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.

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 Euclidean formulation, being a practical means for representing upon just a single piece of paper an entire family of geometric construction patterns which can be exclusively derived from a specific sequence of Euclidean operations; wherein only a double arrow notation would need to be placed at some strategic location upon one of such geometric construction patterns, furthermore referred to as its representative geometric construction pattern, thereby making it possible to observe how the appearance of such drawing would change there due to modifications made to the size of an angle which appears elsewhere upon such drawing that becomes denoted algebraically by the Greek letter θ.

2. A geometric forming process which theorizes that complete pathways which lead from device settings of an angle trisector all the way back to their corresponding trisectors furthermore could be shown to superimpose upon respective geometric construction patterns belonging to a specific Euclidean formulation; thereby validating: that such mechanism actually could portray a wide range of unique motion related solutions for the problem of the trisection of an angle; that the Euclidean limitation of being unable to backtrack upon irreversible geometric construction patterns whose rendered angles amount to exactly three times the size of their respective given angles can be overcome by means of introducing motion; and that the reason why the classical problem of the trisection of an angle cannot be solved is because any devised geometric construction pattern remains impervious to the effects of time.

3. A mathematics demarcation whose geometric forming process portion, being that wherein applied motions can affect outcomes, alone is capable of portraying lengths which are of cubic irrational value; as opposed to its conventional geometric construction portion wherein finite lengths of such magnitudes instead only could be approximated by means of applying a straightedge to a given length of unity.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0436] FIG. 1A is a depiction of the fundamental architecture of a famous Kempe trisecting device, as shown in this particular case to be set to a designated magnitude of 143¼°; thereby divulging the whereabouts of only the longitudinal centerlines of linkages and radial centerlines of interconnecting pivot pins which collectively comprise such prior art, including those which are featured in its three strategically emplaced anti-parallelogram shaped control mechanisms used to strictly regulate the manner in which it such device is permitted to articulate.

[0437] FIG. 1B is a prior art method for determining the trisection of an angle; except for the fact that all of the intersection points appearing therein now are denoted by different letters. It furthermore is representative of prior art, famously known as a marked ruler arrangement, in which the longitudinal centerline of a marked ruler, as denoted therein by straight line MR includes a notch at point N, being located any suitable arbitrary distance away from its tip, as located at point M along such longitudinal centerline; which furthermore sits atop a drawing of angle QPS, as algebraically expressed therein to be of 3θ designated magnitude, an added circle which is drawn about such point P at a radius which is equal in length to that of straight line segment MN, and shows its straight line SP to be extended in such a manner that the longitudinal centerline of such ruler is jockeyed about so that it passes through point Q, has its tip, M, rest somewhere upon straight line SP extended, while its notch additionally becomes located somewhere along the circumference of such drawn circle.

[0438] FIG. 1C is yet another illustration of prior art; shown to have been truncated in order to apply specifically to trisection, as indicated by having its unused linkage depicted in phantom therein; which also expresses the same letter designations which appear on FIG. 1B, thereby making it easier to compare such two drawings in order to recognize that the fundamental architecture of such device, as represented in FIG. 1C, could be reconfigured so that it assumes the very same overall shape as that which is depicted in FIG. 1B, thereby substantiating that, in such particular arrangement, such device would automatically portray a motion related solution for the problem of the trisection of an angle; and which additionally displays θ and 3θ algebraic angular notations for the express purpose of making it perfectly clear that such device is fully capable of trisecting, not only a specific angle of 55° designated magnitude, as actually is depicted therein, but a wide range of other device settings as well.

[0439] FIG. 2 is a flowchart which identifies the various elements of a comprehensive trisection methodology.

[0440] FIG. 3 is a Trisection Mystery Iteration Processes Table which itemizes pertinent ramifications which are considered to underlie the very nature of a plaguing trisection mystery that has persisted for millennia.

[0441] FIG. 4 is a Figure Number Table that cites figure numbers of Euclidean formulations and drawing packages that apply to each of the four embodiments which collective comprise such newly proposed articulating invention.

[0442] FIG. 5 is a Euclidean formulation, easily identified as such because it brandishes a double arrow, as well as bears algebraic angular notations upon it.

[0443] FIG. 6 is a derivative Euclidean formulation, as representing a geometrically constructed extension of FIG. 5.

[0444] FIG. 7 is a second derivative Euclidean formulation, thereby representing a geometrically constructed extension of FIG. 6.

[0445] FIG. 8 is a third derivative Euclidean formulation, thereby representing a geometrically constructed extension of FIG. 7.

[0446] FIG. 9 is a Euclidean Formulation Rendered Angle Relation Table that identifies acute rendered angles which appears in each of such Euclidean formulations, as cited in FIG. 4; furthermore algebraically expressing each of their magnitudes, as shown therein to amount to exactly three times the size of their respective given angles.

[0447] FIG. 10 is a CATEGORY 1, sub-classification B Conforming Aspects Chart which identifies similarities evident within the four constituent embodiments of such newly proposed invention, as tabulated in FIG. 4.

[0448] FIG. 11 is a Mathematics Demarcation Chart, so arranged to divulge exactly which areas of mathematics can be represented only by a newly proposed geometric forming process; thereby exposing where conventional Euclidean practice actually is limited.

[0449] FIG. 12 is Trisecting Emulation Mechanism Flowchart that describes how a trisection emulation invention performs once a designated magnitude becomes specified.

[0450] FIG. 13 is a Euclidean formulation that is representative of the famous algebraic cubic function 4 sin.sup.3 θ−3 sin θ=sin (3θ), wherein for any magnitude which given angle VOO′ might arbitrarily assume. respective lengths, algebraically expressed as 4 sin.sup.3 θ and 3 sin θ, could be drawn solely by conventional Euclidean means, such that the difference noted between them would equal a length that thereby could be algebraically expressed as sin (3θ); in effect, enabling angle VOU′ to be geometrically constructed from such determination with its magnitude amounting to exactly three times the size of such given angle.

[0451] FIG. 14 is a graph of three algebraic functions; wherein the function denoted by the top legend remains continuous within the range −1≤cos θ≤+1, the function denoted by the middle legend remains continuous for all real values of cos θ except when it is equal to zero, and the function denoted by the bottom legend is entirely discontinuous in that it consists of only four discrete points, as noted within the large circles displayed therein; whereby any continuous portions of such curves furthermore could be described by a virtual unlimited number of geometric construction patterns that belong to a particular Euclidean formulation that could be developed in much the same way as that which is represented in FIG. 13.

[0452] FIG. 15 is a table of roots for the quartic equation 80 cos.sup.4 θ−4 cos.sup.3 θ−60 cos.sup.2 θ+6=0, along with other supporting data, as obtained by relating the top and bottom functions denoted in such FIG. 14 legend in order to establish the equality (4 cos.sup.3 θ−6)/(20 cos θ)=4 cos.sup.3 θ−3 cos θ.

[0453] FIG. 16 is a geometric construction pattern showing the process for geometrically solving parabolic equations of the form ax.sup.2+bx+c=0; merely by means of applying such famous Quadratic Formula x=(−b±√{square root over (b.sup.2−4ac)})/2a solely by conventional Euclidean means for the specific case when the coefficients a=−2, b=0.4, and c=0.75.

[0454] FIG. 17 is a geometric solution for the problem of the trisection of an angle whose designated magnitude is algebraically expressed as 3θ and whose tangent, denoted as ζ, is assigned a value of √{square root over (5)}/7. Although not representing a bona fide solution for the classical problem of the trisection of an angle, which cannot be solved, such geometric solution does succeed at resolving a quadratic equation that assumes the algebraic form z.sub.R.sup.2+b′z.sub.R+c′=0, as obtained by means of applying a particular abbreviated version of the Quadratic Formula z.sub.R=(½)(−b′±√{square root over (b′.sup.2−4c′)}) to it for the particular case when b′=(3+γ)/(3ζ+β) and c′=(δ−ζ)/(3ζ+β), thus amounting to b′=−(105+49√{square root over (5)}))/(4√{square root over (5)}+49) and c′=(85√{square root over (5)})/(49+4√{square root over (5)}) for the particular quadratic equation which results when two cubic equations of a singular variable known to share a common root expressed, z.sub.R=tan θ, become simultaneously reduced, solely in algebraic fashion, when each is represented as:

ζ=tan(3θ)=√{square root over (5)}/7=(3z.sub.R−z.sub.R.sup.3)/(1−3z.sub.R.sup.2);and

z.sub.R.sup.3+βz.sub.R.sup.2+γz.sub.R+δ=0 when β=−(√{square root over (5)}+7),γ=7√{square root over (5)}+12, and δ=−12√{square root over (5)}.

[0455] FIG. 18 is a diagram that indicates how an angle of arbitrarily selected designated magnitude, denoted algebraically as 3θ therein, can be trisected by means of geometrically constructing a series of properly arranged successive Euclidean bisections.

[0456] FIG. 19 is a Successive Bisection Convergence Chart that discloses the measure of trisection accuracy which could be obtained by means of increasing the number of properly arranged successive Euclidean bisections that take place within a particular geometric construction process; thereby indicating that just after twenty-one iterations, as indicated in the line item in which n=22 therein, trisection would be performed to an accuracy of six decimal places if the human eye were capable of detecting such activity.

[0457] FIG. 20 is a diagram of a complex number whose arbitrarily selected angular magnitude, algebraically denoted as θ therein, serves both as a trisector for, as well as a cube root of another complex number that becomes geometrically constructed with respect to it such that its magnitude amounts to exactly three times its size, thereby being algebraically designated as 3θ therein.

DETAILED DESCRIPTION

[0458] 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.

[0459] 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 FIG. 2; essentially being to validate that the design of some proposed invention could perform trisection accurately throughout a wide range of device settings and, in so doing, qualify as a legitimate trisecting emulation mechanism that can automatically portray various motion related solutions for the trisection of an angle.

[0460] 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!

[0461] The process box referred to as UNKNOWN GEOMETRIC PROPERTY UNCOVERED 2 is where, in the course of such FIG. 2 flowchart, an entirely new geometric property is to be uncovered which furthermore is considered to be the cause of such identified mathematic limitation. Although presently being unknown, any newly defined geometric property naturally would have to be as basic a shape as a well known straight line or circle; thereby making such trisection mystery that much more intriguing.

[0462] 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.

[0463] 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.

[0464] Next, the decision box entitled DEFICIENCY MITIGATED 5 within such FIG. 2 flowchart serves to verify that certain equipment featured in such proposed articulating devices that are supposed to avail such suspected primary function actually are deemed to be of sufficient designs to suitably mitigate such adverse influence. If it turns out that they are not adequate to perform such identified primary function, then they require redesign. If, instead, it turns out that they perform such primary function, but do not trisect, then such suspected mathematic limitation must be an incorrect selection, and another response thereby needs to be chosen. The recourse for such noted action is indicated by the NO pathway which is shown to exit such decision box.

[0465] The input box entitled TRISECTION RATIONALE 6, as shown in FIG. 2, is where a discussion is presented that accounts for how the correct responses, as indicated in such FIG. 3 Trisection Mystery Iteration Processes Table, were chosen in the very first place.

