heading expressed in
[0914] Such
[0918] As such, dowel and shoulder screw notations cited in such
[0919] Whereby such pivot pins appear only as level IV, V, and level VI entries, it indicates that they are suitably placed at the very top of respective
[0920] Such
[0925] As indicated in
[0926] All embedded feet serve to maintain linkages at level positions throughout their swings. Those depicted in
[0927] Hence, fewer working parts are needed. Such type of design eliminates the need for added washers, functioning as shims, which could easily be lost, especially at location T where shoulder screw 205 becomes unscrewed, and thereby could be easily removed. The idea is only to loosen screws, thereby allowing different given angles to be configured, without actual disengagement. This approach assures that such screws remain attached to the device at all times so that they don't get lost.
[0928] Alternatively, feet could be machined separately and thereafter bonded onto resident linkages; thereby enabling all such elements to be manufactured from a single stock which exhibits a common thickness throughout.
[0929] Based upon such
[0937] Adjustment linkage 202 and opposing adjustment linkage 212, shown to be physically longer than the other linkages which appear in
[0938] Such elongation makes it possible to specifically arrange the vertex angles of such compass and counterbalance compass assemblies from approximately zero to almost 90 degrees; also allowing for their shoulder screws to be properly tightened thereafter. Since each of such magnitudes amounts to 903, this indicates that angle VOU, as represented in
[0939] Such estimate accounts for the fact that various components contained within such first embodiment restrict such angles from being set over the entire ninety degree range; whereby it is concluded that all acute angles can become trisected by such device, except for those which approach: [0940] ninety degrees because reference linkage 201, operating at level III according to such
[0942] Moreover, whereas the lengths of opposing reference linkage 211, interconnecting linkage 230, and the slot cut through such counterclockwise member of positioning linkage 200 are almost the same length, angle OOT as determined before to be of size 2, can be varied from magnitudes of near zero degrees to almost sixty degrees during flexure. Hence, the maximum trisector size, , is almost thirty degrees, amounting to one-third of ninety degrees.
[0943] Such first embodiment is assembled by means of piecing together, interlocking, and thereafter anchoring its constituent compass and compass counterbalance assemblies in the following manner: [0944] such compass assembly becomes pieced together by means of first inserting the shank of main dowel 203 through the circular hole of slightly larger proportion bored through reference linkage 201, and thereafter through the circular hole drilled through the vertex of positioning linkage 200, making sure that at such time both the foot fitted onto its extended integral end piece is facing in a downwards direction, and reference linkage 201 has been rotated about main dowel 203 so that it is repositioned about halfway in between the rigid right angled framework featured by positioning linkage 200; such that the shank of intermediate dowel 204 then becomes inserted through the circular hole bored through adjustment linkage 202, and then becomes press fit until it reaches the very bottom of the vacant circular hole of slightly smaller proportion bored through reference linkage 201 in a manner in which the two extended tips of the large four pointed star imprinted upon its upper face become aligned with the longitudinal centerline of reference linkage 201, wherefore after its shank becomes firmly seated, the cavity created between its chamfer and surrounding hole becomes filled with glue; whereby adjustment linkage 202 becomes rotated about intermediate dowel 204 until some portion of its slot becomes positioned directly over the threaded hole of circular proportions that is machined into the integral end piece of positioning linkage 200 and shoulder screw 205 thereafter becomes tightened into such threaded hole; [0945] likewise, such counterbalance compass assembly becomes pieced together by means of inserting the shank of opposing intermediate dowel 214 through either of the circular holes bored through opposing reference linkage 211, then through the circular hole bored through opposing adjustment linkage 212; and thereafter by means of inserting the shank of opposing main dowel 213 through the vacant circular hole bored through opposing reference linkage 211, and then into the circular hole bored through the rightmost endowed integral end piece of opposing positioning linkage 210, being sure that its foot is facing in an upwards direction; [0946] after which such compass assembly and such counterbalance compass assembly furthermore become interlocked by means of first laying such compass assembly upon a flat surface or table top and then tilting such counterbalance compass assembly in a fashion such that its constituent opposing reference linkage 211 remains above any compass assembly components, but such that a portion of the slot cut into its constituent opposing adjustment linkage 212 is positioned underneath that of adjustment linkage 202; then rotating opposing positioning linkage 210 about opposing main dowel 213 until some portion of the straight line inscribed along the longitudinal centerline upon its upper face becomes observed directly underneath the sight hole bored through intermediate dowel 204; whereupon opposing intermediate dowel 214 then becomes translated so that it resides directly above some portion of the slot that runs along the longitudinal centerline of the entire midsection of the counterclockwise member of positioning linkage 200, thereby enabling the exposed shank portion of opposing intermediate dowel 214 which already was passed through the far end of opposing reference linkage 211 and the circular hole bored through opposing adjustment linkage 212 to become inserted through such positioning linkage 200 slot so that the remaining portion of its shank finally can be press fit through the hole bored into slide mechanism 216, until such time that it bottoms out in such hole, enabling it to be glued into position about its chamfer, thereby enabling slide mechanism 216 to be rotated until the word SLIDE imprinted upon its upper face becomes aligned with the longitudinal centerline running along the upper face of the clockwise member of positioning linkage 200; whereby opposing adjustment linkage 212 thereafter becomes rotated about opposing intermediate dowel 214 until some portion of its slotted midsection aligns directly over the tapped circular hole machined through the leftmost endowed integral end piece of opposing positioning linkage 210 so that opposing shoulder screw 215 thereafter can be tightened into such threaded hole; and [0947] after which such compass assembly and counterbalance compass assembly thereafter become anchored by means of press fitting the exposed shank portion of main dowel 203 which extends beyond the lower surface of positioning linkage 200 through the circular hole bored through interconnecting linkage 230 which precedes the term TRISECTOR imprinted upon its upper face, whereby glue is applied to the intervening space afforded about the chamfer of main dowel 203 and the exposed rim of such circular hole bored through interconnecting linkage 230 preceding the words TRISECTOR, thereby permanently attaching them together; after which interconnecting linkage 230 is rotated about main dowel 203 until the vertical centerline of the vacant circular hole bored through interconnecting linkage 230 coincides with that of opposing main dowel 213, whereupon the exposed shank portion of opposing main dowel 213 becomes press fit through the vacant circular hole bored through interconnecting linkage 230 in a manner such that the two extended tips of the small four pointed star imprinted upon its upper face thereby become aligned along the imaginary longitudinal centerline running along the upper face of interconnecting linkage 230, whereby glue thereafter becomes applied to the intervening space afforded about the chamfer of opposing main dowel 213 and the exposed rim of the vacant circular hole bored through interconnecting linkage 230, thus permanently attaching them together. During such process, extreme care should be exercised to make sure that both main dowel 203 and opposing main dowel 213 bottom out in interconnecting linkage 230.
[0948] The second embodiment, as represented in
[0951] In such Euclidean formulation, as posed in
[0952] Nor has the geometrical function of straight line member 22 of such Euclidean formulation, as illustrated in
[0953] The plan view and associated side elevation views of such second embodiment, as furnished in
[0954] The second embodiment of such newly proposed invention, as posed in
[0957] Upon thorough review, it can be confirmed that such above stated operation for performing trisection is entirely consistent with the procedure which is postulated in such
[0958] Accordingly, an obtuse angle which is algebraically expressed to be of size 1803 can be trisected merely by setting such second embodiment to a designated magnitude which is algebraically denoted to be of size 30!
[0959] Such result can be easily verified for the particular setting which actually is exhibited in
[0960] A further detailed logic serving to verify that such second embodiment furthermore is capable of trisecting obtuse angle VOW, as algebraically expressed to be of size 1803 in
[0963] In
[0964] Wherein the side elevation view of
[0967]
[0968] For example,
[0969] In
[0970] Therein, diagonal hatching depicted upon adjustment linkage 202 indicates where it has been cut by such sectioning process, as posed in
[0971] Such
[0972] As indicated in
[0973] With respect to
[0974] In
[0975] According to
[0976] Such insertion process, as exemplified in
[0977] The four dowels featured in such first and second embodiments also exhibit the very same head and bore diameters. Only their shank lengths vary in size. Whereas the shank of intermediate dowel 204 extends through two respective linkages; the shank of main dowel 203 permeates three elevation levels, and the shanks of opposing main dowel 213 and opposing intermediate dowel 214 travel through four levels of thickness.
[0978] Moreover, the shank diameters of the four dowels featured in such first and second embodiments also are equal in size to the diameters of the shoulders of the screws also included in such designs.
[0979] The head thicknesses of such shoulder screw 205 and opposing shoulder screw 215, as posed in
[0980] With respect to
[0981] In the exploded view shown in
[0982] The third embodiment of such newly proposed articulating trisection invention. as represented in
[1000] Such rhombus configuration has no need for feet, with the exception of right linkage 42 about axis O (as further discussed below); thereby greatly simplifying its overall design. Out of the seven linkages itemized above, only stabilizer linkage 47 is of a different overall span which amounts to approximately double such length.
[1001] Protractor board 40 has been added to such design wherein the radius of the large circle, which readings are imprinted about, is equal to the spans of such other six linkages. Two circular holes of sizes equal to the smaller hole bored through reference linkage 201 also are bored through it whereby their vertical axes pass through axis O and axis T, respectively, as appearing in
[1002] The third embodiment actually represents little more than a repositioning of the components contained in such basic configuration, thereby becoming derived from it. Shoulder screw 205 and opposing shoulder screw 215, as represented in such basic and modified configurations, become removed, and thereafter replaced by less expensive upper left dowel 52 and upper right dowel 56. Such dowels all conform to the previous design standard, as specified in the first and second embodiments of such newly proposed invention, with the only exception being that the overall length of the shank of upper right dowel 56 has been modified from three times the thickness of reference linkage 201 to two times its thickness.
[1003] A detailed accounting of the repositioning of such third embodiment components is provided below: [1004] the clockwise member of positioning linkage 200 appearing in
[1021] In the Third Embodiment Stacking Chart, as presented in
[1022] The second major heading expressed in
[1023] Again, dowel notations, as cited in such
[1024] As further indicated therein, dowel entries are listed at the very top of respective
[1025] An interpretation of
[1026] Such
[1027] Such
[1028] The principal purpose of the side elevation view afforded
[1029] Such third embodiment, as represented in
[1030] To substantiate that such rhombus configuration can, in fact, trisect any and all angles which it might become properly set to require that: [1031] opposite sides of a rhombus, as described by the longitudinal centerlines of left linkage 41, right linkage 42, lower linkage 43, and upper linkage 45 therein, respectively remain parallel throughout device flexure; [1032] the longitudinal centerline of middle linkage 44 furthermore remains parallel to lower linkage 43, as well as upper linkage 45 throughout device flexure; and [1033] a trisector for any setting which axis U could assume thereby would become automatically portrayed by the longitudinal centerline of lower linkage 43.
