A device for dividing an angle into a plurality of smaller equal angles, comprising: a lower translucent sheet; and an upper translucent sheet positioned over the lower translucent sheet and pivotally connected to the lower translucent sheet at a pivot point for alignment to a vertex of the angle, to enable the upper translucent sheet to rotate relative to the lower translucent sheet; wherein the translucent sheets are marked with visual indicia corresponding to: a straight arm line for alignment on an arm of the angle; and a plurality of progressive loci curves using a T-tool defined by a straight line (EF) of a fixed length between two points (E, F) that is bisected by a bisecting straight line (B) at 90 degrees.

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.

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.

Processes for dividing the arc of an angle and its angle into n number of equal parts. The same processes apply to dividing the arc of a wave of any amplitude into n number of equal lengths. The number of equal divisions to be derived may be either an even number or an odd number. The number of degrees in the arc of a circle or the angle of the circle, or in the arc of a wave or its amplitude, need be known.

Multi-part processes and device with elongated flat members for line end-points, which together can divide a straight or curved line of unknown length into specified numbers of equal sub-parts. This invention significantly extends beyond the limited abilities of various devices and construction techniques, including geometric constructions used over millennia, to subdivide a line into equal sub-parts. The length of any straight-line in a plane can be subdivided into any specified positive natural number of equal sub-parts. The length of any 360? or less arc of a circle in a plane and its subtending central angle(s) can be subdivided into any specified positive integer number of equal sub-parts. Each of the four angles formed when two straight-lines intersect in a plane can be subdivided into any specified positive natural number of equal sub-parts. The specified number for subdivision may be any even or odd positive whole number, including a positive prime number. The multi-part processes and device can be used to produce definite specified fractional subdivisions of a line that are not equal sub-parts of the line. They may be applicable to subdividing arcs of constant curvature that are not circular. This invention can be used to avoid the abstractions in reality and illusions created when objects in the three-dimensional real world are represented two-dimensionally in a plane. The same applies to representations in three-dimensional constructions. This application is a continuation of patent application Ser. No. 16/580,542 in which the examiner found in an amended substitute specification that more than one invention had been disclosed.

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.

An educational ruler is a protractor, where the protractor can draw and measure angles. The educational ruler includes a planar disk having a semicircular circumference and labeled in scales along the semicircular circumference; a straight line edge connecting two edges of the semicircular circumference; a through hole on the planar disk; a first edge and a second edge extending from the through hole to the straight line edge; a third edge and a fourth edge extending from the semicircular circumference to the straight line edge. The first edge of the through hole of the educational ruler and the second edge of the through hole of the educational ruler intersect at 90 degrees scale of the planar disk to form the first right angle, and the included angle of the first edge of the through hole of the educational ruler and the straight line edge of the educational ruler is 45 degrees.

Multi-part processes and device with elongated flat members for line end-points, which together can divide a straight or curved line of unknown length into specified numbers of equal sub-parts. This invention significantly extends beyond the limited abilities of various devices and construction techniques, including geometric constructions used over millennia, to subdivide a line into equal sub-parts. The length of any straight-line in a plane can be subdivided into any specified positive natural number of equal sub-parts. The length of any 360 or less arc of a circle in a plane and its subtending central angle(s) can be subdivided into any specified positive integer number of equal sub-parts. Each of the four angles formed when two straight-lines intersect in a plane can be subdivided into any specified positive natural number of equal sub-parts. The specified number for subdivision may be any even or odd positive whole number, including a positive prime number. The multi-part processes and device can be used to produce definite specified fractional subdivisions of a line that are not equal sub-parts of the line. They may be applicable to subdividing arcs of constant curvature that are not circular. This invention can be used to avoid the abstractions in reality and illusions created when objects in the three-dimensional real world are represented two-dimensionally in a plane. The same applies to representations in three-dimensional constructions. This application is a continuation of patent application Ser. No. 16/580,542 in which the examiner found in an amended substitute specification that more than one invention had been disclosed.