[0466] Such trisection rationale discussion specifically directs attention to the first four processes listed in such FIG. 2 flowchart, and proceeds by conjecturing that overlapment points residing within an irreversible geometric construction pattern elude detection from any and all Euclidean interrogations which possibly could be launched exclusively from the sole vantage point of its rendered information.

[0467] 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.

[0468] 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!

[0469] Since the very concept of reversibility is entirely new with regards to conventional Euclidean practice, a validation that isosceles triangle MNP, as posed in FIG. 1B, is a reversible geometric construction pattern is afforded directly below:

[0470] whereas the first three steps of a previously stipulated sequence of Euclidean operations already has accounted for how to geometrically construct isosceles triangle MNP directly from given acute angle RMP, all that is needed in order to demonstrate reversibility is to thereby geometric construct isosceles triangle MNP with respect to its rendered angle PNM instead, as is outlined in the three step sequence of Euclidean operations which follows:

[0471] step 1—from rendered ∠PNM, an arbitrary length NM is marked off along one of the sides with point M becoming assigned to its newly described end;

[0472] step 2—a circular arc is swung about point N whose radius is of length NM; and

[0473] step 3—point P becomes designated at the newly determined intersection of such circular arc with the other side of ∠PNM, whereby straight lines NP and PM become drawn to complete isosceles triangle MNP.

[0474] In order to demonstrate the actual difficulty which an intrusion of overlapment points causes, notice in FIG. 1B that it is impossible to geometrically construct isosceles triangle MNP solely with respect to rendered angle QPS.

[0475] 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 FIG. 1B, never could be located solely with respect to rendered angle QPS by conventional Euclidean means. The reason for such impossibility is furnished below:

[0476] even though it is known that overlapment point M must reside somewhere along straight line SP extended, it cannot be determined solely via straightedge and compass exactly which of the infinite number of possible locations which resides upon it applies when commencing exclusively from rendered angle QPS; and

[0477] the same argument holds true for overlapment point N which is known to reside somewhere along a circle that is drawn about point P that is of radius PQ, but whose exact location cannot be precisely pinpointed exclusively with respect to rendered angle QPS solely via straightedge and compass.

[0478] 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!

[0479] 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.

[0480] 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 FIG. 1B, could not be retraced in complete reverse order by means of attempting to apply only a straightedge and compass with respect to such rendered angle QPS.

[0481] 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!

[0482] 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!

[0483] 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.

[0484] 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.

[0485] 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.

[0486] 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.

[0487] 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 FIGS. 1A and 1B, that also render angles whose magnitudes amount to exactly three times the size of respective given angles. Without being informed as to which particular geometric construction pattern applies in the very first place, resident overlapment points no longer become limited to specific intersection point locations upon a specific pattern.

[0488] 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 FIG. 1B, its given angle NMP or RMP, even when designated to be of a specified size that can be duplicated solely by means of applying a straightedge and compass, still cannot be determined when launching Euclidean operations just with respect to its rendered angle QPS; principally because its resident overlapment points cannot be distinguished.

[0489] 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!

[0490] 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.

[0491] 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.

[0492] 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.

[0493] 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!

[0494] 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!

[0495] 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!

[0496] 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.

[0497] 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.

[0498] 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.

[0499] 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.

[0500] 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!

[0501] Remarkably, only by means of analyzing conventional Euclidean practice from this other seldom viewed perspective could irreversibility be identified as being caused by an intrusion of overlapment points.

[0502] 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.

[0503] The input box entitled IMPROVED DRAWING PRETEXT 7, as posed FIG. 2, is where a new truncated drawing format is to be introduced that can represent an entire family of geometric construction patterns, all upon just a single piece of paper.

[0504] Whereas such FIG. 2 flowchart is meant to apply exclusively to trisection, such improved drawing pretext, as alluded to therein, preferably should be identified as any Euclidean formulation each of whose constituent geometric construction patterns depicts a rendered angle whose magnitude amounts to exactly three times the size of its given angle. By means of suitably designing a trisecting emulation mechanism so that virtually any static image which would become regenerated as the result of its being properly set thereby would automatically portray an overall outline that furthermore could be fully described by a particular geometric construction pattern which belongs to such Euclidean formulation, then it could be substantiated that a motion related solution for the problem of the trisection of a angle could be achieved, merely by means of backtracking upon an irreversible condition that instead would have prevented the classical problem of the trisection of an angle from being solved!

[0505] 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!

[0506] 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!

[0507] 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.

[0508] Such is the very purpose of the input box entitled MATHEMATICS DEMARCATION 8, as posed in FIG. 2 herein. Its key artifact consists of a Mathematics Demarcation Chart, as posed in FIG. 11, which discloses a particular partitioning which should be imposed universally in order to suitably distinguish between geometries which describe stationary patterns, as opposed to those which can quantify disparate motion related geometries.

[0509] As it pertains to trisection matters, the drawing pretext entry appearing in the third column of such FIG. 11 chart, as listed directly under the cell entitled Geometric forming process, quite expectedly, turns out to be that of a Euclidean formulation; hence, limiting overall scope therein to matters in which geometric construction patterns can be associated to one another only through particular sequences of Euclidean operations.

[0510] Headings appearing in FIG. 11, are shown to run along the left side of such chart. Such arrangement enables the two principal listings appearing at the top of the second and third columns therein to serve as minor headings in themselves; thereby making it easy to differentiate between conventional Euclidean practice and a geometric forming process merely by means of comparing such two columns to one another.

[0511] 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!

[0512] 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!

[0513] 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.

[0514] In such FIG. 11 Mathematics Demarcation Chart, notice that cubic irrational numbers are listed only under the heading referred to as geometric forming process. Therein, such partitioning assignment is entirely consistent with the proposed finding that although angular portions within a regenerated static image can become automatically portrayed, even when they consist of cubic irrational trigonometric properties, nevertheless such angles cannot be geometric constructed just from a given length of unity or from another angle whose trigonometric properties are either rational or quadratic irrational!

[0515] 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.

[0516] 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.

[0517] 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!

[0518] An elementary, but nonetheless very revealing example of this concerns attempts to trisect a sixty degree angle solely by conventional Euclidean means!

[0519] 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.

[0520] 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!

[0521] Further note in such FIG. 11 chart that linear, as well as quadratic algebraic equation and associated function format type entries appear under both conventional Euclidean practice, as well as geometric forming process cells. This is because linear straight lines and/or second order circular arcs which remain stationary over time amidst an agitated motion would assume the very same shapes within each and every geometric construction pattern which belongs to any Euclidean formulation that furthermore could be animated in order to replicate such articulation event; thereby applying to both sides of such partitioned FIG. 11 chart.

[0522] 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.

[0523] The most commonly known algebraic continuum is an infinite series whose terms become summed over some specific predetermined range of performance.

[0524] 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.

[0525] 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.

[0526] 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 3θ become associated to those of an angle of size θ.

[0527] 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!

[0528] 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.

[0529] In such case, not only would an entire Euclidean formulation with representative geometric construction pattern as fully described by FIG. 1A geometrically describe such three-to-one angular correspondence during device flexure, but so too would the well known algebraic cubic function which assumes the form cos (3θ)=4 cos.sup.3 θ−3 cos θ.

[0530] That is to say, within such Euclidean formulation, angle ABC, when amounting to virtually any designated magnitude 3θ within the limits of 20°≤∠ABC≤120°, furthermore would algebraically relate to an angle ABD therein, of resulting size θ, by such aforementioned famous algebraic cubic function.

[0531] 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°:

[00019]$\angle .Math..Math.\mathrm{ABC}=3.Math.\theta ={60}^{\prime }$$\theta ={60}^{\prime }/3=20.Math.°=\angle .Math..Math.\mathrm{ABD};$$\cos .Math..Math.\left(\angle .Math..Math.\mathrm{ABC}\right)=\cos .Math..Math.\left(3.Math.\theta \right)=\cos .Math..Math.60.Math.°=0.5;$$\cos .Math..Math.\left(\angle .Math..Math.\mathrm{ABD}\right)=\cos .Math..Math.\theta =\cos .Math..Math.20.Math.°=0.93969262.Math..Math.\mathrm{.Math.}.Math.;$$\begin{array}{c}4.Math..Math.{\cos }^3.Math..Math.\theta -3.Math..Math.\cos .Math..Math.\theta =.Math.4.Math.{(0.9.Math.3969262.Math..Math.\mathrm{.Math.}.Math.)}^3-3.Math.(0.9.Math.3969262.Math..Math.\mathrm{.Math.}.Math.)\\ =.Math.3.319077.Math.8.Math.6.Math.2-2.8.Math.1.Math.9.Math.0.Math.7.Math.7.Math.862\\ =.Math.0.5.\end{array}$

[0532] Additionally, a specific nature that is found to be evident within algebraic continuums furthermore shall become addressed, wherein:

[0533] 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 θ−4 sin.sup.3 θ; thereby conforming to the famous cubic function 3 sin θ−4 sin.sup.3 θ=sin (3θ); and

[0534] a graph is to become developed that distinguishes between the continuity of such well known cubic function 4 cos.sup.3 θ−3 cos θ=cos (3θ) and the discontinuity that clearly is evident within a function that otherwise assumes the form (4 cos.sup.3 θ−6)/(20 cos θ)=cos (3θ).

[0535] 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.

[0536] 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.

[0537] 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+β.sub.1z.sub.R.sup.2+γ.sub.1z.sub.R+δ.sub.1;

y.sub.2=0=z.sub.R.sup.3+β.sub.2z.sub.R.sup.2+γ.sub.2z.sub.R+δ.sub.2;

z.sub.R.sup.3+β.sub.1z.sub.R.sup.2+γ.sub.1z.sub.R+δ.sub.1=0=z.sub.R.sup.3+β.sub.2z.sub.R.sup.2+γ.sub.2z.sub.R+δ.sub.2;

β.sub.1z.sub.R.sup.2+γ.sub.1z.sub.R+δ.sub.1=0=β.sub.2z.sub.R.sup.2+γ.sub.2z.sub.R+δ.sub.2;

0=(β.sub.2−β.sub.1)z.sub.R.sup.2+(γ.sub.2−γ.sub.1)z.sub.R+(δ.sub.2−δ.sub.1); and

0=az.sub.R.sup.2+bz.sub.R+c.

[0538] 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.2−4ac)})/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.

[0539] In such FIG. 11 Mathematics Demarcation Chart, algebraic equations and their associated functions are addressed interchangeably. Such association between them easily can be recognized when considering that by reformatting the function stipulated above into equation format, its overall content in no way changes, but only becomes perceived from a completely different perspective, such that:

z.sup.3+βz.sup.2+γz+δ=y; and

z.sup.3+βz.sup.2+γz+(δ−y)=0

[0540] In such first case, the variable z can change in value, thereby promoting a new corresponding value for y.

[0541] 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 β, its linear coefficient γ, and its scalar coefficient δ−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+γz+δ=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 other rudimentary elements, as contained within an all-encompassing newly proposed equation sub-element theory.