[1034] For such validation, the four centerlines which comprise such rhombus are denoted in
[1035] Whereas both the lower left dowel 49 and upper left dowel 52 are permanently affixed to such underlying protractor board 40, as posed in
[1036] Accordingly, lower linkage 43 can be rotated about lower left dowel 49 in a completely unobstructed manner; whereas, upper linkage 45 also is free to be rotated about upper left dowel 52 in much the same manner.
[1037] Two solutions for a quadratic equation, as arrived at below, algebraically indicate two possible locations where axis Y could reside in such rhombus configuration, with one of such solutions being ignored because it is located at axis O, therein recognized as the origin of such
[1038] In effect, such determination consists of: [1039] designating and thereby identifying coordinate values with respect to axis O for each of the following listed axes: [1040] for U(x.sub.U; y.sub.U)=( cos 3; sin 3); [1041] for axis T(x.sub.T; y.sub.T)=(0; ; [1042] for axis O(x.sub.O; y.sub.O)=( cos ; sin ); [1043] for axis T(x.sub.T; y.sub.T)=(0; 2 sin ); [1044] for axis Y(x.sub.Y; y.sub.Y); [1045] for axis Y(x.sub.Y; y.sub.Y); [1046] since axis Y must exist somewhere upon the circumference of a circle of radius whose center is situated at axis O its coordinates would satisfy the equation (xx.sub.O).sup.2+(yy.sub.O).sup.2=.sup.2; [1047] since axis Y also must exist somewhere upon the circumference of a circle of radius whose center is situated at axis T, its coordinates furthermore would satisfy the second order equation x.sup.2+(yy.sub.T).sup.2=.sup.2: [1048] equating such two mathematical relationships avails a determination of their respective intersection points, thereby identifying exactly where axis Y is located. This is accomplished by means of substituting x.sub.Y for the x designations and y.sub.Y for the y designations in such above two equations, whereby:
(x.sub.Yx.sub.O).sup.2+(y.sub.Yy.sub.O).sup.2=x.sub.Y.sup.2+(y.sub.Yy.sub.T).sup.2
(x.sub.Yx.sub.O).sup.2+(y.sub.Yy.sub.O).sup.2=x.sub.Y.sup.2+(y.sub.Y).sup.2
(x.sub.O.sup.2+y.sub.O.sup.2)2(x.sub.Yx.sub.O+y.sub.Yy.sub.O)=2y.sub.Y+.sup.2
.sup.22(x.sub.Yx.sub.O+y.sub.Yy.sub.O)=2y.sub.Y+.sup.2
2(x.sub.Yx.sub.O+y.sub.Yy.sub.O)=2y.sub.Y
x.sub.Yx.sub.O+y.sub.Yy.sub.O=y.sub.Y
x.sub.Yx.sub.O=y.sub.Y(y.sub.O);
but since x.sub.O.sup.2+y.sub.O.sup.2=.sup.2
x.sub.O.sup.2=.sup.2y.sub.O.sup.2
x.sub.Ox.sub.O=(+y.sub.O)(y.sub.O): [1049] by means of comparing the respective terms of such above two results, it turns out that when x.sub.Y=x.sub.O, y.sub.Y=+y.sub.O:
[1050] Checking such two above cited second order equations for correctness renders:
(x.sub.Yx.sub.O).sup.2+(y.sub.Yy.sub.O).sup.2=.sup.2;
(x.sub.Ox.sub.O).sup.2+(+y.sub.Oy.sub.O).sup.2=.sup.2
.sup.2=.sup.2; and
x.sub.Y.sup.2+(y.sub.Yy.sub.T).sup.2=.sup.2;
x.sub.O.sup.2+(+y.sub.O).sup.2=.sup.2;
x.sub.O.sup.2+y.sub.O.sup.2=.sup.2.
[1051] Above, since x.sub.Y=x.sub.O, it indicates that phantom straight line segment OY must remain parallel to straight line segment OT for all possible positions that axis O might assume; thereby indicating that right linkage 42 remains parallel to left linkage 41, as well as the +y-axis, for all possible third embodiment configuration engagements.
[1052] Secondly, by means of fabricating middle linkage 44 so that its longitudinal centerline also is of span p, it thereby also must remain parallel to the longitudinal centerlines of lower linkage 43 and upper linkage 45 throughout device flexure. This is because such straight line segments OO, TY, and TY, as posed in
[1053] By means of arranging upper linkage 45 parallel to lower linkage 43, as can be accomplished when rotating them properly with respect to stationary left linkage 41, as posed in
[1054] Otherwise, if right linkage 42 were not situated in a perfectly vertical attitude, by nevertheless being bound to terminate somewhere along the circumference of a circle of radius p produced about axis T, it thereby would terminate at some location other than at axis Y. However, in order to do so, right linkage 42 then would have to be either shorter or longer; consequently violating the requirements of being a rhombus in the very first place!
[1055] Lastly, an additional proof thereby relies upon the fact that since such straight line segment TY remains parallel to straight line segment OO throughout device flexure, it thereby must subtend an angle with respect to straight line segment OT that amounts to a magnitude of 90+. Hence, by the Law of Sines, it can be determined that:
[1056] For the above analysis to be valid, angle OTU therein, being algebraically expressed to be of size 90+, with its straight line segment OT furthermore residing upon the +y-axis, as posed in
[1057] Such observation is to be confirmed based upon the understanding that the reading upon the circumference of protractor board 40 which just so happens to appear within the longitudinal centerline strip imprinted on either side of the longitudinal centerline of middle linkage 44 actually defines the very location of axis U; thereby furthermore specifying a particular designated magnitude of 3 which such third embodiment, at such very moment, trisects.
[1058] As such, the following identity can be derived wherein sin (3) is to be denoted by and cos(3) is to be designated as :
2=3;
sin(2)=sin(3)
2 sin cos =sin(3)cos cos(3)sin
2 sin cos = cos sin ; and
(2 sin cos )=( cos sin )
2 sin cos = cos sin
2 sin = tan
tan =2 sin
tan =(2 sin )/.
[1059] Such result easily can be confirmed by means of referring to
[1060] As axis U approaches ninety degrees with respect to the +x-axis, such third embodiment reaches its design limit; thereby enabling lower linkage 43 to operate only between 0 and 3 degrees.
[1061] As such, the remaining phantom lines shown in
[1062] As such, it just has been proven that the longitudinal centerline of lower linkage 43, as represented in the plan view of
[1063] Hence such designated reading, as algebraically is expressed as 3 and represented by angle VOU therein, is shown to be trisected by angle VOO, thereby amounting to a magnitude .
[1064] Therefore, the overall outline of any static image that becomes regenerated by means of properly setting such third embodiment, because it furthermore could be fully described by a geometric construction pattern which belongs to such second derivative Euclidean formulation, as represented in
[1065] The purpose of stabilizer linkage 47, as represented in plan view in
[1066] Such rhombus configuration is strictly regulated during device flexure by an expanse of stabilizer linkage 47 which extends from axis O to axis Y; thereby furthermore describing a diagonal of a parallelogram whose respective sides, as posed in
[1067] Since the midpoints of diagonals of a rhombus must cross each other, cross dowel 53 which resides midway along cross linkage 46 must translate through the slot afforded in stabilizer linkage 47 to a location such that span OY always is equal to span YY.
[1068] Accordingly, axis Y becomes located, not by one, but by two simultaneous, independent movements which serve to reinforce one another, and thereby increase device accuracy, described as follows: [1069] the first motion is that of such parallelogram OOYT, as formed by aforementioned portions of left linkage 41 and right linkage 42, along with the entire spans of lower linkage 43, and middle linkage 44, which thereby describes the positioning of axis Y during device articulation; and [1070] the second motion is that of middle right dowel 55 whose center point, not only describes axis Y, but also must reside inside of the slot afforded by right linkage 42, no matter what angle it becomes rotated to with respect to the x-axis during the entire articulation process.
[1071] The reason that both lower linkage 43 and middle linkage 44 are of transparent design is so that angle VOU and angle VOO readings can be accurately deciphered upon a protractor board 40 which resides underneath them.
[1072] In order to trisect given obtuse angle VOW of magnitude 1803, the following procedure shall become administered: [1073] given obtuse angle VOW of magnitude 1803 first is to be superimposed upon protractor board 40, as represented in the plan view of
[1077]
[1078]
[1079] As also is indicated in
[1080]
[1087]
[1091] The fourth embodiment of such newly proposed articulating trisection invention, is comprised of a slotted linkage arrangement, as well as a slider arrangement.
[1092] Because detail discussions on such arrangements are rather lengthy, each is presented separately, one after the other.
[1093] The overall layout of components which comprise such slotted linkage arrangement is clearly delineated in
[1113] The protocol of designing linkages which exhibit constant cross-sections, as was applied to such first, second, and third embodiments, carries over into such slotted linkage arrangement. Therefore, every cross-section is to consist a rectangle of the same size, each of whose who shorter opposing sides furthermore serves as the diameter of a half circle appended onto it whose remaining periphery faces away from such rectangle, thereby residing outside of its periphery. Just as before, each linkage is to be of the very same thickness and assumes the overall shape of a bar with rounded extremities.
[1114] The cutout patterns and spans of such linkages are to be in accordance with those depicted in the plan view of
[1115] Notice that some of the find numbers represented in the front elevation view of
[1116]
[1117] Therein, the depth of the lower portion of foot fitted onto control slotted linkage 61 is to be equal to the thickness given obtuse angle solid linkage 62 (not shown), thereby maintaining a separation at axis T which is the same as that afforded between the lower face of control slotted linkage 61 and the upper surface of protractor strip 57 about axis O, as depicted in
[1118]
[1119] Whereas dowel at axis Z 75 is shown in
[1120] The overall design of the device is such that during its articulation, all linkage portions are to remain confined within specific elevations; that is, they are permitted to translate only at designated vertical distances above the upper surface of protractor strip 57.
[1121] Such approach assures that all linkage spans remain entirely parallel to one another during device flexure; thus confirming that they do not pose any potential for introducing an obstruction that, if otherwise permitted, very well might impede acceptable trisection performance.
[1122] As such, each specific portion of any constituent linkage, consisting of its overall span, as well as any incorporated foot it might feature is duly accounted for in the Slotted Linkage Arrangement Stacking Chart, as presented in
[1123] Just as before, the first column therein, under the heading entitled LEVEL, is reserved for itemizing levels in chronological order away from a table top that such device can be laid upon. The second major heading expressed in
[1124] Such
[1131] As such, bisector slotted linkage 60 and control slotted linkage 61 both articulate within the same level. This does not pose a problem because trisector solid linkage 58 can rotate about axis O only from approximately zero to thirty degrees relative to the +x-axis.
