In geometry, a polycon is a kind of a developable roller. It is made of identical pieces of a cone whose apex angle equals the angle of an even sided regular polygon. In principle, there are infinitely many polycons, as many as there are even sided regular polygons. Most members of the family have elongated spindle like shapes. The polycon family generalizes the sphericon. It was discovered by the Israeli inventor David Hirsch in 2017.
Construction Two adjacent edges of an even sided regular polygon are extended till they reach the polygon's axis of symmetry that is furthest from the edges' common vertex. By rotating the two resulting line segments around the polygon's axis of symmetry that passes through the common vertex, a right circular cone is created. Two planes are passed such that each one of them contains the normal to the polygon at its center point and one of the two distanced vertices of the two edges. The cone part that lies between the two planes is replicated n 2 − 1 {\displaystyle {\frac {n}{2}}-1} times, where n {\displaystyle {n}} is the number of the polygon's edges. All n 2 {\displaystyle {\frac {n}{2}}} parts are joined at their planer surfaces to create a spindle shaped object. It has n {\displaystyle {n}} curved edges which pass through alternating vertices of the polygon. The obtained object is cut in half at its plane of symmetry (the polygon's plane). The two identical halves are reunited after being rotated at an offset angle of 2 π n {\displaystyle {\frac {2\pi }{n}}}
Edges and vertices A polycon based on a regular polygon with n {\displaystyle {n}} edges has n + 2 {\displaystyle {n+2}} vertices, n {\displaystyle {n}} of which coincide with the polygon's vertices, with the remaining two lying at the extreme ends of the solid. It has n {\displaystyle {n}} edges, each one being half of the conic section created where the cone's surface intersects one of the two cutting planes. On each side of the polygonal cross-section, n 2 {\displaystyle {\frac {n}{2}}} edges of the polycon run (from every second vertex of the polygon) to one of the solid's extreme ends. The edges on one side are offset by an angle of 2 π n {\displaystyle {\frac {2\pi }{n}}} from those on the other side. The edges of the sphericon ( n = 4 {\displaystyle {n=4}} ) are circular. The edges of the hexacon ( n = 6 {\displaystyle {n=6}} ) are parabolic. All other polycons' edges are hyperbolic.
The sphericon as a polycon
The sphericon is the first member of the polycon family. It is also a member of the poly-sphericon and the convex hull of the two disc roller (TDR convex hull) families. In each of the families, it is constructed differently. As a poly-sphericon, it is constructed by cutting a bicone with an apex angle of π 2 {\displaystyle {\frac {\pi }{2}}} at its plane of symmetry and reuniting the two obtained parts after rotating them at an offset angel of π 2 {\displaystyle {\frac {\pi }{2}}} . As a TDR convex hull it is the convex hull of two perpendicular 180° circular sectors joined at their centers. As a polycon, the starting point is a cone created by rotating two adjacent edges of a square around its axis of symmetry that passes through their common vertex. In this specific case there is no need to extend the edges because their ends reach the square's other axis of symmetry. Since, in this specific case, the two cutting planes coincide with the plane of the cone's base, nothing is discarded and the cone remains intact. By creating another identical cone and joining the two cones together using their flat surfaces, a bicone is created. From here the construction continues in the same way described for the construction of the sphericon as a poly-sphericon. The only difference between the sphericon as a poly-sphericon and sphericon as a polycon is that as a poly- sphericon it has four vertices and as a polycon it is considered to have six. The additional vertices are not noticeable because they are located in the middle of the circular edges, and merge with them completely.
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