[0542] In such FIG. 11 chart, it further is indicated that only certain real numbers can reside within specific algebraic equations types, as well as their associated functions; thereby even further evidencing an overall order that exists within a farther reaching mathematics!

[0543] 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”.

[0544] Such technical position doesn't address higher order equations merely because they represent byproducts of cubic relationships which are fashioned in a singular variable.

[0545] 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.

[0546] An example of this follows with respect to the parabolic equation presented below, followed by an associated abbreviated form of the Quadratic Formula:

[00020]$a.Math.x^2+b.Math.x+c=0$$x^2+\frac{b}{a}.Math.x+\frac{c}{a}=0$$x^2+b^{\prime }.Math.x+c^{\prime }=0;\mathrm{and}$$\begin{array}{c}x=.Math.\frac{-b\pm \sqrt{b^2-4.Math..Math.\mathrm{ac}}}{2.Math.a}\\ =.Math.\frac{-b/a\pm \left(1/a\right).Math.\sqrt{b^2-4.Math..Math.\mathrm{ac}}}{2.Math.a/a}\\ =.Math.\frac{-b/a\pm \sqrt{b^2/a^2-4.Math..Math.\mathrm{ac}/a^2}}{2}\\ =.Math.\frac{-b/a\pm \sqrt{{\left(b/a\right)}^2-4.Math.c/a}}{2}\\ =.Math.\frac{-b^{\prime }\pm \sqrt{b^{\mathrm{\prime 2}}-4.Math..Math.c^{\prime }}}{2}.\end{array}$

[0547] 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.

[0548] As such, a corollary furthermore states, “Cubic irrational root pairs which appear in parabolic equations or their associated functions require supporting cubic irrational coefficients”.

[0549] 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 FIG. 2, such entries pertain to the input box entitled SET OF RULES 9.

[0550] 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:

[0551] 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:

[0552] 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

[0553] 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 FIG. 1C, notice that the triangle whose vertices are described by axis M, axis N, and axis P must remain isosceles no matter what magnitude becomes applied to angle RMS. Such is the case because the constant distance between axis N and axis M always must be equal to that which lies between axis N and axis P therein;

[0554] a second rule which more particularly pertains to trisection is that the various shapes that collectively comprise an entire family of geometric construction patterns all bear a distinct geometric relationship to one another based upon the fact that they all stem from the very same sequence of Euclidean operations. A few examples of how such rule can be administered are presented below:

[0555] the radial centerlines of interconnecting pivot pins which become constrained within trisecting emulation mechanism linkage slots must remain aligned along the straight line, or even curved paths of their respective longitudinal centerlines during device flexure; as based upon the design principle that the constant width of such slot, whose longitudinal centerline also remains coincidental with that of such slotted linkage, is just slightly larger than the diameter of the shanks of the interconnecting pins which are constrained within it;

[0556] static images which become regenerated whenever a CATEGORY I sub-classification A trisecting emulation mechanism becomes cycled over its wide range of device settings automatically portray a virtually unlimited number of unique overall shapes which furthermore fully can be fully described by a Euclidean formulation; more particularly meaning that for any discrete device setting, the longitudinal centerlines of linkages and radial centerlines of interconnecting pivot pins which constitute its fundamental architecture furthermore can be described by the respective straight lines and intersection points of a geometric construction pattern which belongs to such Euclidean formulation; and

[0557] as a CATEGORY I sub-classification A trisecting emulation mechanism becomes cycled over a wide range of device settings, any change which can be observed in the magnitude of the intrinsic angles of its fundamental architecture furthermore fully can be described by those which become exhibited between corresponding straight lines within a Euclidean formulation which describes the overall shapes of its portrayed static images;

[0558] another rule is that both rational, as well as quadratic irrational numbers can be algebraically equated to specific sets of cubic irrational numbers. The procedure for accomplishing this consists of first selecting a specific rational or quadratic irrational number that is to be characterized and then setting it equal to the left-hand side of one of nine equations presented in the preceding definition of a cubic irrational number, wherein:

[0559] for the particular case when a rational number of ⅛ is to be further characterized, the first of such nine equations can be applied in order to determine a value for 3θ.sub.1 as follows wherein each concluding three dot notation indicates that such number extends an infinite number of decimal places to the right, thereby being indicative of an actual cubic irrational number:

[00021]$\cos \left(3.Math..Math.{\theta }_1\right)/4=1/8=\cos .Math..Math.{\theta }_1.Math..Math.\cos .Math..Math.{\theta }_2.Math..Math.\cos .Math..Math.{\theta }_3;$$\cos \left(3.Math.{\theta }_1\right)=1/2$$\begin{array}{c}3.Math.{\theta }_1=.Math.\mathrm{arc}.Math..Math.\mathrm{cosine}.Math..Math.1/2\\ =.Math.60.Math.°;\end{array}$$\begin{array}{c}{\theta }_1=.Math.60.Math.°/3\\ =.Math.20.Math.°\end{array}$$\begin{array}{c}\cos .Math..Math.{\theta }_1=.Math.\cos .Math..Math.20.Math.°\\ =.Math.0.93969262.Math..Math.\mathrm{.Math.}.Math.;\end{array}$$\begin{array}{c}{\theta }_2=.Math.{\theta }_1+120.Math.°\\ =.Math.20.Math.°+120.Math.°\\ =.Math.140.Math.°;\end{array}$$\begin{array}{c}\cos .Math..Math.{\theta }_2=.Math.\cos .Math..Math.140.Math.°\\ =.Math.-0.7.Math.66044443.Math..Math.\mathrm{.Math.}.Math.;\end{array}$$\begin{array}{c}{\theta }_3=.Math.{\theta }_1+240.Math.°;\\ =.Math.20.Math.°+240.Math.°\\ =.Math.260.Math.°\end{array}$$\begin{array}{c}\cos .Math..Math.{\theta }_3=.Math.\cos .Math..Math.260.Math.°\\ =.Math.-0.1.Math.73648177.Math..Math.\mathrm{.Math.}.Math.;\mathrm{and}\end{array}$$\begin{array}{c}\cos .Math..Math.\left(3.Math.{\theta }_1\right)/4=.Math.\cos .Math..Math.{\theta }_1.Math..Math.\cos .Math..Math.{\theta }_2.Math..Math.\cos .Math..Math.{\theta }_3\\ =.Math.\left(\cos .Math..Math.{\theta }_{1.Math.}.Math.\cos .Math..Math.{\theta }_2\right).Math.\cos .Math..Math.{\theta }_3\\ =.Math.(-0.7.Math.1984631.Math..Math.\mathrm{.Math.}.Math.).Math.(-0.1.Math.73648177.Math..Math.\mathrm{.Math.}.Math.)\\ =.Math.1/8;\end{array}$

[0560] when a rational number of magnitude 0, −¾, or −3 is to be further characterized, then the fourth, fifth, seventh, eighth, and ninth of such nine equations would apply; whereby those that pertain to the sin θ are validated for the particular case when a value of 34.3° becomes assigned to θ.sub.1 as follows:

[00022]$\begin{array}{c}\sin .Math..Math.{\theta }_1=.Math.\sin .Math..Math.34.3.Math.°\\ =.Math.0.563526048.Math..Math.\mathrm{.Math.}.Math.;\end{array}$$\begin{array}{c}{\theta }_2=.Math.{\theta }_1+120.Math.°\\ =.Math.34.3.Math.°+120.Math.°\\ =.Math.154.3.Math.°;\end{array}$$\begin{array}{c}\sin .Math..Math.{\theta }_2=.Math.\sin .Math..Math.154.3.Math.°\\ =.Math.0.433659084.Math..Math.\mathrm{.Math.}.Math.;\end{array}$$\begin{array}{c}{\theta }_3=.Math.{\theta }_1+240.Math.°;\\ =.Math.34.3.Math.°+240.Math.°\\ =.Math.274.3.Math.°;\mathrm{and}\end{array}$$\begin{array}{c}\sin .Math..Math.{\theta }_3=.Math.\sin .Math..Math.274.3.Math.°\\ =.Math.-0.9.Math.97185133.Math..Math.\mathrm{.Math.}.Math.;\end{array}$$\begin{array}{c}0=.Math.\sin .Math..Math.{\theta }_1+\sin .Math..Math.{\theta }_2+\sin .Math..Math.{\theta }_3\\ =.Math.0.563526048.Math..Math.\mathrm{.Math.}+0.4336590.Math..Math.\mathrm{.Math.}+\sin .Math..Math.{\theta }_3\\ =.Math.0.997185133.Math..Math.\mathrm{.Math.}+\sin .Math..Math.{\theta }_3\\ =.Math.0.997185133)-0.997185133.Math..Math.\mathrm{.Math.}\\ =.Math.0;\end{array}$$\begin{array}{c}-3/4=.Math.\sin .Math..Math.{\theta }_1.Math.\sin .Math..Math.{\theta }_2+\sin .Math..Math.{\theta }_1.Math.\sin .Math..Math.{\theta }_3+\sin .Math..Math.{\theta }_2.Math.\sin .Math..Math.{\theta }_3\\ =.Math.\sin .Math..Math.{\theta }_1.Math.\sin .Math..Math.{\theta }_2+\left(\sin .Math..Math.{\theta }_1.Math.\sin .Math..Math.{\theta }_2\right).Math.\sin .Math..Math.{\theta }_3\\ =.Math.\sin .Math..Math.{\theta }_1.Math.\sin .Math..Math.{\theta }_2+\left(-\sin .Math..Math.{\theta }_3\right).Math..Math.\sin .Math..Math.{\theta }_3\\ =.Math.\sin .Math..Math.{\theta }_1.Math.\sin .Math..Math.{\theta }_2-{\sin }^2.Math..Math.{\theta }_3\\ =.Math.0.24437819.Math..Math.\mathrm{.Math.}-{(-0.9.Math.97185133.Math..Math.\mathrm{.Math.}.Math.)}^2\\ =.Math.0.24437819.Math..Math.\mathrm{.Math.}-(0.99437819.Math.).Math..Math.\mathrm{.Math.}\\ =.Math.0.24437819.Math..Math.\mathrm{.Math.}-\left(0.2.Math.4.Math.4.Math.3.Math.7.Math.8.Math.1.Math.9+3/4\right)\\ =.Math.-3/4.Math.;.Math.\mathrm{and}.Math..Math.\mathrm{as}.Math..Math.a.Math..Math.\mathrm{check}\end{array}.Math.-\sin .Math..Math.\left(3.Math.{\theta }_1\right)/4=\sin .Math..Math.{\theta }_1.Math..Math.\sin .Math..Math.{\theta }_2.Math..Math.\sin .Math..Math.{\theta }_3$$\begin{array}{c}\sin .Math..Math.\left(3.Math.{\theta }_1\right)=.Math.-4.Math.\sin .Math..Math.{\theta }_{1.Math.}.Math.\sin .Math..Math.{\theta }_2.Math..Math.\sin .Math..Math.{\theta }_3\\ =.Math.+0.9.Math.74761194.Math..Math.\mathrm{.Math.}\end{array}$$\begin{array}{c}3.Math.{\theta }_1=.Math.\mathrm{arc}.Math..Math.\sin .Math..Math.(0.974761194.Math..Math.\mathrm{.Math.}.Math.)\\ =.Math.77.1.Math.°,\mathrm{or}.Math..Math.102.9.Math.°\end{array}$$\begin{array}{c}{\theta }_1=.Math.77.1.Math.°/3.Math..Math.\mathrm{or},102.9.Math.°/3\\ =.Math.25.7.Math.°,\mathrm{or}.Math..Math.34.3.Math.°;\end{array}$

and

[0561] lastly, an example is afforded for the particular case when a quadratic irrational number is to be further characterized, such that when:

[00023]$\tan .Math..Math.\left(3.Math.{\theta }_1\right)=-\sqrt{3};$$3.Math.{\theta }_1=\arctan \left(-\sqrt{3}\right);$$3.Math.{\theta }_1=-60.Math.°$${\theta }_1=-20.Math.°$$\begin{array}{c}\tan .Math..Math.{\theta }_1=.Math.\tan .Math..Math.\left(-20.Math.°\right)\\ =.Math.-0.3.Math.63970234.Math..Math.\mathrm{.Math.}.Math.;\end{array}$$\begin{array}{c}{\theta }_2=.Math.{\theta }_1+120.Math.°\\ =.Math.-20.Math.°+120.Math.°\\ =.Math.100.Math.°;\end{array}$$\begin{array}{c}\tan .Math..Math.{\theta }_2=.Math.\tan .Math..Math.100.Math.°\\ =.Math.-5.6.Math.7128182.Math..Math.\mathrm{.Math.}.Math.;\end{array}$$\begin{array}{c}{\theta }_3=.Math.{\theta }_1+240.Math.°;\\ =.Math.-20.Math.°+240.Math.°\\ =.Math.220.Math.°\end{array}$$\begin{array}{c}\tan .Math..Math.{\theta }_3=.Math.\tan .Math..Math.220.Math.°\\ =.Math.0.839099631.Math..Math.\mathrm{.Math.}.Math.;\mathrm{and}\end{array}$$\begin{array}{c}-\tan .Math..Math.\left(3.Math..Math.{\theta }_1\right)=.Math.\tan .Math..Math.{\theta }_1.Math..Math.\tan .Math..Math.{\theta }_2.Math..Math.\tan .Math..Math.{\theta }_3.\\ =.Math.\left(\tan .Math..Math.{\theta }_1.Math..Math.\tan .Math..Math.{\theta }_2\right).Math..Math.\tan .Math..Math.{\theta }_3\\ =.Math.(2.0641777.Math..Math.\mathrm{.Math.}.Math.).Math..Math.(0.8390996.Math..Math.\mathrm{.Math.}.Math.)\\ =.Math.1.732050808.Math..Math.\mathrm{.Math.}\\ =.Math.\sqrt{3};\end{array}$

and

[0562] conversely, whenever trigonometric values of triads θ.sub.1, θ.sub.2, and θ.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 θ°, (θ+120°), and (θ+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:

[00024]$\begin{array}{c}0=.Math.\sin .Math..Math.{\theta }_1+\sin .Math..Math.{\theta }_2+\sin .Math..Math.{\theta }_3\\ =.Math.\sin .Math..Math.{\theta }_1+\sin .Math..Math.\left({\theta }_1+1.Math.2.Math.0\right)+\sin .Math..Math.\left({\theta }_1+2.Math.4.Math.0\right)\\ =.Math.\sin .Math..Math.{\theta }_1+\left(\sin .Math..Math.{\theta }_1.Math..Math.\cos .Math..Math.120+\cos .Math..Math.{\theta }_1.Math.\sin .Math..Math.120\right)+\\ .Math.\left(\sin .Math..Math.{\theta }_1.Math..Math.\cos .Math..Math.240+\cos .Math..Math.{\theta }_1.Math..Math.\sin .Math..Math.240\right)\\ =.Math.\sin .Math..Math.\theta +\sin .Math..Math.\theta \left(-1/2\right)+\cos .Math..Math.\theta \left(\sqrt{3}/2\right)+\\ .Math.\sin .Math..Math.\theta \left(-1/2\right)+\cos .Math..Math.\theta \left(\sqrt{3}/2\right);\mathrm{such}.Math..Math.\mathrm{that}\end{array}$$\begin{array}{c}0=.Math.\sin .Math..Math.\theta \left(1-1/2-1/2\right)+\cos .Math..Math.\theta \left(\sqrt{3}/2-\sqrt{3}/2\right)\\ =.Math.\sin .Math..Math.\theta \left(0\right)+\cos .Math..Math.\theta \left(0\right)\\ =.Math.0.\end{array}$

[0563] 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!

[0564] 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.

[0565] 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!

[0566] 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.

[0567] 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.

[0568] 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!

[0569] 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!

[0570] 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.

[0571] 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.

[0572] 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.

[0573] The input box appearing in such FIG. 2 flowchart, entitled PROBABILISTIC PROOF OF MATHEMATIC LIMITATION 10, refers to the specific results which can be obtained by realizing that a given angle within a geometric construction pattern furthermore must serve the dual role of also being a trisector for any rendered angle therein whose magnitude amounts to exactly three times its size; thereby signifying that a trisection event successfully has been performed solely by conventional Euclidean means!

[0574] 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!

[0575] 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.

[0576] 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!

[0577] 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.

[0578] 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.

[0579] In FIG. 2, notice that all five of such parallelogram shaped listed input boxes, along with the non-iterative, or YES output portion of such DEFICIENCY MITIGATED 5 decision box, all funnel into a downstream process box which is entitled GEOMETRIC FORMING PROCESS DEVISED 11.

[0580] 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.

[0581] In 1893, Thomas Alva Edison at long last showcased his kinetoscope. Obviously, such discovery spurred on the development of a cinematic projector by the Lumiere 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 Lumiere'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 modem 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.

[0582] 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.

[0583] 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!

[0584] 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!

[0585] 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.

[0586] 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.

[0587] 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!

[0588] 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!

[0589] 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.

[0590] Accordingly, Euclidean formulations can be of service in motion related problems which cannot be fully interpreted by a singular geometric construction pattern.

[0591] 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.

[0592] 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.

[0593] 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 FIG. 2 flowchart.

[0594] 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

[0595] 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!

[0596] 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.

[0597] 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.

[0598] 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.

[0599] In conclusion, if the logic proposed in such FIG. 3 Trisection Mystery Iteration Processes Table turns out to be entirely true, meaning that an inability to solve the classical problem of the trisection of an angle results because it impossible to backtrack upon any irreversible geometric construction pattern, then it would be utterly senseless to attempt drawing any type of singular geometric construction pattern whatsoever, solely by conventional Euclidean means, in an effort to achieve such ends!

[0600] 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!

[0601] 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 FIG. 2, where two distinct, independent Euclidean trisection approaches are to be discredited. Below, it should become rather obvious that such second listed approach is an entire reversal of the first:

[0602] not only is it impossible to fully backtrack upon any rendered angle within a geometric construction pattern whose magnitude amounts to exactly three times the size of its given angle, thereby explaining why the classical problem of the trisection of an angle cannot be solved; but

[0603] conversely, the probability that the magnitude of a rendered angle matches the designated magnitude of an angle that is intended to be trisected approaches zero whenever such rendered angle becomes geometrically constructed such that its magnitude amounts to exactly three times the arbitrarily selected size of a given angle.

[0604] 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.

[0605] 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.

[0606] 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!

[0607] 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!

[0608] 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!

[0609] The process box entitled SINGULAR DRAWING SOLUTION DISPELLED 13 is included in such FIG. 2 flowchart to address the fact that although a singular drawing solution can be described for any regenerated static image that automatically becomes portrayed once a trisecting emulation mechanism becomes properly set, designing a device of that type which has only one discrete setting would be entirely impractical!

[0610] 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!

[0611] The process box described as SUPPLEMENTAL DEVICE CAPABILITIES SPECIFIED 14 is the principal location in such FIG. 2 flowchart where information pertaining to such MATHEMATICS DEMARCATION 8 input box contributes to an understanding that trisecting emulation mechanisms additionally have the affinity to portray exact lengths that only could be approximated when performing geometric construction upon a given length of unity.

[0612] Such fact is duly reflected in such FIG. 11 Mathematics Demarcation Chart wherein cubic irrational real number types appear only in its third column, as headed by the geometric forming process cell; thereby indicating that deliberate motions must be imparted in order portray them. They can appear either as the ratios of portrayed lengths with respect to a given length of unity, or as trigonometric properties inherent within trisecting angles which become portrayed during certain trisection events.

[0613] For the particular case of the fourth embodiment of such newly proposed invention, a supplemental device leveling capability also is to be thoroughly described.

[0614] 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!

[0615] 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.

[0616] 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!

[0617] 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.

[0618] 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.

[0619] 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!

[0620] FIG. 1B presents a very good example of such capability to portray angles which exhibit cubic irrational trigonometric properties. Therein, angle QPS amounts to exactly sixty degrees. Such sixty degree angle QPS was chosen because, as stated earlier, its magnitude can be represented exactly by any of the included angles within an equilateral triangle, and thereby can be drawn solely by conventional Euclidean means.

[0621] 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°.

[0622] Upon interpreting FIG. 1B to be representative of a famous as a marked ruler arrangement, angle NMP would be twenty degrees in magnitude.

[0623] Moreover consider that the notch appearing in its ruler resides away from its endpoint, M, one unit of measurement.

[0624] 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:

[00025]$\stackrel{\_}{\mathrm{MP}}/\stackrel{\_}{\mathrm{MN}}=2.Math..Math.\cos \left(\angle .Math..Math.\mathrm{NMP}\right)$$\stackrel{\_}{\mathrm{MP}}/1=2.Math..Math.\cos .Math..Math.20.Math.°;\mathrm{and}$$\begin{array}{c}\stackrel{\_}{\mathrm{MP}}=.Math.2.Math.(0.9.Math.3969262.Math..Math.\mathrm{.Math.}.Math.)\\ =.Math.1.879385242.Math..Math.\mathrm{.Math.}\end{array}$

[0625] 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.

[0626] 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.

[0627] 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 CATEGORIZED 15 process box represents the location within such FIG. 2 flowchart where associations of this nature are to be carried out.

[0628] 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 FIG. 2 flowchart, including the decision box entitled DESIGN REQUIREMENTS MET 18, have been added for the express purpose of specifying that all of such posed requirements must be satisfied in order for a proposed design, as described by the process box entitled PROPOSED INVENTION DESIGN REFINEMENT 17, to fully qualify as a trisection emulation mechanism, as itemized in the TRISECTING EMULATION MECHANISM SUBSTANTIATED 19 process box described therein.

[0629] 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!

[0630] 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!

[0631] Accordingly it is concluded that a geometric forming process thereby eclipses a rather limited conventional Euclidean practice that has been in vogue for millennia!