[1132] Moreover, the following determination has been made regarding the location of linkage feet: [1133] trisector solid linkage 58 incorporates a foot which occupies levels I and II at axis O; [1134] given acute angle slotted linkage 59 incorporates a foot which occupies levels I, II, III and IV at axis U; [1135] bisector slotted linkage 60 incorporates no feet, whereby it sits atop retaining ring at axis Y 74 and supports spacer at axis Y 73 about axis Y; [1136] control slotted linkage 61 incorporates a lower foot which occupies level I at axis T, and an upper foot which occupies levels III, IV and V at axis T; [1137] given obtuse angle solid linkage 62 incorporates a foot which resides both in levels II and III at axis Z; and [1138] transverse slotted linkage 63 has no feet, whereby it is perched atop spacer at axis T 68 at axis T, atop trisector solid linkage 58 at axis O, atop spacer at axis Y 73 at axis Y; and atop of the foot incorporated into given obtuse angle solid linkage 62 at axis Z, as can be easily verified by means of referring to
[1139] As indicated in such
[1146] For any particular axis that is depicted in
[1159] By adopting the same convention as formerly was applied to the previous stacking charts, dowel at axis O 65, dowel at axis T 67, shoulder screw 70, dowel at axis O 71, dowel at axis Y 72, and dowel at axis Z 75 notations enumerated in such
[1160] In particular, such
[1161] The functions of the spacers and retaining rings listed in such
[1167] As further indicated in such
[1168] Whereas bisector slotted linkage 60 doesn't repeat itself in any row in such
[1169] Hence, such
[1170] Next, an explanation is to be furnished pursuant to such
[1175] In such above described scenario, trisection thereby mechanically becomes achieved because as slide linkage 64 becomes translated either upwards or downwards inside of the slot afforded within control slotted linkage 61, as depicted in
[1179] Whereas the magnitude of virtually any angle which could be swept out about axis O in this prescribed manner, as extending from the +x-axis to the longitudinal centerline of given acute angle slotted linkage 59 furthermore would have to amount to the sum of the magnitudes of the angles simultaneously extending from such +x-axis to the longitudinal centerline of trisector solid linkage 58, and then from the longitudinal centerline of trisector solid linkage 58 to that of given acute angle slotted linkage 59, it would have to be of a size algebraically calculated to amount to as +2=3; meaning that for any reading which appears at axis U, a corresponding reading which appears at axis O would have to amount to one-third of its size, thereby representing its trisector.
[1180] It thereby can be concluded that the slotted linkage arrangement of the car jack configuration of such newly proposed invention, as duly depicted in plan view in
[1181] Substantiating such capability would consist merely of demonstrating that virtually any static image which could be regenerated by means of properly setting such slotted linkage arrangement would automatically portray an overall shape that furthermore fully could be described by a geometric construction pattern in which the magnitude of its rendered angle amounts to exactly three times the size of its given angle.
[1182] In certain circumstances it can be shown that such rather cumbersome trisection substantiation process, as described above, could be dramatically reduced by means of taking advantage of the understanding that the fundamental architecture of each defining embodiment drawing, as cited in
[1183] Such disclosure becomes quite apparent when referring to
[1184] Next, an explanation is to be furnished pursuant to such
[1193] Such slotted linkage arrangement can be specifically arranged in such manner because both given acute angle slotted linkage 59 and control slotted linkage 61 exhibit hole cutouts of the same size, Hence, shoulder screw 70 could be inserted through the hole featured by either linkage at will.
[1194] Moreover, since such
[1195] In such above described scenario, trisection thereby mechanically becomes achieved because as slide linkage 64 becomes translated inside of the slot afforded within control slotted linkage 61, as depicted in
[1199] Whereas the magnitude of virtually any angle which could be swept out about axis O in this prescribed manner, as extending from the +x.sub.T-axis to the longitudinal centerline of given obtuse angle solid linkage 62 furthermore would have to amount to the sum of the magnitudes of the angles simultaneously extending from such +x.sub.T-axis to the longitudinal centerline of trisector solid linkage 58, and then from the longitudinal centerline of trisector solid linkage 58 to that of given obtuse angle solid linkage 62, it would have to be of a size of (902)+(1804)=2706; meaning that for any reading which appears at axis Z, a corresponding reading which appears at axis O would have to amount to one-third of its size, thereby representing its trisector.
[1200] It thereby can be concluded that the slotted linkage arrangement of the car jack configuration of such newly proposed invention, as duly depicted in plan view in
[1201] Substantiating such capability would consist merely of demonstrating that virtually any static image which could be regenerated by means of properly setting such slotted linkage arrangement would automatically portray an overall shape that furthermore fully could be described by a geometric construction pattern in which the magnitude of its rendered angle amounts to exactly three times the size of its given angle.
[1202] In much the same manner as described previously, substantiating such capability could be very much simplified for the particular case posed in
[1203] Notice that the first column of
[1204]
[1205] The exploded view of such slotted linkage arrangement, as depicted in
[1214]
[1215]
[1216] Lastly, in order to enable such slotted linkage arrangement to furthermore function as a level, it very easily could be fitted with additional provisions which, depending upon the whims of any particular consumer, would become available as optional accoutrements consisting of slotted linkage 305, adapter 306, castellated nut 307 and added dowel 308, as posed in
[1217] That explains why the overall envelopes of such components appear as phantom lines therein. The upper surface of slotted linkage 305, as shown to reside farthest away from such protractor strip 57 in
[1218] Proposed leveling provisions of this nature might prove suitable for associated applications, as well, possibly becoming of value in novel transit designs, or in airplane attitude measurements.
[1219] In
[1222] During flexure, slotted linkage 305 thereby would remain parallel to the x-axis at all times, as depicted in the plan view
[1223] As represented in
[1228] Such deficiencies have been rectified by a more sophisticated slider arrangement, as described below.
[1229] The slider arrangement is a design variation of the fourth embodiment of such newly proposed invention that elaborates upon trisection capabilities of such slotted linkage arrangement by means of miniaturizing it, designing it to be easily transportable, featuring operating instructions, and adding scales to its x- and y-axes which are to appear as ruled divisions in order to precisely measure lengths associated with trisected angles whose trigonometric properties are of cubic irrational values that otherwise only could be approximated when performing geometric construction upon a given length of unity.
[1230] Such capability applies to exact lengths whose ratios with respect to a unit length are of cubic irrational value, or even transcendental value; thereby establishing an entirely new gateway which overcomes the dilemma that normally is experienced when unsuccessfully attempting to trisect an angle solely via straightedge and compass whose trigonometric properties are either of rational or quadratic value, but whose trisector instead exhibits cubic irrational trigonometric properties!
[1231] Such slider arrangement, as represented in
[1276] Such slider arrangement, as represented in
[1277] Only solid linkages are permitted throughout such slider arrangement. Without featuring slots, solid linkages not only are stronger, but their fabrication becomes easier because it excludes detailed stamping operations which otherwise would be required; therefore, being more cost effective in the long run.
[1278] Such improved device also features sliders which surround and support respective linkages; thereby permitting their unobstructed movement within them, while still effectively constraining them as necessary.
[1279] The opaque linkages depictions, as posed in
[1280] For example, upon examining
[1283] In such
[1286] The operating instructions which are posted upon such protractor/instructions piece of paper 76 duly reflect these differences. Naturally another set of operating instructions could have been incorporated onto protractor strip 57, as posed in
[1287] What remains fundamental, however, is that acute angle trisection and obtuse angle trisection operating instructions are posted separately upon such protractor/instructions piece of paper 76, as posed in
[1288] Whereas such slotted linkage arrangement, as posed in
[1289] Whereas a miniaturized slider arrangement exhibits much finer features than those afforded by its slotted linkage arrangement counterpart, in order to suitably depict the proper proportions between rivet head thicknesses and their respective shank lengths, the scale of its front and side views would have to be so enormous, as not to fit upon a single drawing page. Were such views thereby to be represented upon multiple pages instead, the very purpose of showing entire linkage spans as placements upon a single plan view naturally would become defeated.
[1290] Hence, rather than provide massive plan views of such slider arrangement, being much larger than those now represented in
[1291] Such drawings differentiate shims from washers, whereby the latter exhibit heights which are either equal to or multiples of a standard overall linkage thickness. For example, washer 91, as posed in
[1292] Whereas levels are not specified in the third derivative Euclidean formulation, as posed in
[1301] With particular regard to
[1302] A now fully sectioned
[1303] As such,
[1304] Therein, notice that slider 97 already has been bent to fit snugly about transverse linkage 82, as the later is depicted only in
[1305] The purpose of such rivet gun is to bear upon such rivet center pin 113 in order to pull it upwards with respect to such rivet head. Such pull up operation is described as follows: [1306] rivet center pin 113 becomes pulled up, being careful to leave sufficient play within it to permit the captive portion of slider 97 to freely rotate about it in uninhibited fashion, while nevertheless applying enough force upon sandwiched slider 97, given obtuse angle linkage 81 and washer 96 in order to clamp them firmly together. If slider 97 were to become over-tightened during such pull up operations by mistake, thereby no longer being free to rotate about rivet center pin 113 in an uninhibited manner, most times such problem can be remedied simply by exerting a torque which can overcome the contact resistance afforded between the tiny amount of slider 97 surface area which bears upon the head of rivet 98. During such process, rivet 98 would become loosened a miniscule amount, whereas a small amount of surface area also could be shaved away from slider 97; thereby promoting relative rotation in a completely unobstructed manner. Such pull up force serves to bend the lower portion of the rim of rivet 98 outwards, thereby occupying part of the chamfer, or recess previously machined into the lower portion of washer 96, as thereafter is depicted in a reshaped form in
[1313] Notice that such process enables a portion of slider 97 to reside in between the head of rivet 98 and the lower surface of transverse linkage 82; thereby affording a smooth area for transverse linkage 82 to glide over without being eroded by the slightly projecting head of rivet 98.
[1314] Such gluing operation is typical for all sliders, whereby: [1315]
[1320] As shown in
[1321] Such above cited illustrations depict rivets to be made out of a solid material such as aluminum. However, they just as easily could have been transparent plastic extrusions. By displaying them as solid objects, an ideal contrast is afforded with respect to any transparent linkages and sliders represented.
[1322] A suitable material for such slider arrangement linkages is a clear polycarbonate because it is durable, as well as inexpensive in small amounts.
[1323] Shims and washers also appear as solid objects.
[1324] Any obstruction to viewing, as normally posed by solid rivets is greatly mitigated because: [1325] they exhibit a very small surface area; and [1326] their pin holes provide a point of reference by exposing relevant locations upon the protractor circle imprinted on such protractor/instructions piece of paper 76.
[1327] Pin 108 also is made of a solid material so that it is not easily lost upon removal from the toploader during its disengagement from clutch 109.