[0632] Now that new definitions have been provided, and a resulting comprehensive methodology, as presented in FIG. 2 has been suitably described, it is due time to account for exactly how a trisecting emulation mechanism operates.

[0633] FIG. 12 has been prepared just for this purpose. Such flowchart commences by means of supplying details to an input box, as entitled DESIGNATED ANGLE SPECIFIED 120 therein. Such specific activity consists merely of selecting the designated magnitude of an angle that is intended to become trisected.

[0634] 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:

[0635] 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; or

[0636] if such third embodiment were to be chosen, then the NO route would apply, thereby leading to a process box entitled, DEVICE IS SET 123.

[0637] 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.

[0638] 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.

[0639] 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.

[0640] 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.

[0641] In connection with such input box entitled MATHEMATICS DEMARCATION 8, as posed in FIG. 2 herein, it previously was mentioned that 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 the sine of any of such rendered angles described by a length of 3 sin θ−4 sin.sup.3 θ; thereby conforming to a famous function expressed as 3 sin θ−4 sin.sup.3 θ=sin (3θ).

[0642] 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!

[0643] 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:

[00026]$\cos .Math..Math.\left(3.Math.\theta \right)=4.Math.{\cos }^3.Math.\theta -3.Math.\cos .Math..Math.\theta ;$$\sin .Math..Math.\left(3.Math..Math.\theta \right)=3.Math..Math.\sin .Math..Math.\theta -4.Math..Math.{\sin }^3.Math..Math.\theta ;\mathrm{and}$$\tan .Math..Math.\left(3.Math..Math.\theta \right)=\frac{3.Math.\tan .Math..Math.\theta -{\tan }^3.Math.\theta }{1-3.Math..Math.{\tan }^2.Math.\theta }.$

[0644] Whenever the magnitude of an angle that is algebraically denoted to be of size 3θ becomes supplied as a given quantity in any of such three cubic expressions, then such algebraic relationship truly would typify trisection!

[0645] This is because, a corresponding magnitude of θ, being an exact trisector of such given 3θ 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.

[0646] For example, for the particular condition when it is given that:

[00027]$3.Math.\theta =75.Math.°$$\begin{array}{c}\theta =.Math.75.Math.°/3\\ =.Math.25.Math.°\end{array}$$\cos .Math..Math.\theta =0.906307787$$3.Math..Math.\cos .Math..Math.\theta =2.718.Math.9.Math.23361$$4.Math..Math.{\cos }^3.Math..Math.\theta =2.97.Math.7.Math.742406;$$\begin{array}{c}\cos \left(3.Math.\theta \right)=.Math.4.Math.{\cos }^3.Math..Math.\theta -3.Math..Math.\cos .Math..Math.\theta \\ =.Math.2.9777.Math.4.Math.2.Math.406-2.7.Math.1.Math.8.Math.9.Math.2.Math.3.Math.361\\ =.Math.0.258819095.Math.;\end{array}$

and

[0647] As a check, 3θ=75°√

[0648] Conversely, if an infinite number of magnitudes of θ 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!

[0649] This is because all three of such above cited cubic expressions are continuous and their respective right-hand terms furthermore are geometrically constructible.

[0650] To aptly demonstrate this, a Euclidean formulation, as posed in FIG. 13, has been developed to suitably represent such famous cubic relationship sin (3θ)=3 sin θ−4 sin.sup.3 θ; wherein any geometric construction pattern belonging to thereby would exhibit a discrete value of sin (3θ) for each and every selected real sin θ value existing within the range of −1 to +1.

[0651] The governing sequence of Euclidean operations for such new Euclidean formulation is specified as follows:

[0652] given angle VOO′ is geometrically constructed of an arbitrarily selected magnitude that algebraically is denoted as θ such that its side OO′ exhibits the same length as its side OV;

[0653] side OV is designated to be the x-axis;

[0654] 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;

[0655] 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′;

[0656] point T thereafter becomes designated as the intersection between such UNIT CIRCLE ARC and such geometrically constructed y-axis;

[0657] a straight line which passes through point O is drawn at forty-five degree angle counterclockwise to such x-axis;

[0658] another straight line which passes through point O is drawn making a three-to-one slope with the +x-axis;

[0659] a horizontal straight line is drawn which passes through point O′ and thereby lies parallel to the x-axis;

[0660] the juncture between such horizontal straight line and the y-axis becomes designated as “sin θ”, thereby denoting its vertical distance above such x-axis;

[0661] 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;

[0662] the horizontal distance such vertical straight line resides to the right of such y-axis also thereby is to be designated as “sin θ” along such x-axis;

[0663] a second vertical straight line is drawn which passes through coordinate point V, thereby being tangent to such previously drawn UNIT CIRCLE ARC;

[0664] 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;

[0665] the angle which such slanted straight line makes with the x-axis becomes designated as “ω”, not to be confused with angle VOO′ amounting to a slightly larger magnitude of θ;

[0666] a second horizontal straight line is draw which passes through the intersection point made between such slanted straight line and such vertical straight line;

[0667] 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;

[0668] 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;

[0669] the angle which such second slanted straight line makes with the x-axis thereafter becomes designated as “φ”;

[0670] 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;

[0671] 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;

[0672] 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;

[0673] the juncture which such fourth horizontal straight line makes with the y-axis becomes denoted as “3 sin θ”, thereby distinguishing its vertical distance above point O;

[0674] 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”;

[0675] the juncture which is made between such fifth horizontal straight line and the y-axis becomes designated as “sin (3θ)”, thereby denoting its vertical distance above point O; and

[0676] the intersection point of such fifth horizontal straight line with such UNIT CIRCLE ARC becomes designated as point U′.

[0677] The proof for such FIG. 13 Euclidean formulation is provided below:

[00028]$\begin{array}{c}\tan .Math..Math.\omega =h_1/\sin .Math..Math.\theta =.Math.\sin .Math..Math.\theta /1\\ h_1=.Math.{\sin }^2.Math..Math.\theta ;\end{array}.Math.\hspace{1em}.Math.\begin{array}{c}\tan .Math..Math.\varphi =h_2/\sin .Math..Math.\theta =.Math.h_1/1\\ =.Math.{\sin }^2.Math..Math.\theta /1\end{array}.Math..Math.h_2={\sin }^3.Math..Math.\theta .Math..Math.4.Math.h_2=4.Math..Math.{\sin }^3.Math..Math.\theta ;.Math.\begin{array}{c}\sin .Math..Math.\left(3.Math.\theta \right)=.Math.3.Math..Math.\sin .Math..Math.\theta -4.Math..Math.{\sin }^3.Math..Math.\theta \\ =.Math.3.Math.\sin .Math..Math.\theta -4.Math.h_2;\end{array}$

and

[0678] since point U′ lies upon such UNIT CIRCLE ARC and exhibits a sin (3θ) ordinate value, radius OU′ must reside at an angle of 3θ with respect to the x-axis.

[0679] Accordingly, FIG. 13 distinguishes an entire family of geometric construction patterns, all generated from the very same sequence of Euclidean operations as stipulated above; with the only exception being that the respective magnitudes of given angle VOO′ becomes slightly altered each time a new geometric construction pattern becomes drawn.

[0680] Based upon a reasoning that such famous cubic relationship sin (3θ)=3 sin θ−4 sin.sup.3 θ actually can be fully distinguished by an entire family of geometric construction patterns which together comprise such newly proposed Euclidean formulation, as posed in FIG. 13, it theoretically might become possible to devise yet another rather crude, or cumbersome, trisecting emulation mechanism which, due to a considerable increase in its number of overall working parts, obviously would be considered to lie far beyond the very scope of this presentation. In order to become feasible, however, a newly fashioned device of such type would have to be designed so that when it becomes articulated by means of rotating its axis U′ circumferentially about axis O in accordance with such double arrow notation as expressed in FIG. 13, such motion additionally could be replicated by means of animating the conglomeration of geometric construction patterns which belong to such Euclidean formulation in successive order.

[0681] 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.

[0682] For example, if a particular value of 1.119769515 radians were to be accorded to θ, then an algebraic determination could be made, as follows of 3θ, which furthermore fully could be described by a singular geometric pattern which belongs to such newly proposed Euclidean formulation, as posed in FIG. 13:

[00029]$\theta =1.1.Math.19769515.Math..Math.\mathrm{radians}$$\sin .Math..Math.\theta =0.9;\mathrm{and}$$\begin{array}{c}\sin .Math..Math.\left(3.Math.\theta \right)=.Math.3.Math.\sin .Math..Math.\theta -4.Math.{\sin }^3.Math..Math.\theta \\ =.Math.3.Math.\left(0.Math.\mathrm{.9}\right)-4.Math.{\left(0.9\right)}^3\\ =.Math.2.7-4.Math.\left(0.7.Math.2.Math.9\right)\\ =.Math.2.7-2.916\\ =.Math.-0.Math.\mathrm{.216};\end{array}$$\begin{array}{c}3.Math..Math.\theta =.Math.\pi +0.2.Math.1.Math.7.Math.7.Math.1.Math.5.Math.891\\ =.Math.3.3593.Math.0.Math.8.Math.545\\ =.Math.3.Math.\left(1.1.Math.1.Math.9.Math.7.Math.6.Math.9515\right).\end{array}$

[0683] Such above furnished overall detailed accounting explains exactly why all three of such previously cited famous cubic expressions remain incredibly important!

[0684] 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.

[0685] Such scenario is far different than what transpires with respect to discontinuous functions, as are about to be discussed in detail next.

[0686] Also in connection with such input box entitled MATHEMATICS DEMARCATION 8, as posed in FIG. 2, it previously was mentioned that a graph is to become developed that distinguishes between the continuity of such well known cubic function 4 cos.sup.3 θ−3 cos θ=cos (3θ) and the discontinuity that very clearly accompanies the function (4 cos.sup.3 θ−6)/(20 cos θ)=cos (3θ).

[0687] FIG. 14 is intended to make clear such distinction.

[0688] Its top legend identifies the path charted by a curve for such first famous cubic function, algebraically expressed as y=4 cos.sup.3 θ−3 cos θ=cos (3θ) wherein:

[0689] abscissa values in x signify cos θ magnitudes; and

[0690] ordinate values in y signify cos (3θ) magnitudes.

[0691] Such well known curve is shown to be continuous within the specific range of −1≤x≤+1, thereby accounting for all real number values of cos θ.

[0692] The second legend therein identifies the particular function y=(4 cos.sup.3 θ−6)/(20 cos θ) wherein abscissa values in x again signify cos θ 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 θ, 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.

[0693] 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.3 θ−6)/(20 cos θ)=cos (3θ), as typified by the third legend, as displayed in FIG. 14.

[0694] Hence, such intersection points, shown to be positioned at the centers of such four large circles drawn therein, locate positions where (4 cos.sup.3 θ−6)/(20 cos θ)=cos (3θ).

[0695] By then substituting 4 cos.sup.3 θ−3 cos θ for cos (3θ), as shown below, the following fourth order equation can be obtained, along with a determination of the four associated roots for cos θ and other relevant quantitative details:

[00030]$4.Math.{\cos }^3.Math.\theta -3.Math.\cos .Math..Math.\theta =\frac{4.Math.{\cos }^3.Math.\theta -6}{20.Math..Math.\cos .Math..Math.\theta };$

and

[0696] via cross multiplication,

(4 cos.sup.3θ−3 cos θ)(20 cos θ)=4 cos.sup.3θ−6;

80 cos.sup.4θ−60 cos.sup.20=4 cos.sup.3θ−6;

80 cos.sup.4θ−4 cos.sup.3θ−60 cos.sup.2θ+6=0; and

cos.sup.4θ− 1/20 cos.sup.3θ−¾ cos.sup.2θ+ 3/40=0.

[0697] Values of the roots of such quartic equation are provided in FIG. 15. The first column therein, as headed by the term VALUE, contains various entries of algebraic significance. For each of such five listed entries, corresponding values are cited each of the four the roots θ.sub.1, θ.sub.2, θ.sub.3, and θ.sub.4 which appear as headings in the following four columns. Notice that for each of such particular values of θ, as specified in the second line item therein, a respective value of cos (3θ) appears in the fifth line item therein which is equal to the value of (4 cos.sup.3 θ−6)/(20 cos θ), as it appears in the sixth line item therein.

[0698] In conclusion, the cos (3θ)=(4 cos.sup.3 θ−6)/(20 cos θ) quartic function clearly qualifies as being discontinuous because it consists of only four distinct points, as are identified by circles appearing in such of FIG. 14.

[0699] With particular regard to the two continuous curve representations drawn in FIG. 14, a Euclidean formulation could be generated, whereby each of the singular geometric construction patterns which belong to it can be algebraically determined; three examples of which are presented directly below:

[00031]$\mathrm{at}.Math..Math.x=\cos .Math..Math.\theta =1;$$\begin{array}{c}y=.Math.\left(4.Math..Math.{\cos }^3.Math..Math.\theta -6\right)/\left(20.Math..Math.\cos .Math..Math.\theta \right)\\ =.Math.\left[4.Math.\left(1\right)-6\right]/\left[2.Math.0.Math.\left(1\right)\right]\\ =.Math.\left(4-6\right)/20\\ =.Math.-2/20\\ =.Math.-1/10;\end{array}$$\mathrm{at}.Math..Math.x=\cos .Math..Math.\theta =1/2;$$\begin{array}{c}y=.Math.\left(4.Math..Math.{\cos }^3.Math..Math.\theta -6\right)/\left(20.Math..Math.\cos .Math..Math.\theta \right)\\ =.Math.\left[\left(4\right).Math.{\left(1/2\right)}^3-6\right)]/[\left(20.Math.x\left(1/2\right)\right]\\ =.Math.\left[\left(4\right).Math.\left(1/8\right)-6\right)]/10\\ =.Math.\left(1/2-6\right)]/10\\ =.Math.-5.5)]/10\\ =.Math.-0.Math.\mathrm{.55};\end{array}$$\mathrm{at}.Math..Math.x=\cos .Math..Math.\theta =-1/2;\mathrm{and}$$\begin{array}{c}y=.Math.\left(4.Math..Math.{\cos }^3.Math..Math.\theta -6\right)/\left(20.Math..Math.\cos .Math..Math.\theta \right)\\ =.Math.[\left(4\right).Math.{\left(-1/2\right)}^3-6/[\left(20.Math.x\left(-1/2\right)\right]\\ =.Math.\left[\left(4\right).Math.\left(-1/8\right)-6\right)]/-10\\ =.Math.\left(-1/2-6\right)]/-1.Math.0\\ =.Math.-\left(6.Math.\mathrm{.5}\right)/-10\\ =.Math.0.65.\end{array}$

[0700] 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 FIG. 14.

[0701] Above, the length (½).sup.3 would be geometrically constructed in much the same fashion as was the sin.sup.3 θ in FIG. 13. The development of such envisioned Euclidean formulation would encompass first generating a length which is equal to (½).sup.2, solely by conventional Euclidean means; produced in similar manner to length h.sub.1, as it appears therein. From such length, another length representative of the algebraic expression (½).sup.3 would become drawn, similar to h.sub.2, as it appears therein.

[0702] 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 θ−6)/(20 cos θ). 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 θ−6)/(20 cos θ); whereby only the value of cos θ would be altered in during such development.

[0703] Each respective length of the ordinate value y then could be drawn by way of the proportion y/1=(4 cos.sup.3 θ−6)/(20 cos θ), thereby producing such length ‘y’ by means of applying only a straightedge and compass.

[0704] As such, the function y=(4 cos.sup.3 θ−6)/(20 cos θ) 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.

[0705] Obviously, such types of involvements inevitably should serve as building blocks for mathematics!

[0706] 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!

[0707] In this vein, prior claims made in connection with such FIG. 11 Mathematics Demarcation Chart, now are to be somewhat bolstered by theorizing that the very formats expressed by algebraic equations give clear indication of the types of geometric construction practices they support.

[0708] Such explanation begins with what clearly is known concerning any linear function of the form y=mx+b.

[0709] 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=6x−3, 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.

[0710] 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.

[0711] 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.

[0712] 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.

[0713] By means of referring back to the previous discussion regarding such input box entitled MATHEMATICS DEMARCATION 8, as posed in FIG. 2, note that it was mentioned that a geometric construction pattern that is representative of the famous Quadratic Formula z.sub.R=(−b±√{square root over (b.sup.2−4ac)})/2a would be created to resolve the parabolic equation of −0.2x.sup.2+0.4x+0.75=0 belonging to such ax.sup.2+bx+c=0 format type.

[0714] Herein, FIG. 16 represents such very solution.

[0715] The very sequence of Euclidean operations from which such singular geometric construction pattern is derived is provided directly below:

[0716] a square each whose sides is of unit length is drawn;

[0717] a right triangle is inscribed within it such that:

[0718] its first side begins at one of the corners of such square, extends a length of 0.75, or ¾ 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;

[0719] 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

[0720] its hypotenuse then is to become drawn;

[0721] 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:

[0722] 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;

[0723] 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;

[0724] 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

[0725] 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;

[0726] 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;

[0727] 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;

[0728] 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;

[0729] 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.2−4ac)} 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)};

[0730] 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.2−4ac)};

[0731] 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 length −b+√{square root over (b.sup.2−4ac)};

[0732] 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;

[0733] with respect to such last drawn straight line:

[0734] a straight line is drawn perpendicular to its left termination point; and

[0735] 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;

[0736] 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.2−4ac)})/−2a, therefore amounting to x.sub.1=(−b−√{square root over (b.sup.2−4ac)})/2a; and

[0737] 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.2−4ac)})/−2a, thus amounting to x.sub.2=(−b+√{square root over (b.sup.2−4ac)})/2a.

[0738] 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 FIG. 14.

[0739] Notice that when interpreting such continuous cubic function y=(4 cos.sup.3 θ−6)/(20 cos θ):

[0740] 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 θ, becomes slightly altered; but

[0741] 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!

[0742] 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.

[0743] It is in this area of discussion that perhaps the greatest confusion abounds concerning trisection!

[0744] 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.

[0745] 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?

[0746] Again, such answer, most emphatically, turns out to be a resounding no!

[0747] 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.

[0748] Naturally, such algebraic approach cannot solve the classical problem of the trisection of an angle!

[0749] 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!

[0750] As a relevant example of this, one of the three famous cubic functions cited above is to undergo such simultaneous reduction process, wherein ζ is to denote the particular value of the tangent of a designated magnitude of an angle, 3θ, that is about to be trisected; thereby becoming algebraically expressed as tan (3θ). 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 θ, thereby enabling the following algebraic cubic equation expressions to be reformatted as follows:

[0751] whereas, tan (3θ)=(3 tan θ−tan.sup.3 θ)/(1−3 tan.sup.2θ);

then, ζ=(3z.sub.R−z.sub.R.sup.3)/(1−3z.sub.R.sup.2)

ζ(1−3z.sub.R.sup.2)=3z.sub.R−z.sub.R.sup.3

z.sub.R.sup.3=3z.sub.R−ζ(1−.sup.3z.sub.R.sup.2).

[0752] In order to perform such simultaneous reduction, a generalized cubic equation format type of the form z.sup.3+βz.sup.2+γz+δ=0 now is to become introduced, as well.

[0753] 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+βz.sup.2+γz+δ=0;

z.sub.R.sup.3+βz.sub.R.sup.2+γz.sub.R+δ=0; and

z.sub.R.sup.3=−(βz.sub.R.sup.2+γz.sub.R+δ).

[0754] 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!

[0755] Since such famous tangent cubic function can be arranged as z.sub.R.sup.3−3ζz.sub.R.sup.2−3z.sub.R+ζ=0, it must be a subset of such generalized cubic equation for the specific case when coefficient β=−3ζ, γ=−3, and δ=ζ.

[0756] 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!

[0757] 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!

[0758] 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.

[0759] By equating z.sub.R.sup.3 terms, the following quadratic equation relationships can be obtained by means of removing mutual cubic parameters:

[00032]$3.Math.z_R-\zeta \left(1-3.Math.z_R^2\right)=-\left({\mathrm{\beta z}}_R^2+\gamma .Math.z_R+\delta \right)=z_R^3.Math.\left(3.Math.\zeta +\beta \right).Math.z_R^2+\left(3+\gamma \right).Math.z_R+\left(\delta -\zeta \right)=0$$a.Math..Math.z_R^2+b.Math..Math.z_R+c=0;\mathrm{and}.Math.\left(3.Math.\zeta +\beta \right).Math.z_R^2+\left(3+\gamma \right).Math.z_R+\left(\delta -\zeta \right)=0$$z_R^2+\frac{3+\gamma }{3.Math.\zeta +\beta }.Math.z_R+\frac{\delta -\zeta }{3.Math.\zeta +\beta }=0$$z_R^2+b^{\prime }.Math..Math.z_R+c^{\prime }=0.$

[0760] 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.2−4ac)})/2a=[−b′+√{square root over (b′.sup.2−4(1)(c′))}]/2(1)(½)(−b′±√{square root over (b′.sup.2−4c′)}).

[0761] Obviously, such abbreviated Quadratic Formula then applies only to quadratic equations of a singular variable whose squared term coefficients are equal to unity!

[0762] In order to simultaneously reduce two cubic equations in a single variable which share a common root, their remaining root values must be different.

[0763] 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.

[0764] 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.

[0765] As such, the magnitude of (could be determined merely by means of computing the overall value associated with (3z.sub.R−z.sub.R.sup.3)/(1−3z.sub.R.sup.2) (3√{square root over (5)}−5√{square root over (5)})/(1−3×5)=√{square root over (5)}/7.

[0766] 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.

[0767] Accordingly, such famous cubic relationship in a single variable z.sub.R.sup.3−3ζz.sub.R.sup.2−3z.sub.R+ζ=0 would assume the particular form z.sub.R.sup.3−3 (√{square root over (5)}/7) z.sub.R.sup.2−3z.sub.R+√{square root over (5)}/7=0.