[1328]
[1329] By geometrically constructing an altitude from vertex Y to base OO of isosceles triangle YOO, as illustrated in
[1330] Accordingly, the magnitude of each angle of such congruent right triangle residing at vertex Y must be complementary to such 2 value, thereby amounting to an algebraically expressed value of 902.
[1331] From these two angles, two more distinct vertical angles thereby become distinguished about point Y, also being of magnitude 902 as further indicated in
[1332] Next, point Y becomes selected along such extended geometrically constructed altitude such that it is positioned a suitable distance away from point Y; realizing that both point Y and point Y now must reside upon the perpendicular bisector of straight line OO.
[1333] Straight line YY.sub.a thereafter is geometrically constructed perpendicular to straight line OT, and straight line YY.sub.b then becomes drawn perpendicular to straight line OU.
[1334] Since right triangle YYY.sub.a and right triangle YYY.sub.b each exhibit respective angles of magnitude ninety degrees, and of size 902, they must be similar to one another.
[1335] Moreover, since such right triangles each contain side of YY in common, they also must be congruent to one another by the geometric proof of having corresponding angle-side-angle (ASA) components of equal magnitude; whereby side YY.sub.a of one congruent triangle must be equal in length to corresponding side YY.sub.b of the other. As illustrated in
[1336] Thereafter, such radii become algebraically designated to be of length s in
[1337] Recognizing that such three straight lines which pass through point Y, as posed in
[1338]
[1339] Such control becomes regulated as follows: [1340] the longitudinal centerline of slider 101 is shown to align with such perpendicular bisector, as drawn in
[1343]
[1344]
[1345]
[1346] Rivet 89 next is installed into the center portion of intermediate assembly 117, through its axis O, for purposes of pulling it up in order to firmly secure preliminary assembly 115 and linkage assembly 116 within toploader 114.
[1347] Adjustment linkage 83 becomes maneuvered so that axis T of intermediate assembly 117 thereafter aligns upon the ninety degree mark inscribed upon protractor/instructions piece of paper 76.
[1348] Removable pin 108 then is installed through toploader 114 about axis T, then through the vacant hole afforded by rivet 100, thereby specifically arranging such device so that it can trisect angles of acute designated magnitude, then through the underside of toploader 114, whereby it finally can be secured by clutch 109, as indicated in
[1349] The above described procedure enables the slider arrangement to be assembled within the confined space afforded by toploader 114, even after acknowledging that its overall height necks down about its sides. Hence, such process abets development of totally transportable, miniaturized trisection device.
[1350] Next, an explanation is to be furnished pursuant to such
[1355] In such above described scenario, trisection thereby mechanically becomes achieved because as adjustment linkage 83 becomes manipulated from outside of toploader 114, as depicted in
[1359] Whereas the magnitude of virtually any angle which could be swept out about axis O in this prescribed manner, as extending from the +x-axis to the longitudinal centerline of given acute angle linkage 79 furthermore would have to amount to the sum of the magnitudes of the angles simultaneously extending from such +x-axis to the longitudinal centerline of trisector linkage 77, and then from the longitudinal centerline of trisector linkage 77 to that of given acute angle linkage 79, it would have to be of a size algebraically calculated to amount to as +2=3; meaning that for any reading which appears at axis U, a corresponding reading which appears at axis O would have to amount to one-third of its size, thereby representing its trisector.
[1360] It thereby can be concluded that slider arrangement of the car jack configuration of such newly proposed invention, as duly depicted in plan view in
[1361] Substantiating such capability would consist merely of demonstrating that virtually any static image which could be regenerated by means of properly setting such slider arrangement would automatically portray an overall shape that furthermore fully could be described by a geometric construction pattern in which the magnitude of its rendered angle amounts to exactly three times the size of its given angle.
[1362] In much the same manner as described previously, substantiating such capability could be very much simplified for the particular case posed in
[1363] Next, an explanation is to be furnished pursuant to such
[1372] In such above described scenario, trisection thereby mechanically becomes achieved because as adjustment linkage 83 becomes manipulated from outside of toploader 114, as depicted in
[1376] Whereas the magnitude of virtually any angle which could be swept out about axis O in this prescribed manner, as extending from the +x.sub.T-axis to the longitudinal centerline of given obtuse angle linkage 81 furthermore would have to amount to the sum of the magnitudes of the angles simultaneously extending from such +x.sub.T-axis to the longitudinal centerline of trisector linkage 77, and then from the longitudinal centerline of trisector linkage 77 to that of given obtuse angle linkage 81, it would have to be of a size of (902)+(1804)=2706; meaning that for any reading which appears at axis Z, a corresponding reading which appears at axis O would have to amount to just one-third of its size, thereby representing its trisector.
[1377] It thereby can be concluded that the slider arrangement of the car jack configuration of such newly proposed invention, as duly depicted in plan view in
[1378] Substantiating such capability would consist merely of demonstrating that virtually any static image which could be regenerated by means of properly setting such slotted linkage arrangement would automatically portray an overall shape that furthermore fully could be described by a geometric construction pattern in which the magnitude of its rendered angle amounts to exactly three times the size of its given angle.
[1379] In much the same manner as described previously, substantiating such capability could be very much simplified for the particular case posed in
[1380] Wherein
[1384] Now that new definitions have been provided, a resulting comprehensive methodology, as presented in
[1385]
[1386] The decision box entitled DEVICE NEEDS TO BE SPECIFICALLY ARRANGED 121 is where it is to be determined which particular embodiment is to be utilized to perform such anticipated trisection; whereby: [1387] if either such first, second, or fourth embodiment were to be chosen, then the YES route would apply, thereby leading to a process box entitled DEVICE IS SPECIFICALLY ARRANGED 122 which is where such device is to be specifically arranged in accordance with applicable provisions, as specified in such
[1389] At this stage in the flowchart, such chosen device now should be properly set to a magnitude which matches the designated magnitude which first was specified.
[1390] The next process box entitled, STATIC IMAGE BECOMES REGENERATED 124 refers to the fact that by having properly set such device, a specific static image became regenerated, a particular portion of which assumed the overall outline of an actual trisector for such device setting; thereby automatically portraying a motion related solution for the problem of the trisection of an angle.
[1391] Activities which appear inside of the large square shaped dotted line are those which are to be performed exclusively by any trisecting emulation mechanism which might be placed into use, thereby being considered as properties that are intrinsic to it.
[1392] Outside of such trisecting emulation mechanism dotted box, the process box entitled, TRISECTOR AUTOMATICALLY PORTRAYED 125 is where such motion related solution for the trisection of an angle thereafter can be witnessed.
[1393] Although all embodiment designs of such newly proposed invention are quite similar in the respect that they share common fan portion linkage designs, as specified in such
[1398] In connection with such input box entitled MATHEMATICS DEMARCATION 8, as posed in
[1399] With regard to the very limited scope of trisection covered in this presentation, it should suffice to say that discussions below are to begin by significantly pointing out that the pretext of a Euclidean formulation just so happens to be conducive to physically describing various equations which have an infinite number of solutions!
[1400] Perhaps the most relevant of these, as specified below, assume the form of three very famous cubic expressions which address trisection by means of relating trigonometric properties of one angle of variable size to another whose magnitude always amounts to exactly three times its size:
[1401] Whenever the magnitude of an angle that is algebraically denoted to be of size 3 becomes supplied as a given quantity in any of such three cubic expressions, then such algebraic relationship truly would typify trisection!
[1402] This is because, a corresponding magnitude of , being an exact trisector of such given 3 value, then could be computed simply by means of dividing such given value by a factor of three; thereby enabling a determination of the constituent trigonometric properties, as specified above.
[1403] For example, for the particular condition when it is given that:
and
[1404] As a check, 3=75
[1405] Conversely, if an infinite number of magnitudes of were to become supplied as given values instead, each of such three algebraic relationships thereby could be suitably represented by means of developing a newly established Euclidean formulation that fully could distinguish it!
[1406] This is because all three of such above cited cubic expressions are continuous and their respective right-hand terms furthermore are geometrically constructible.
[1407] To aptly demonstrate this, a Euclidean formulation, as posed in
[1408] The governing sequence of Euclidean operations for such new Euclidean formulation is specified as follows: [1409] given angle VOO is geometrically constructed of an arbitrarily selected magnitude that algebraically is denoted as such that its side OO exhibits the same length as its side OV; [1410] side OV is designated to be the x-axis; [1411] a y-axis is drawn, hereinafter represented as a straight line which passes through vertex O of given angle VOO and lies perpendicular to such x-axis; [1412] a UNIT CIRCLE ARC becomes geometrically constructed, hereinafter to be represented as a portion of the circumference of a circle drawn about center point O whose radius is set equal in length to OV, thereby enabling it to pass through points V and O, both of which previously have been designated as respective termination points of angle VOO; [1413] point T thereafter becomes designated as the intersection between such UNIT CIRCLE ARC and such geometrically constructed y-axis; [1414] a straight line which passes through point O is drawn at forty-five degree angle counterclockwise to such x-axis; [1415] another straight line which passes through point O is drawn making a three-to-one slope with the +x-axis; [1416] a horizontal straight line is drawn which passes through point O and thereby lies parallel to the x-axis; [1417] the juncture between such horizontal straight line and the y-axis becomes designated as sin , thereby denoting its vertical distance above such x-axis; [1418] a vertical straight line is drawn so that it remains parallel to the y-axis while passing through the intersection made between such forty-five degree straight line and such horizontal straight line; [1419] the horizontal distance such vertical straight line resides to the right of such y-axis also thereby is to be designated as sin along such x-axis; [1420] a second vertical straight line is drawn which passes through coordinate point V, thereby being tangent to such previously drawn UNIT CIRCLE ARC; [1421] a slanted straight line is drawn which originates at point O and passes through the intersection point made between such second vertical straight line and such horizontal straight line; [1422] the angle which such slanted straight line makes with the x-axis becomes designated as , not to be confused with angle VOO amounting to a slightly larger magnitude of ; [1423] a second horizontal straight line is draw which passes through the intersection point made between such slanted straight line and such vertical straight line; [1424] the juncture of such second horizontal straight line with the y-axis becomes designated as h.sub.1, thereby denoting its unknown vertical distance above point O; [1425] a second slanted straight line is drawn which extends from point O to the intersection point made by such second horizontal straight line with such second vertical straight line; [1426] the angle which such second slanted straight line makes with the x-axis thereafter becomes designated as ; [1427] a third horizontal straight line is drawn so that it passes through the intersection point made between such second slanted straight line and such vertical straight line; [1428] the juncture of such third horizontal straight line with the y-axis becomes designated as h.sub.2, thereby denoting its unknown vertical distance above point O; [1429] a fourth horizontal straight line is drawn so that it passes through the intersection point made between such straight line which exhibits a 3:1 slope with respect to the x-axis and such vertical straight line; [1430] the juncture which such fourth horizontal straight line makes with the y-axis becomes denoted as 3 sin , thereby distinguishing its vertical distance above point O; [1431] a fifth horizontal straight line is drawn at a distance directly below such fourth horizontal straight line which measures four times the height which such third horizontal straight line resides above such x-axis, algebraically denoted therein as 4h.sub.2; [1432] the juncture which is made between such fifth horizontal straight line and the y-axis becomes designated as sin (3), thereby denoting its vertical distance above point O; and [1433] the intersection point of such fifth horizontal straight line with such UNIT CIRCLE ARC becomes designated as point U.
[1434] The proof for such
and [1435] since point U lies upon such UNIT CIRCLE ARC and exhibits a sin (3) ordinate value, radius OU must reside at an angle of 3 with respect to the x-axis.
[1436] Accordingly,
[1437] Based upon a reasoning that such famous cubic relationship sin (3)=3 sin 4 sin.sup.3 actually can be fully distinguished by an entire family of geometric construction patterns which together comprise such newly proposed Euclidean formulation, as posed in
[1438] In conclusion, any algebraic determination that can be made by means of relating like trigonometric properties that exist between one value and another that amounts to exactly three times its magnitude, as specified in such three cited famous cubic expressions, furthermore can be fully described by a geometric construction pattern which belongs to one of three Euclidean formulations which could be developed to characterize them.
[1439] For example, if a particular value of 1.119769515 radians were to be accorded to , then an algebraic determination could be made, as follows of 3, which furthermore fully could be described by a singular geometric pattern which belongs to such newly proposed Euclidean formulation, as posed in
[1440] Such above furnished overall detailed accounting explains exactly why all three of such previously cited famous cubic expressions remain incredibly important!
[1441] More particularly, this is because each of such three expressions can be considered to be a distinctive format type, in itself, one that furthermore can be broken down into an infinite number of unique relationships that have three cubic roots each.
[1442] Such scenario is far different than what transpires with respect to discontinuous functions, as are about to be discussed in detail next.
[1443] Also in connection with such input box entitled MATHEMATICS DEMARCATION 8, as posed in
[1444]
[1445] Its top legend identifies the path charted by a curve for such first famous cubic function, algebraically expressed as y=4 cos.sup.3 3 cos =cos (3) wherein: [1446] abscissa values in x signify cos magnitudes; and [1447] ordinate values in y signify cos (3) magnitudes.
[1448] Such well known curve is shown to be continuous within the specific range of 1x+1, thereby accounting for all real number values of cos .
[1449] The second legend therein identifies the particular function y=(4 cos.sup.36)/(20 cos ) wherein abscissa values in x again signify cos magnitudes. Such curve also is shown to be continuous in the same range, except for the fact that it is discontinuous at x=0. Notice that as the value of x, or cos , nears zero from a negative perspective, the corresponding value of y approaches positive infinity, and as it nears zero from the positive side, the corresponding value of y approaches negative infinity; thereby maintaining a one-to-one relationship between x and y values all along its overall path.
[1450] Where the curves identified by such first and second legends intersect, they can be equated due to the fact that they exhibit both x values of equal magnitude, as well as y values of equal size. Algebraically this can be expressed by the equation y=(4 cos.sup.36)/(20 cos )=cos (3), as typified by the third legend, as displayed in
[1451] Hence, such intersection points, shown to be positioned at the centers of such four large circles drawn therein, locate positions where (4 cos.sup.36)/(20 cos )=cos (3).
[1452] By then substituting 4 cos.sup.3 3 cos for cos (3), as shown below, the following fourth order equation can be obtained, along with a determination of the four associated roots for cos and other relevant quantitative details:
and [1453] via cross multiplication,
[1454] Values of the roots of such quartic equation are provided in
[1455] In conclusion, the cos (3)=(4 cos.sup.36)/(20 cos ) quartic function clearly qualifies as being discontinuous because it consists of only four distinct points, as are identified by circles appearing in such of
[1456] With particular regard to the two continuous curve representations drawn in
[1457] Naturally any geometric construction pattern which possibly could be drawn which belongs to such Euclidean formulation would identify just a single point which lies upon the two curve potions represented by the second legend in
[1458] Above, the length ().sup.3 would be geometrically constructed in much the same fashion as was the sin.sup.3 in
[1459] From the above calculations, it should become rather clear that an entire family of geometric construction patterns could be drawn for the function y=(4 cos.sup.3 6)/(20 cos ). The corresponding sequence of Euclidean operations needed to conduct such activity could be obtained merely by administering the formula represented on the right hand side of the equation given above, thereby represented as (4 cos.sup.3 6)/(20 cos ); whereby only the value of cos would be altered in during such development.
[1460] Each respective length of the ordinate value y then could be drawn by way of the proportion y/1=(4 cos.sup.3 6)/(20 cos ), thereby producing such length y by means of applying only a straightedge and compass.
[1461] As such, the function y=(4 cos.sup.3 6)/(20 cos ) could be fully described by yet another entirely separate Euclidean formulation. Even though each of such generated geometric construction patterns belonging to such Euclidean formulation most certainly would not relate trigonometric values of angles to those of angles which amount to exactly one-third their respective size, it nevertheless would be possible to design an entirely new invention whose distinctive flexure, maybe even being a harmonic motion, could be replicated by means of animating the entire family of geometric construction patterns which belong to such newly devised Euclidean formulation in successive order.
[1462] Obviously, such types of involvements inevitably should serve as building blocks for mathematics!
[1463] More specifically stated, a novel assortment of sundry mechanical devices that exhibit capabilities well beyond those of trisecting emulation mechanisms whose fundamental architectures during flexure regenerate static images that automatically portray overall geometries that furthermore can be fully described by Euclidean formulations additionally can be quantified algebraically!
[1464] In this vein, prior claims made in connection with such
[1465] Such explanation begins with what clearly is known concerning any linear function of the form y=mx+b.
[1466] Its geometric construction counterpart consists merely of locating a second point which lies a magnitude that algebraically is denoted as b either directly above or below a first point, depending upon the sign placed in front of such coefficient. For example, in the equation y=6x3, such second point would be situated exactly three units of measurement below such first point. In order to complete such singular geometric construction pattern, a straight line next would need to be drawn which passes through such second described point and furthermore exhibits a slope, m, whose rise and run values could be depicted as the sides of a right triangle, the ratios of whose mutual lengths amount to such magnitudes.
[1467] Second order functions of a singular variable cannot be fully described by a geometric construction process, thereby necessitating instead that they be fully charted by means of plotting a y value that appears upon a Cartesian coordinate system that becomes algebraically determined for each x value belonging to such function.
[1468] However, conventional Euclidean practice can be of assistance in determining the roots of quadratic functions. For example, consider an entire set of parabolic functions whose overall format type thereby could be expressed as ax.sup.2+bx+c=y.
[1469] For any specific values which its coefficients might be respectively assigned, a singular algebraic function belonging to such format type would become specified. Its roots would indicate where such singular curve crosses the x-axis; but only could when the variable y within such function amounts to zero; hence becoming representative of a quadratic equation which instead would belong to another simplified format type, algebraically expressed as ax.sup.2+bx+c=0 which would typify a subset of such parabolic function format type.
[1470] By means of referring back to the previous discussion regarding such input box entitled MATHEMATICS DEMARCATION 8, as posed in
[1471] Herein,
[1472] The very sequence of Euclidean operations from which such singular geometric construction pattern is derived is provided directly below: [1473] a square each whose sides is of unit length is drawn; [1474] a right triangle is inscribed within it such that: [1475] its first side begins at one of the corners of such square, extends a length of 0.75, or of a unit from it, and becomes drawn so that it aligns upon a side of such square, thereafter becoming algebraically denoted as being of length c therein; [1476] its second side, drawn at a right angle away from the endpoint of such first side, is to be of unit length also such that its endpoint resides somewhere along the opposite side of such previously drawn square; and [1477] its hypotenuse then is to become drawn; [1478] a straight line of length of 0.8 units which extends from a point which resides somewhere upon the first side of such previously drawn right triangle that is parallel to its second side, and terminates somewhere along its hypotenuse is to be drawn as follows: [1479] a straight line reference becomes drawn that lies parallel the first side of such previously drawn right triangle and resides 0.8 units in length above it; [1480] from the intersection point of such straight line reference and the hypotenuse of such previously drawn right triangle, another straight line is drawn that is perpendicular to such straight line reference; [1481] such 0.8 units in length which spans the distance between the first side of such previously drawn right triangle and such straight line reference is to be algebraically denoted as 4a therein; and [1482] the span of the first side of such previously drawn right triangle which extends from its beginning point to where it intersects such straight line which was drawn to be of 0.8 units in length thereby can be algebraically denoted to be of a length 4ac due to the fact that it represents a corresponding side belonging to another right triangle which is similar such previously drawn right triangle, thereby meeting the proportion c/1=4ac/4a; [1483] a semicircle is drawn whose diameter aligns upon the side of such square that the first side of such previously drawn right triangle also aligns with whose circumferential portion lies outside of such square; [1484] such 0.8 unit straight line next is to be extended below the side of such square until it meets such previously drawn circumferential portion, from which two more straight lines are to be drawn, each terminating at a lower corner of such square, thereby describing a second right triangle whose hypotenuse then can be denoted as {square root over (4ac)}, since is squared value is equal to the area of the rectangle inscribed in such square whose sides are of unit and 4ac respective lengths by virtue of the Pythagorean Theorem; [1485] the remaining side of such newly drawn right triangle, as appearing within such previously drawn semicircle, becomes extended a distance that amounts to 0.4 units in length such that the circumference of a whole circle can be drawn about its new endpoint, being of a radius that thereby can be algebraically denoted to be of length b therein; [1486] a straight line then is drawn which extends from the beginning of the first side of such previously drawn right triangle that terminates at the center point of such whole circle, thereby being algebraically denoted to be of length {square root over (b.sup.24ac)} as determined by Pythagorean Theorem, once realizing that it represents the hypotenuse of yet another right triangle whose respective sides are of lengths b and {square root over (4ac)}; [1487] such newly drawn straight line then becomes extended until it reaches the far circumference of such circle, thereby to become algebraically denoted to be of overall length b+{square root over (b.sup.24ac)}; [1488] its span extending from the beginning of the first side of such previously drawn right triangle to the near circumference of such circle thereby becomes algebraically denoted to be of lengthb+{square root over (b.sup.24ac)}; [1489] another straight line then is drawn which passes through the corner of such previously drawn square upon which the vertex of such previously drawn right triangle was geometrically constructed, and its first side began, which furthermore lies perpendicular to the diameter of such newly drawn circle which is shown, being a total length of unity such that 0.4 units of such overall length resides to right side of such diameter, thereby becoming algebraically denoted to be of length 2a; [1490] with respect to such last drawn straight line: [1491] a straight line is drawn perpendicular to its left termination point; and [1492] two more straight lines are drawn emanating from its rightmost termination point, each of which passes through respective locations where the diameter drawn for such circle intersects its circumference; [1493] the longer cutoff made upon such lastly drawn perpendicular straight line thereby is algebraically denoted to be of length x.sub.1, signifying an overall length whose magnitude is equal to the value of the first root of such given quadratic function 0.2x.sup.2+0.4x+0.75=y, as determined by the respective sides of two right triangles that establish the proportion x.sub.1/1=(b+{square root over (b.sup.24ac)})/2a, therefore amounting to x.sub.1=(b{square root over (b.sup.24ac)})/2a; and [1494] the shorter cutoff made upon such lastly drawn perpendicular straight line thereby is algebraically denoted to be of length x.sub.2, signifying an overall length whose magnitude is equal to the negative value of the second root of such given quadratic, as determined by the respective sides of two right triangles that establish the resulting proportion x.sub.2/1=(b+{square root over (b.sup.24ac)})/2a, thus amounting to x.sub.2=(b+{square root over (b.sup.24ac)})/2a.