[0768] As for such generalized cubic equation, since it can be stated that:

z−z.sub.R=0;

z−z.sub.S=0; and

z−z.sub.T=0.

[0769] By thereafter multiplying such three equations together, the following algebraic expression could become obtained:

(z−z.sub.R)(z−z.sub.S)(z−z.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)z−z.sub.Rz.sub.Sz.sub.T=0; and

z.sup.3+βz.sup.2+γz+δ=0.

[0770] By equating coefficients of like terms, the following three relationships can be determined:

[00033]$\begin{array}{c}\beta =.Math.-\left(z_R+z_S+z_T\right)\\ =.Math.-\left(\sqrt{5}+3+4\right)\\ =.Math.-\left(\sqrt{5}+7\right);\end{array}$$\begin{array}{c}\gamma =.Math.z_R.Math.z_S+z_R.Math.z_T+z_S.Math.z_T\\ =.Math.\left(\sqrt{5}\right).Math.\left(3+4\right)+3.Math.\left(4\right)\\ =.Math.7.Math.\sqrt{5}+12;\mathrm{and}\end{array}$$\begin{array}{c}\delta =.Math.-z_R.Math.z_S.Math.z_T\\ =.Math.-\left(\sqrt{5}\right).Math.(3.Math.(\left(4\right)\\ =.Math.-12.Math.\sqrt{5}.\end{array}$

[0771] Such generalized cubic equation format would be z.sup.3−(√{square root over (5)}+7) z.sup.2+(7√{square root over (5)}+12) z−12√{square root over (5)}=0.

[0772] Accordingly:

[00034]$\begin{array}{c}{b}^{\prime }=.Math.\frac{3+\gamma }{3.Math.\zeta +\beta }\\ =.Math.\frac{3+\left(1.Math.2+7.Math.\sqrt{5}\right)}{3.Math.\left(\sqrt{5}/7\right)-\left(\sqrt{5}+7\right)}\\ =.Math.\frac{1.Math.5+7.Math.\sqrt{5}}{-\left(4.Math..Math.\sqrt{5}/7+7\right)}\\ =.Math.-\left(\frac{105+49.Math.\sqrt{5}}{4.Math.\sqrt{5}+49}\right);\end{array}$$\begin{array}{c}{b}^{{\prime }^2}=.Math.\frac{1.Math.0.Math.5^2+2.Math.1.Math.0.Math.\left(49\right).Math.\sqrt{5}+4.Math.9^2.Math.\left(5\right)}{1.Math.6.Math.\left(5\right)+8.Math.\left(49\right).Math.\sqrt{5}+4.Math.9^2}\\ =.Math.\frac{23.Math.\text{,}.Math.0.Math.3.Math.0+10.Math.\text{,}.Math.290.Math.\sqrt{5}}{2.Math.\text{,}.Math.4.Math.8.Math.1+3.Math.9.Math.2.Math.\sqrt{5}};\end{array}$$\begin{array}{c}{c}^{\prime }=.Math.\frac{\delta -\zeta }{3.Math.\zeta +\beta }\\ =.Math.\frac{-1.Math.2.Math.\sqrt{5}-\sqrt{5}/7}{3.Math.\left(\sqrt{5}/7\right)-\left(\sqrt{5}+7\right)}\\ =.Math.\frac{8.Math.5.Math.\sqrt{5}}{4.Math.\sqrt{5}+49};\end{array}$$\begin{array}{c}-4.Math.c^{\prime }=.Math.-\left(\frac{340.Math.\sqrt{5}}{49+4.Math.\sqrt{5}}\right).Math.\left(\frac{49+4.Math.\sqrt{5}}{49+4.Math.\sqrt{5}}\right)\\ =.Math.-\left(\frac{6.Math.8.Math.0.Math.0+16.Math.\text{,}.Math.660.Math.\sqrt{5}}{2.Math.\text{,}.Math.4.Math.8.Math.1+3.Math.9.Math.2.Math.\sqrt{5}}\right);\end{array}$$\begin{array}{c}{b}^2-4.Math.c^{\prime }=.Math.\frac{\left(23.Math.\text{,}.Math.0.Math.3.Math.0+10.Math.\text{,}.Math.290.Math.\sqrt{5}\right)-\left(6800+16.Math.\text{,}.Math.660.Math.\sqrt{5}\right)}{{\left(49+4.Math.\sqrt{5}\right)}^2}\\ =.Math.\frac{16.Math.\text{,}.Math.2.Math.3.Math.0-6.Math.\text{,}.Math.370.Math.\sqrt{5}}{{\left(49+4.Math.\sqrt{5}\right)}^2};\end{array}.Math.\pm \sqrt{b^2-4.Math.c^{\prime }}=\frac{\pm \sqrt{16.Math.\text{,}.Math.230-6.Math.\text{,}.Math.370.Math.\sqrt{2}}}{49+4.Math.\sqrt{5}};\mathrm{and}$$\begin{array}{c}{z}_R=.Math.\frac{-b^{\prime }\pm \sqrt{b^{{\prime }^2}-4.Math.c^{\prime }}}{2}\\ =.Math.\frac{1.Math.0.Math.5+49.Math.\sqrt{5}\pm \sqrt{16.Math.\text{,}.Math.230-6.Math.\text{,}.Math.370.Math.\sqrt{5}}}{9.Math.8+8.Math.\sqrt{5}}\\ =.Math.\frac{1.Math.0.Math.5+49.Math.\sqrt{5}\pm \sqrt{{\left(-65+49.Math.\sqrt{5}\right)}^2}}{9.Math.8+8.Math.\sqrt{5}}\\ =.Math.\frac{1.Math.0.Math.5+49.Math.\sqrt{5}\mp \left(6.Math.5-49.Math.\sqrt{5}\right)}{9.Math.8+8.Math.\sqrt{5}}\\ =.Math.\frac{4.Math.0+98.Math.\sqrt{5}}{9.Math.8+8.Math.\sqrt{5}.Math.};\frac{170}{98+8.Math.\sqrt{5}}\\ =.Math.\left(\frac{\sqrt{5}}{\sqrt{5}}\right)\left[\frac{8.Math.{\left(\sqrt{5}\right)}^2+9.Math.8.Math.\sqrt{5}}{9.Math.8+8.Math.\sqrt{5}}\right];\frac{170}{98+8.Math.\sqrt{5}}\\ =.Math.\frac{\sqrt{5}.Math.\left(8.Math.\sqrt{5}\right)+98)}{9.Math.8+8.Math.\sqrt{5}.Math.};\frac{170}{98+8.Math.\sqrt{5}}\\ =.Math.\sqrt{5};\frac{1.Math.7.Math.0}{9.Math.8+8.Math.\sqrt{5}}.\end{array}$

[0773] 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.

[0774] This could be achieved simply by realizing that once a value of (becomes designated, an angle of 3θ 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 θ, thereafter also could be trigonometrically determined.

[0775] 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!

[0776] 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=(½)(−b′±√{square root over (b′2−4c′)}), much in the same manner as was employed earlier when quadratic roots first were determined by means of geometric construction in FIG. 16.

[0777] 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 FIG. 17:

[0778] two right triangles are drawn in the lower right corner which share a common side of length (49+4√{square root over (5)})/100, and whose other mutual sides are of respective lengths:

(105+49√{square root over (5)})/100; and

85√{square root over (5)}/100;

[0779] such common side is extended to a unit length;

[0780] a perpendicular straight line is drawn above the newly formed endpoint of such extension;

[0781] 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;

[0782] 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:

[00035]$b^{\prime }=-\left(\frac{1.Math.0.Math.5+49.Math.\sqrt{5}}{4.Math.\sqrt{5}+49}\right).Math.-b^{\prime }=\left(\frac{1.Math.0.Math.5+49.Math.\sqrt{5}}{4.Math.\sqrt{5}+49}\right).Math.\left(\frac{1/1.Math.0.Math.0}{1/1.Math.0.Math.0}\right)$$\frac{-b^{\prime }}{1}=\frac{\left(1.Math.0.Math.5+49.Math.\sqrt{5}\right)/100}{\left(49+4.Math.\sqrt{5}\right)/100};$$c^{\prime }=\frac{85.Math.\sqrt{5}}{4.Math.\sqrt{5}+49}.Math.\left(\frac{1/100}{1/100}\right)$$\frac{c^{\prime }}{1}=\frac{\left(8.Math.5.Math.\sqrt{5}\right)/100}{\left(49+4.Math.\sqrt{5}\right)/1.Math.0.Math.0};$

[0783] next, a square whose sides are of length −b′ is to be drawn, as indicated in the lower left-hand corner of FIG. 17;

[0784] a rectangle then becomes drawn whose base of unit length is to align along the lower side of such square and whose left lower corner is to share the very position which the left hand lower corner of such square occupies;

[0785] a straight line then is to become drawn which extends from such newly identified common corner, passes through an intersection point which is made between the upper side of such previously drawn square and the right side of such newly drawn rectangle, and thereafter continues as a large diagonal until it intersects with the right side of such previously drawn square;

[0786] the distance between such newly determined intersection point above the lower side of such square of base dimension −b′ is to become denoted as b′.sup.2, as determined by the proportion established between the corresponding sides of two new similar right triangles whose respective hypotenuses align upon such just drawn long diagonal, whereby such proportion becomes calculated as b′2/−b′=−b′/1;

[0787] a horizontal line next is set off a distance of b′.sup.2 above the based of such previously drawn square;

[0788] another horizontal line of is set off a distance of 4c′ above the based of such previously drawn square;

[0789] the intervening length existing between them, amounting to a magnitude of b′.sup.2−−4c′, must constitute the entire area of the small rectangle they furthermore describe, as bounded by the two opposite side of such previously drawn rectangle whose base is equal to a length of unity;

[0790] a second square of unit base dimension then becomes described such that its lower portion aligns directly upon such previously described rectangle of area equal to b′.sup.2−−4c′;

[0791] a semicircle thereby can be drawn to the right of such square whose diameter aligns upon its left side;

[0792] straight lines thereafter are drawn from the respective ends of such semicircle diameter to the point residing upon its circumference which intersects the horizontal straight line which resides at a distance of b′.sup.2 above the base of such previously drawn square whose respective sides each are ‘b’ in length;

[0793] by virtue of the Pythagorean Theorem, such lower straight line, as drawn from the lower extremity of the diameter of such semicircle and extending to a point lying upon its circumference, must amount to a length which is equal to the square root of the b′.sup.2−4c′ area of such previously described rectangle;

[0794] such length thereafter is reproduced as an extension to the horizontal straight line previously drawn which resides a distance of 4c′ above the base of the previously drawn square whose sides each equal −b′ in length;

[0795] such new straight line extension is notated as being of overall length 2z.sub.R=−b′+√{square root over (b′.sup.2−4c′)}, as is indicated both at the very the top and very bottom of such drawing; and

[0796] such overall length thereafter becomes bisected in order to distinguish and thereby designate a length z.sub.R which amounts to one-half such magnitude.