[1495] Likewise, a cubic functions of a single variable also cannot be fully described by a single geometric construction pattern, but instead requires an entire Euclidean formulation to describe what otherwise would need to become fully plotted by means of algebraically determining a value of y for each x value belonging to such function; as is the case for the either of the continuous cubic curves which are charted in
[1496] Notice that when interpreting such continuous cubic function y=(4 cos.sup.3 6)/(20 cos ): [1497] when reading from right to left, it indicates an entire family of unique geometric construction patterns, each of which can be generated by means of applying the very same sequence of Euclidean operations, whereby only the magnitude of its given value, cos , becomes slightly altered; but [1498] when otherwise going from left to right, it becomes indicative of a certain motion which could be imparted by some mechanical device whose fundamental architecture during flexure can be replicated by means of animating a Euclidean formulation which could fully describe its constituent overall shapes. That is to say, a geometric forming process which should be incorporated into the fold of mathematics can characterize trisection for virtually any of the equations contained within the three very famous cubic curves expressed above!
[1499] As such, a sequel, or follow-on development, being one that presently is considered to be well beyond the very limited scope postulated herein, might entail placing parameters of time within continuous algebraic cubic functions, thereby opening up an entirely new gateway for mathematical investigation; principally because motion cannot transpire without it.
[1500] It is in this area of discussion that perhaps the greatest confusion abounds concerning trisection!
[1501] In order to suitably avoid its pitfalls, it becomes necessary to pose one last riddle which finally should fully expose any disturbing myths that yet might be perpetuated by such great trisection mystery.
[1502] The last riddle is: Can the classical problem of the trisection of an angle actually be solved after gaining an understanding of the role which algebraic expressions play in the determination of the magnitude of a trisector for an angle of virtually any designated magnitude?
[1503] Again, such answer, most emphatically, turns out to be a resounding no!
[1504] Such above proposed determination can be substantiated by examining the proceedings associated with a cubic equation containing a single variable which becomes resolved by means of simultaneously reducing it with respect to another cubic equation of a single variable which harbors a common root, whereby such algebraic process enables vital information to be converted into second order form.
[1505] Naturally, such algebraic approach cannot solve the classical problem of the trisection of an angle!
[1506] However, it can serve to justify that there is a certain order within mathematics that most certainly should be exposed for the benefit of mankind!
[1507] As a relevant example of this, one of the three famous cubic functions cited above is to undergo such simultaneous reduction process, wherein is to denote the particular value of the tangent of a designated magnitude of an angle, 3, that is about to be trisected; thereby becoming algebraically expressed as tan (3). Since such famous cubic equations can track trigonometric relationships which exist between various given angles and those amounting to exactly three times their respective sizes, such previously mentioned common root, denoted as z.sub.R, is to represent corresponding values of tan , thereby enabling the following algebraic cubic equation expressions to be reformatted as follows:
whereas, tan(3)=(3 tan tan.sup.3)/(13 tan.sup.2);
then, =(3z.sub.Rz.sub.R.sup.3)/(13z.sub.R.sup.2)
(13z.sub.R.sup.2)=3z.sub.Rz.sub.R.sup.3
z.sub.R.sup.3=3z.sub.R(13z.sub.R.sup.2).
[1508] In order to perform such simultaneous reduction, a generalized cubic equation format type of the form z.sup.3+z.sup.2+z+=0 now is to become introduced, as well.
[1509] In order to determine what common root values any of such equations which belong to such generalized cubic equation format type share in common, in such above equation:
z.sup.3+z.sup.2+z+=0;
z.sub.R.sup.3+z.sub.R.sup.2+z.sub.R+=0; and
z.sub.R=(z.sub.R+z.sub.R+).
[1510] Such format type is to be referred to as the generalized cubic equation because its accounts for virtually every possible equation that a cubic equation of a single variable could possibly assume!
[1511] Since such famous tangent cubic function can be arranged as z.sub.R.sup.33z.sub.R.sup.23z.sub.R+=0, it must be a subset of such generalized cubic equation for the specific case when coefficient =3; =3, and =.
[1512] As, I'm sure the reader by now must have guessed, the significance of such association is that both equation formats thereby must bear a common root!
[1513] Moreover, the term format, as addressed above, applies to a whole family of equations that exhibit identical algebraic structures, but differ only in respect to the particular values of the algebraic coefficients they exhibit!
[1514] Such mathematical phenomenon occurs because the uncommon roots of each particular equation belonging to such generalized cubic equation format, when arranged in certain combinations with common roots, z.sub.R, which they share with respective equations that belong to such famous tangent cubic equation format, actually determine such other coefficient values, as will be more extensively explained below.
[1515] By equating z.sub.R.sup.3 terms, the following quadratic equation relationships can be obtained by means of removing mutual cubic parameters:
[1516] Such last alteration, amounting to the division of each contained coefficient by a factor of a, gives an indication of how to further manipulate algebraic equation results in order to realize their geometric solutions in a more efficient manner, leading to an abbreviated Quadratic Formula of the form z.sub.R=(b+{square root over (b.sup.24ac)})/2a=[b+{square root over (b.sup.24(1)(c))}]/2(1)=()(b{square root over (b.sup.24c)}).
[1517] Obviously, such abbreviated Quadratic Formula then applies only to quadratic equations of a singular variable whose squared term coefficients are equal to unity!
[1518] In order to simultaneously reduce two cubic equations in a single variable which share a common root, their remaining root values must be different.
[1519] To demonstrate how this works, a generalized cubic equation is to be determined whose uncommon roots, for the sake of simplicity exhibit values of z.sub.S=3 and z.sub.T=4.
[1520] For the example which is about to be presented below, a common root value of z.sub.R={square root over (5)} is to be assigned because it is of quadratic irrational magnitude, and thereby can be geometrically constructed directly from a given length of unity, thereby representing the length of the hypotenuse of a right triangle whose sides are of lengths 1 and 2, respectively.
[1521] As such, the magnitude of could be determined merely by means of computing the overall value associated with (3z.sub.Rz.sub.R.sup.3)/(13z.sub.R.sup.2)=(3{square root over (5)}5{square root over (5)})/(135)={square root over (5)}/7.
[1522] Notice that such calculation furthermore must be of quadratic irrational magnitude, thereby enabling such length to be represented as the very starting point within an upcoming geometric construction process.
[1523] Accordingly, such famous cubic relationship in a single variable z.sub.R.sup.33z.sub.R.sup.23z.sub.R+=0 would assume the particular form z.sub.R.sup.33({square root over (5)}/7) z.sub.R.sup.23z.sub.R+{square root over (5)}/7=0.
[1524] As for such generalized cubic equation, since it can be stated that:
zz.sub.R=0;
zz.sub.S=0; and
zz.sub.T=0.
[1525] By thereafter multiplying such three equations together, the following algebraic expression could become obtained:
(zz.sub.R)(zz.sub.S)(zz.sub.T)=0; or
z.sup.3(z.sub.R+z.sub.S+z.sub.T)z.sup.2+(z.sub.Rz.sub.S+z.sub.Rz.sub.T+z.sub.Sz.sub.T)zz.sub.Rz.sub.Sz.sub.T=0; and
z.sup.3+z.sup.2+z+=0.
[1526] By equating coefficients of like terms, the following three relationships can be determined:
[1527] Such generalized cubic equation format would be z.sup.3({square root over (5)}+7)z.sup.2+(7{square root over (5)}+12)z12{square root over (5)} 0.
[1528] Accordingly:
[1529] Naturally, the last of such three famous continuous cubic equations, as stipulated above, alternatively could have been resolved algebraically without having to resort to such cumbersome simultaneous reduction process.
[1530] This could be achieved simply by realizing that once a value of becomes designated, an angle of 3 magnitude that it is representative of very easily could be determined trigonometrically; whereby, a value for z.sub.R which corresponds to its trisector, computed as being one-third of such value, and thereby algebraically expressed merely as , thereafter also could be trigonometrically determined.
[1531] Unfortunately, the pitfall that accompanies such shortened algebraic process is that such common root, z.sub.R, does not become identified solely by conventional Euclidean means!
[1532] The method to do so would be to draw straight lines whose lengths are of magnitudes which are equal to the value of roots belonging to such abbreviated Quadratic Formula z.sub.R=()(b{square root over (b.sup.24c)}), much in the same manner as was employed earlier when quadratic roots first were determined by means of geometric construction in
[1533] For such algebraic determination, as made above, the magnitude of a trisector for an angle whose tangent is of a designated magnitude {square root over (5)}/7 could be geometrically constructed by means of applying the following sequence of Euclidean operations; thereby rendering a particular pattern, as is depicted in
(105+4{square root over (5)})/100; and
85{square root over (5)}/100; [1535] such common side is extended to a unit length; [1536] a perpendicular straight line is drawn above the newly formed endpoint of such extension; [1537] the hypotenuses appearing in such two previously drawn right triangles are extended until they intersect such newly drawn perpendicular straight line, thereby depicting two more similar right triangles; [1538] whereby, the lengths of the unknown sides of such two newly drawn right triangles can be determined by virtue of the proportions established between the known lengths of corresponding sides of their respective similar right triangles and their common side of unit length, thereby enabling designations of b length and c to be notated upon such drawing to reflect the following determinations:
[1553] Obviously, such geometric construction approach cannot pose a solution for the classical problem of the trisection of an angle; simply because the generalized cubic equation format that contributes to its very determination, specifically being z.sup.3({square root over (5)}+7)z.sup.2+(7{square root over (5)}+12)z12{square root over (5)}=0, could not be derived without a prior awareness of the very solution itself.
[1554] A second less complicated example demonstrating that it is possible to apply algebraic information in order to create a geometric solution for the problem of the trisection of an angle pertains to a generalized cubic equation whose coefficients and are set to zero, and whose coefficient amounts to a value of +1, thereby establishing the specific cubic equation z.sub.R.sup.3+1=0.
[1555] From such information, the following details can be gleaned:
[1556] Such algebraic determination, as made above, thereby enables the trisection of an angle to be geometrically constructed as follows: [1557] from a designated value of =tan (3)=+1, an angle designated as 3 which amounts to exactly 45 in magnitude first becomes geometrically constructed with respect to the +x-axis; and [1558] from an algebraically determined common root value of z.sub.R=1, a trisecting angle designated as which amounts to exactly 135 in magnitude thereafter becomes geometrically constructed with respect to the +x-axis.
[1559] Needless to say, such geometric construction, as posed above, although representing geometric solution for the problem of the trisection of an angle, nevertheless does not pose a solution for the classical problem of the trisection of an angle. This is because a value for such common root z.sub.R cannot be ascertained solely by means of a geometric construction which proceeds exclusively from a given value of =tan (3)=+1.
[1560] Although a straight line of slope z.sub.R=1 could be geometrically constructed rather easily from another line of given slope =+1, such geometric construction pattern represents just one out of an infinite number of straight line possibilities which otherwise could be distinguished geometrically from a given value of =+1.
[1561] Hence, the sequence of Euclidean operations which governs such trisection can be completed with certainty only by incorporating such algebraic determination that z.sub.R=+1, or else simply by algebraically dividing such geometrically constructed 405 angle by a factor of three.
[1562] In either case, since both of such algebraic results are tied only to such 135 trisector of slope z.sub.R=1, the only way to determine such information solely via straightedge and compass from a geometrically constructed 45 angle would be to distinguish them from the results of a Euclidean trisection which has not yet been performed.
[1563] Such process entails knowledge of the results of a geometric construction before it actually becomes conducted, thereby violating the rules of conventional Euclidean practice which require that geometric construction can proceed only from a given set of previously defined geometric data.
[1564] In order to further emphasize just how the use of aforehand knowledge inadvertently creeps into conventional Euclidean practice, thereby grossly violating its very rules, a last rather telling example is afforded below whereby given angle NMP, as depicted in
[1568] Such above analysis reveals that with respect to the particular geometry represented in such famous
[1569] Were this above assertion not to be true, it would be tantamount to trisecting such sixty degree angle QPS solely by means of applying a straightedge and compass to it; thereby solving the classical problem of the trisection of an angle without having any other predisposed knowledge and, in so doing, accomplishing a feat that is entirely impossible!
[1570] With regard to a prior discussion concerning the input box entitled PROBABILISTIC PROOF OF MATHEMATIC LIMITATION 10, it was mentioned that trisection can be achieved by means of performing a multitude of consecutive angular bisections, all geometrically constructed upon just a single piece of paper.
[1571] Such approach generates a geometric construction pattern that is indicative of a geometric progression whose: [1572] constant multiplier, m, is set equal to ; and [1573] first term, f, is algebraically denoted as 3.
[1574] Moreover, the overall sum, s, of such geometric progression consisting of an n number of terms can be represented by the common knowledge formula:
whereby [1575] for an infinite number of terms, such equation thereby reduces to,
[1576] Such result indicates that after conducting an infinite number of successive bisection operations, it becomes possible to geometrically construct an angle that amounts to exactly the size of an angle of designated 3 magnitude, whereby their difference then must distinguish its trisector.
[1577] Below, a method is furnished which describes how to geometrically construct the first five terms appearing in such governing geometric progression; and in so doing thereby assuming the form 33/2+3/43/8+3/16=33/16.
[1578] In such development, the value of the first term, algebraically denoted as 3, can be set equal to virtually any designated magnitude that is intended to be trisected. By inspection, it furthermore becomes apparent that the numerical value of each succeeding term is equal to one-half the magnitude of its predecessor. As such, values for such diminishing magnitudes can be geometrically constructed merely by means of bisecting each of such preceding angles.
[1579] Lastly, wherein positive values could applied in a counterclockwise direction, negative magnitudes would appear in a completely opposite, or clockwise direction, with respect to them.
[1580] The specific details which pertain to a
[1586] Quite obviously, it remains possible to continue such activity until such time that the naked eye no longer could detect a bisector for an arc that invariably becomes smaller and smaller with each subsequent bisection operation.
[1587] In this regard, the resolution of the naked eye is considered to be limited to about one minute of arc, thereby amounting to 1/60.sup.th of a degree, whose decimal equivalent is 0.01667.
[1588] Once the human eye no longer can detect gradations resulting from such bisectors process, they could be located erroneously or even superimposed upon prior work.
[1589] Since the use of a microscope might increase such perception capabilities, it might enable a few additional bisections to become accurately determined. However, being that an infinite number of bisections are needed in order to generate a precise trisector in this manner, such enhancement only would serve to slightly improve upon the overall approximation of any trisector which becomes produced.
[1590] The Successive Bisection Convergence Chart, as presented in
[1591] The second column therein is devoted to calculations which apply to such geometric progression, based upon the number of terms it contains. In each line item, the last value provided indicates the overall size of the angle which would become geometrically constructed by means of conducting such successive bisection process.
[1592] Notice that
[1593] Since the only time that a bisection operation is not conducted is when n=1, each successive line item within such
[1594] Hence, an accuracy of one-millionth could be obtained by means of conducting twenty-one successive bisections.
[1595] The analysis presented below discloses that for a 20 trisector, such above summarized process of successive angular bisections would have to be disbanded during the twelfth bisection operation due to the naked eye no longer being able to discern the exact placement of its bisector.
[1596] As such, the number of terms this condition would apply to, as indicated in such
[1597] From such
[1598] Therefore, since such 0.01464 needed separation clearly is smaller than the 0.01667 which the naked eye is capable of perceiving; it means that such twelfth bisector could be located erroneously.
[1599] When referring to
[1600] Additionally, four subsequent bisections are depicted, each of which is considered to have been performed solely by conventional Euclidean means.
[1601] The purpose of the shading therein is to suitably distinguish between each of such bisection activities as follows: [1602] such angle of magnitude +3 is bisected in order to distinguish two separate arcs, each being of 3/2 size; [1603] with the upper portion of such bisected angle, amounting to a size of 3/2, then itself becoming bisected, the determination made as to the location of such second bisector would place it at an angle of 3/4 counterclockwise of such first bisector position; [1604] with the angle formed between such first bisector and second bisector, amounting to a size of 3/4, then itself becoming bisected, the determination made as to the location of such third bisector would place it at an angle of 3/8 clockwise of such second bisector position, ad denoted by the minus sign notation; and [1605] with the angle formed between such second bisector and third bisector, amounting to a size of 3/8, then itself becoming bisected, the determination made as to the location of such fourth bisector would place it at an angle of 3/16 counterclockwise of such third bisector position.
[1606] As to the role which cube roots could play in a geometric solution of the problem of the trisection of an angle, below it is shown how to determine the length of a straight line, half which amounts to its cube root value, whereby it could be algebraically stated that:
3{square root over (l)}=l/2; such that by cubing both sides;
l=l.sup.3/8
8l=l.sup.3
4(2)=l.sup.2
2{square root over (2)}=l
{square root over (2)}=l/2; and [1607] relevant information then is to be introduced in the form of an angle whose complement furthermore turns out to be its trisector, algebraically determined as follows:
[1611] The algebraic cubic equation which correlates to this geometric construction process assumes the form of z.sub.R.sup.3+3z.sub.R.sup.2+3z.sub.R+(32)=0; as determined below:
tan(3)={square root over (2)}+1=
{square root over (2)}=1; and
tan =z.sub.R={square root over (2)}1
z.sub.R+1={square root over (2)}
(z.sub.R+1).sup.3=({square root over (2)}).sup.3
(z.sub.R+1).sup.3=2{square root over (2)}
(z.sub.R+1).sup.3=2(1)
(z.sub.R.sup.3+3z.sub.R.sup.2+3z.sub.R+1)2(1)=0
z.sub.R.sup.3+3z.sub.R.sup.2+3z.sub.R+(32)=0.
[1612] To finalize a discussion raised earlier,
[1613] To elaborate upon this, complex numbers typically are represented geometrically as straight lines which appear upon an xy plane known as the complex plane.
[1614] Each straight line featured therein commences from the origin of a rectilinear coordinate system, and contains an arrow at its termination point to express direction.
[1615] The convention used to specify a complex number is first to indicate its real numerical magnitude, followed by its imaginary component. Such imaginary aspect is represented by an Arabic letter, i, used to denote an imaginary term {square root over (1)}, followed by its magnitude.
[1616] As such, the coordinate values of complex number termination points designate their respective imaginary and real number magnitudes; thereby fully describing them.
[1617] In
[1618] Conversely, since the ratio between the magnitudes of the real and imaginary portions of such first complex number is (sin 3)/(cos 3)=tan 3, the straight line which represents it, by exhibiting such slope, thereby must pass through the origin while forming an angle of 3 with such x-axis.
[1619] Likewise, the straight line which represents such second complex number, by exhibiting a slope of tan , thereby must pass through the origin while instead forming an angle of with respect to the x-axis and, in so doing, trisecting such angle of 3 magnitude.
[1620] The fact that the complex number cos i sin also turns out to be the cube root of the first complex number cos (3)+i sin (3) furthermore is to be verified algebraically by applying the binomial expansion (A+B).sup.3=A.sup.3+3A.sup.2B+3AB.sup.2+B.sup.3 for the express condition when the A=cos , and B=i sin as follows:
[1621]
[1622] Therein, ramp 128 and ramp 129 are included for purposes of driving a four wheel motor vehicle onto near side skirt 130 and far side skirt 131 simultaneously. Notice that the stationary front ends prevent such motor vehicle from being driven too far and thereby falling off, while the webs of their channeled cross-sections prevent such motor vehicle from tipping off either side.
[1623] Near pin 132 and far pin 133 thereafter are inserted to secure such motor vehicle in place before lifting operations commence. As indicated, such mechanisms are no different than the designs which support toilet tissue in a bathroom.
[1624] As indicated in
[1628] The single hydraulic actuator 137 which powers such device by varying the distance between attachment strut 135 and attachment strut 136 is controlled by wall remote 139 as a safety provision, whereby there is no chance of being caught underneath such mechanism while it becomes activated; thereby preventing possible injury.
[1629] Such simple actuation approach precludes having to provide two separate power supply sources which otherwise would have to be regulated with respect to each other at all times, thereby necessitating additional equipment.
[1630] The twofold advantage of such design is that it remains level at any set height, while the load which it supports always is maintained so that its center of gravity aligns very close to such center strut location, thereby permitting it to remain balanced during lifting operations.
[1631] To afford an example of such advantage, consider a forklift which supports a particular load upon a pallet. Were the distance between its prongs to become reduced for any reason, such as to clear an obstacle that they might encounter during lifting operations, the center of gravity of such load might shift to another location where it might become subject to tipping.