[0797] 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)z−12√{square root over (5)}=0, could not be derived without a prior awareness of the very solution itself.

[0798] 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 β and γ are set to zero, and whose coefficient δ amounts to a value of +1, thereby establishing the specific cubic equation z.sub.R.sup.3+1=0.

[0799] From such information, the following details can be gleaned:

[00036]$z_R^5+1=0$$z_R^5=-1$$\begin{array}{c}{z}_R=.Math.\sqrt[3]{-1}\\ =.Math.-1\end{array}$$\tan .Math..Math.\theta =-1$$\begin{array}{c}\theta =.Math.\mathrm{arc}.Math..Math.\tan .Math..Math.\left(-1\right)\\ =.Math.135.Math.°;\end{array}$$\begin{array}{c}3.Math.\theta =.Math.3.Math.\left(\theta \right)\\ =.Math.3.Math.\left(135.Math.°\right)\\ =.Math.405.Math.°;\mathrm{and}.Math..Math.\mathrm{as}.Math..Math.a.Math..Math.\mathrm{check}\end{array}$$\begin{array}{c}\zeta =.Math.\left(3.Math.z_R-z_R^3\right)/\left(1-3.Math.z_R^2\right)\\ =.Math.\left(-3+1\right)/\left(1-3\right)\\ =.Math.-2/-2\end{array}$$\tan \left(3.Math..Math.\theta \right)=+1$$\begin{array}{c}3.Math.\theta =.Math.\mathrm{arc}.Math..Math.\tan .Math..Math.\left(+1\right)\\ =.Math.\left(3.Math.6.Math.0+45\right).Math.°\\ =.Math.405.Math.°.\end{array}$

[0800] Such algebraic determination, as made above, thereby enables the trisection of an angle to be geometrically constructed as follows:

[0801] from a designated value of =tan (3θ)=+1, an angle designated as 3θ which amounts to exactly 450 in magnitude first becomes geometrically constructed with respect to the +x-axis; and

[0802] from an algebraically determined common root value of z.sub.R=−1, a trisecting angle designated as θ which amounts to exactly 135° in magnitude thereafter becomes geometrically constructed with respect to the +x-axis.

[0803] 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 (3θ)=+1.

[0804] 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.

[0805] 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.

[0806] 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 450 angle would be to distinguish them from the results of a Euclidean trisection which has not yet been performed.

[0807] 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.

[0808] 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 FIG. 1B, is to be of the very size which actually appears in such figure; thereby very closely amounting to twenty degrees. As such:

[0809] angle QPS, being geometrically constructed to three times that size, must be exactly sixty degrees. It becomes very easy to draw such rendered angle because the internal angle of a geometrically constructed equilateral triangle is that same size;

[0810] the next step is to determine whether or not Euclidean operations can be launched exclusively from such designated angle QPS in order to locate the correct positions of points M and N; and

[0811] as it turns out, intersection points M and N cannot be distinguished solely via straightedge and compass solely from such rendered angle QPS. That is to say, there is absolutely no geometric construction that can be performed with respect to such sixty degree angle QPS which can locate points M and N, short of having aforehand awareness of their respective locations.

[0812] Such above analysis reveals that with respect to the particular geometry represented in such famous FIG. 1B Archimedes Euclidean formulation, when commencing only from angle QPS of designated sixty degree magnitude, points M and N truly qualify as overlapment points.

[0813] 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!

[0814] 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.

[0815] Such approach generates a geometric construction pattern that is indicative of a geometric progression whose:

[0816] constant multiplier, “m”, is set equal to −½; and

[0817] first term, “f”, is algebraically denoted as 3θ.

[0818] Moreover, the overall sum, “s”, of such geometric progression consisting of an “n” number of terms can be reresented by the common knowledge formula:

[00037]$\begin{array}{c}s=.Math.f\left(m^n-1\right)/\left(m-1\right)\\ =.Math.3.Math..Math.\theta \left(-1/2^n-1\right)/\left(-1/2-1\right)\\ =.Math.-2.Math..Math.\theta \left(-1/2^n-1\right);\end{array}\hspace{1em}$

whereby

[0819] for an infinite number of terms, such equation thereby reduces to,

[00038]$\begin{array}{c}s=.Math.-2.Math..Math.\theta \left(-1/2^{\infty }-1\right)\\ =.Math.-2.Math..Math.\theta \left(0-1\right)\\ =.Math.2.Math..Math.\theta .\end{array}\hspace{1em}$

[0820] 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 ⅔ the size of an angle of designated 3θ magnitude, whereby their difference then must distinguish its trisector.

[0821] 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 3θ−3θ/2+3θ/4−3θ/8+3θ/16=33θ/16.

[0822] In such development, the value of the first term, algebraically denoted as 3θ, 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.

[0823] 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.

[0824] The specific details which pertain to a FIG. 18 drawing of this nature are itemized as follows:

[0825] an angle of magnitude 3θ is drawn such that its vertex aligns upon the origin of an orthogonal coordinate system with its clockwise side residing along its +x-axis;

[0826] such given angle, being of magnitude 3θ, becomes bisected, whereby such bisector resides at an angle relative to such +x-axis that amounts to ½(3θ)=3θ/2;

[0827] the upper portion of such bisected angle, amounting to a size of 3θ/2, then itself becomes bisected, whereby a determination made as to the location of such second bisector would place it at an angle of 3θ/2+3θ/4=9θ/4 with respect to the +x-axis;

[0828] the angle formed between such first bisector and second bisector next becomes bisected, whereby a determination made as to the location of such third bisector would place it at an angle of 9θ/4−3θ/8=15θ/8 with respect to the +x-axis; and

[0829] the angle formed between such second bisector and third bisector then itself becomes bisected, whereby a determination made as to the location of such fourth bisector would place it at an angle of 15θ/8+3θ/16=33θ/16 with respect to the +x-axis.

[0830] 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.

[0831] 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°.

[0832] 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.

[0833] 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.

[0834] The Successive Bisection Convergence Chart, as presented in FIG. 19, describes the results produced by such geometric progression as the number of terms is shown to increase in its first column, as headed by the term n.

[0835] 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.

[0836] Notice that FIG. 19 is discontinued at a value of n=22. This is because, at this point in such overall geometric construction process, an accuracy of six decimal places, amounting to (2.000000)θ would become realized.

[0837] Since the only time that a bisection operation is not conducted is when n=1, each successive line item within such FIG. 19 chart depicts a geometric construction pattern that could be generated by means of performing a total of n−1 bisection operations.

[0838] Hence, an accuracy of one-millionth could be obtained by means of conducting twenty-one successive bisections.

[0839] 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.

[0840] As such, the number of terms this condition would apply to, as indicated in such FIG. 19 chart, would be when n=13.

[0841] From such FIG. 19 chart, the separation needed to be distinguished when performing such twelfth bisection is calculated to be

[00039]$\begin{array}{c}2.000244.Math..Math.\theta -1.999512.Math..Math.\theta =.Math.0.000732.Math.\theta \\ =.Math.0.000732.Math.\left(20.Math.°\right)\\ =.Math.0.01464.Math.°.\end{array}\hspace{1em}$

[0842] 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.

[0843] When referring to FIG. 18, notice that an angle of size 3θ whose vertex is placed at the origin of a Cartesian Coordinate System such that its clockwise side aligns upon its +x-axis is indicative of such geometric progression for the particular condition when n=1.

[0844] Additionally, four subsequent bisections are depicted, each of which is considered to have been performed solely by conventional Euclidean means.

[0845] The purpose of the shading therein is to suitably distinguish between each of such bisection activities as follows:

[0846] such angle of magnitude +3θ is bisected in order to distinguish two separate arcs, each being of 3θ/2 size;

[0847] with the upper portion of such bisected angle, amounting to a size of 3θ/2, then itself becoming bisected, the determination made as to the location of such second bisector would place it at an angle of 3θ/4 counterclockwise of such first bisector position;

[0848] with the angle formed between such first bisector and second bisector, amounting to a size of 3θ/4, then itself becoming bisected, the determination made as to the location of such third bisector would place it at an angle of 3θ/8 clockwise of such second bisector position, ad denoted by the minus sign notation; and

[0849] with the angle formed between such second bisector and third bisector, amounting to a size of 3θ/8, then itself becoming bisected, the determination made as to the location of such fourth bisector would place it at an angle of 3θ/16 counterclockwise of such third bisector position.

[0850] 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:

[00040]$\sqrt[3]{}=/2;$

such that by cubing both sides;

=.sup.3/8

8=.sup.3

4(2)=.sup.2

2√{square root over (2)}=

√{square root over (2)}=/2; and

[0851] 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:

[00041]$\theta =90.Math.°-3.Math..Math.\theta$$3.Math.\theta +\theta =90.Math.°$$4.Math..Math.\theta =90.Math.°$$\theta =22.5.Math.°$$2.Math..Math.\theta =45.Math.°$$3.Math.\theta =67.5.Math.°;$$\sin .Math..Math.\left(3.Math.\theta \right)=3.Math..Math.\sin .Math..Math.\theta -4.Math..Math.{\sin }^3.Math..Math.\theta$$\cos .Math..Math.\left(90-3.Math..Math.\theta \right)=\sin .Math..Math.\theta .Math..Math.\left(3-4.Math..Math.{\sin }^2.Math..Math.\theta \right)$$\begin{array}{c}\cos .Math..Math.\theta =.Math.\sin .Math..Math.\theta .Math.\left[\left(2\right).Math.\left(1-2.Math..Math.{\sin }^2.Math..Math.\theta \right)+1\right]\\ =.Math.\sin .Math..Math.\theta .Math.\left[2.Math.\cos .Math..Math.\left(2.Math..Math.\theta \right)+1\right]\\ =.Math.\sin .Math..Math.\theta .Math..Math.\left(2.Math..Math.\cos .Math..Math.45.Math.°+1\right)\\ =.Math.\sin .Math..Math.\theta \left[\left(2\right).Math.\left(\frac{\sqrt{2}}{2}\right)+1\right]\\ =.Math.\sin .Math..Math.\theta \left(\sqrt{2}+1\right)\end{array}$$\frac{1}{\sqrt{2}+1}=\tan .Math..Math.\theta$$\frac{1}{\sqrt{2}+1}.Math.\left(\frac{\sqrt{2}-1}{\sqrt{2}-1}\right)=\tan .Math..Math.\theta$$\frac{\sqrt{2}-1}{2-1}=\tan .Math..Math.\theta$$\sqrt{2}-1=\tan .Math..Math.\theta$$\sqrt{2}-1=\frac{1}{\tan \left(3.Math..Math.\theta \right)}$$\begin{array}{c}\tan \left(3.Math.\theta \right)=.Math.\frac{1}{\sqrt{2}-1}.Math.\left(\frac{\sqrt{2}+1}{\sqrt{2}+1}\right)\\ =.Math.\frac{\sqrt{2}+1}{2-1}\\ =.Math.\sqrt{2}+1;\end{array}$