[1632] However, with regard to the design of the device proposed in
[1633] Lastly, one final justification is about to be put forth, essentially claiming that only an availability of overlapment points can fully account for why the classical problem of the trisection of an angle cannot be solved!
[1634] Public sentiment on this topic, as highly influenced by the earlier discoveries of Wantzel and Galois dating all the back to the mid 1800's, instead generally leans to attributing an inability to geometrically construct cube roots as being the principal cause which prevents trisection.
[1635] Moreover, at the very heart of this matter lies a fundamental issue of constructability.
[1636] To openly dispute such issue, upon drawing an angle of arbitrarily selected magnitude, there is a good chance that its trigonometric properties will turn out to be cubic irrational. This is because a far greater number of angles exist which exhibit cubic irrational trigonometric properties than do other angles whose trigonometric properties are of rational and quadratic irrational value.
[1637] From such initial angle, an entire geometric construction pattern could be generated which belongs to the Euclidean formulation, as posed in
[1638] The basic problem with such scenario is that such drawing, although fully constructible by a process of sheer random selection, never could be repeated; thereby becoming relegated to approximation when attempting to reproduce it.
[1639] More particularly stated, although the likelihood of drawing an angle which exhibits cubic irrational trigonometric properties is quite high, as due to a substantial availability of them, the probability of geometrically constructing a specific angle, even one which might feature a particular transcendental trigonometric property such a pi for example, nevertheless approaches zero; being entirely consistent with the previously stipulated premise that absolutely no cubic irrational length can be geometrically constructed, but only approximated, from a given unit length.
[1640] To further emphasize this outstanding difficulty, consider the largely unknown fact that even the rarified transcendental number, , can be approximated by means of geometric construction well beyond what the naked eye could detect.
[1641] To demonstrate this, a rational number very easily can be described by the ratio of two cubic irrational numbers by an algebraic manipulation such as:
[1642] Similarly, the actual transcendental value of can be multiplied to the sin 80 in order to produce another transcendental length as follows:
sin 80=3.093864802 . . . ; and
(0.9848077530 . . . )=4(0.77346620052 . . . ).
[1643] Moreover, all of the stated values in such above equation, except for , furthermore very closely could be approximated as actual rational numbers, down to a significance of at least ten decimal places; being well beyond the accuracy of what the naked eye could detect.
[1644] Such estimated result is furnished directly below, whereby all constructible rational numbers thereby could be algebraically expressed as follows:
[1645] Notice that such above described rational lengths 4, T, and L now can be geometrically constructed from an arbitrarily applied, or given length of unity.
[1646] In the above example, there is little need to attempt to reduce the rational length T any further than is indicated. This is because it is necessary only to know that a rational length of T=19,336,655,013/25,000,000,000 could be made use of to geometrically construct another length that very closely approximates the actual value of pi.
[1647] From such equation L=4T, as determined above, the proportion
readily could be established; whereby a very close estimation of the length pi thereby could be identified from the geometric construction of two similar right triangles whose sides respectively consist of drawn rational lengths 4, T, and L. Understandably, the level of accuracy attributed would amount to only three, or perhaps four at the very most, significant digits.
[1648] To conclude, since transcendental lengths describe decimal sequences which are considered to continue on indefinitely, they cannot be exactly geometrically constructed from any long-hand division computation that is indicative of a pair of rational numbers whose quotients begin to repeat themselves.
[1649] In the past, such difficulty merely was bypassed by means of considering only geometric construction patterns which could be redrawn.
[1650] Such process simply entails selecting a given angle whose trigonometric properties are either rational or quadratic irrational. For example, upon considering a given angle VOO whose sine is equal to , the following algebraic relationship could be obtained:
[1651] Obviously the sin (3) also must be a rational value because it amounts to the sum of three times such selected rational value of plus four times the value of its cube; meaning that all coefficients within such resulting equation 23/27=3 sin 4 sin.sup.3 very handily would consist of only rational numbers!
[1652] Accordingly, an associated geometric solution for the problem of the trisection of an angle very easily could be drawn merely geometrically constructing an angle whose sine equals .
[1653] Notice, however, that such particular drawing would remain entirely irreversible, despite being characterized by that very geometric construction pattern, as just described, belonging to the Euclidean formulation, as posed in
[1654] Next, the issue of attempting to extract cube roots is to be addressed. In order to do this, consider that some Euclidean formulation someday might become devised, each of whose constituent geometric construction patterns would be fully reversible, as well as exhibit a rendered length that amounts to the cube of its given length. In so doing, it naturally would follow that for each of such singular drawings, a cube root of such rendered length value thereby could be geometrically constructed without having to introduce any additional relevant information.
[1655] Now, if a Euclidean formulation of such nature truly could be devised, an overriding question then would be whether such capability could in some way overcome the irreversible nature of any geometric construction pattern in which the magnitude of a rendered angle amounts to exactly three times the size of its given angle. For instance, could such magical Euclidean cube root capability enable angle VOU, as appearing upon the irreversible representative geometric construction pattern for such Euclidean formulation, as posed in
[1656] Naturally, an activity of this nature would be severely limited in that some far-fetched reversible Euclidean cube root capability only could be applied to any known aspect of such rendered angle VOU. Such is the case because when attempting to solve the classical problem of the trisection of an angle, other lengths in
[1657] Accordingly, it is conjectured that some as yet undeveloped Euclidean capability to extract cube roots would have little to no impact whatsoever upon enabling the classical problem of the trisection of an angle to become solved; as based upon the fact that such hypothetical cube root development couldn't possibly offset the irreversibility of such
[1658] In closing, it is important to note that vital input leading to the very discovery of significant findings, as presented herein, never even would have been obtained had it not been for one strange incident which occurred in 1962. It was then, that my high school geometry teacher informed me that it was impossible to perform trisection solely by conventional Euclidean means. Her disclosure moved me greatly. I become intrigued; thereby fueled with a relentless curiosity to ascertain secrets needed to unlock a trisection mystery that had managed to baffle mathematicians for millennia!
[1659] Naturally, during such prolonged fifty-five year investigation, certain critical aspects pertaining to trisection became evident well ahead of others. For example, I realized that a general perception of geometry dating back all the way to the time of Archimedes perhaps might be better served by means of now considering a much needed extension to it; one that would transcend beyond the confines of conventional Euclidean practice, and amplify even upon Webster's own definition of such word; whereby from an availability of straight lines, intersection points, circles, triangles, rectangles and parallelograms, leading to an overall profusion of spheres, prisms and even pyramids, eventually would emerge the far greater understanding that any visualization which could be mathematically interpreted diagrammatically should be considered to be of a geometric nature!
[1660] Such enhanced perception would apply to real world events wherein certain articulating mechanisms, even those capable of performing trisection, would be credited for accomplishing specific geometric feats that otherwise could not be matched solely by conventional Euclidean means. Certain famous convolutions then would comprise known geometric shapes, such as the Conchoid of Nicomedes, the Trisectrix of Maclaurin, the catenary or hyperbolic cosine, the elliptical cone, the parabola, the Folium of Decartes, the Limacon of Pascal, the Spiral of Archimedes, the hyperbolic paraboloid, as well as logarithmic and even exponential curves; as previously were considered to be taboo within an otherwise limited realm of conventional Euclidean practice.
[1661] Revolutionary material, as presented herein, consists largely of a wealth of information that can be traced directly to a newly established methodology that, in turn, is predicated upon a proposed extension to conventional Euclidean practice. In order to succeed at developing such rather unconventional output, it became essential to take good notes over extended periods of time. Moreover, copyrights conveniently served to document dates pertaining to significant discoveries.
[1662] Many concepts, as expressed herein, stem from a far broader pretext which previously was referred to as equation sub-element theory Upon reading my unfinished treatise entitled, The Principles of Equation Sub-element Theory; United States Copyright Number TXu 1-960-826 granted in April of 2015, it would become apparent that such purported new field of mathematics unfortunately only is in its embryonic stage of development. By no means should it be considered to be complete! In fact, such document already was amended under United States Copyright Number TXu 1-976-071 during August of 2015, and presently is undergoing yet another revision in order to keep abreast with recent findings, some of which are to be disseminated to the public for the very first time herein. Such copyright process permits premature theories to become documented, and thereafter revised without difficulty in order to suitably become refined into viable output.
[1663] Any prior art issue which might arise concerning the concurrent preparation of two documents which might contain somewhat similar, or even closely related information could be reconciled by means of controlling which becomes published and/or disseminated first.
[1664] In this regard, such above described copyrighted material should pose no problem because it never before was published, nor even disseminated to the general public in any manner whatsoever. Hence, there is no compelling reason to suspect that information contained therein might qualify as prior art material. Such position is predicated upon one basic understanding; being, that because the exclusive right granted by such copyrights to reproduce and/or distribute never before was exercised, it becomes impossible for anyone to be aware of the very nature of such material.
[1665] Conversely, if the argument that such copyrighted material actually should qualify as prior art otherwise were to persist in some thoroughly unabated manner, it then would require a review by some expert who, by gaining access in some surreptitious manner to undisclosed information, thereby independently only would collaborate that such unfinished copyrighted information is seriously flawed. For example, such hypothetical review would reveal that the term transcendental was used inappropriately throughout such copyright and amendment thereto. Today such mistake can be easily explained by mentioning that a thorough understanding of Al-Mahani's work was gained only after such copyrighted information first became amended. Therefore, the correct replacement term, being cubic irrational, couldn't possibly have appeared in earlier forms of such copyrights. Moreover, had such copyrighted information been released to the public, well before it completion, then inaccurate information stating that only transcendental values, as consisting of a limited subset of all cubic irrational numbers, could be automatically portrayed by means of performing trisection; thereby contradicting correct details as presented herein.
[1666] Regarding the 2 year interim which elapsed between the granting of such two 2015 copyrights and the present day completion of this disclosure, such period of time is indicative of an expected turnaround needed to effectively update information that well should be construed to include complex revolutionary material, thereby exceeding that of evolutionary projects by some considerable degree; whereby more leniency should be extended for their proper update.
[1667] By means of documenting what might appear to be similar theory concurrently in dual records, a process of leap frog would unfold, whereby what might have seemed to be credible information appearing in a copyrighted document, when worked upon earlier, soon would become outdated by a subsequent accounting, such as this one; thereby necessitating yet another revision of such copyrighted document to be completed before its release in order to remain totally consistent with refinements now incorporated herein.
[1668] Accordingly, by means of publishing the contents of this disclosure well ahead of any portion of such, as yet undisclosed 500+ page copyrighted treatise, this document shall be the first to become disseminated anywhere on earth. Lastly, whereas such copyrights, as identified directly above, evidently do not appear to qualify as prior art, it thereby should not be necessary to furnish a copy of them along with the submittal of this patent disclosure.