Listen To A Shape

RUPA’s Twisted Box


"Math Traverse", Math Seminar, vol.12:58-62 (2020)/Nippon Hyorin sha co.,Ltd

This is about a RUPA’s twisted box. The idea of this shape was developed during exploration of polyhedra having 3- and 4-degree vertices and 4-gonal faces. Some of them appeared very unique, since their approximate shapes can be represented by a simple net. The net consists of two squares joined at the edge, but not aligned, with some additional edges inside (in general, two congruent convex polygons with rotational symmetry). This is surely the one disregarded in school. It’s easy to construct. Bend the internal edges, and then join the external edges sequentially. A mountain-fold or valley-fold makes a chiral form. It is a curved solid (shown below is isomorphic to a truncated square trapezohedron*). Interestingly, the curved faces of the solid are created solely by straight edges, not curved edges, drawn on the net. Briefly, the center areas for the top and bottom faces are two squares, which are surround by four pairs of rectangle-shaped pentagons in a wave. A pair of pentagons joins at a dihedral angle about 90 degrees, which turn gradually as they connect the lower and upper parts of the solid. The solid is twisted in two ways, the center areas around a vertical axis and the side areas around a horizontal axis. This twisted box resembles in part the Penrose square, an impossible object (Penrose & Penrose, British Journal of Psychology 49:31-33, 1958) about the side areas, while connections of internal edges are different. Above all, if you blow, it rotates around its rotational axis (total 3) like a pin wheel!
ROOM No.:V16[316]F10[4258]

Truncated square trapezohedron

V10[3842]F8[48]Truncated square trapezohedron
It has 16 vertices (3-deg x16) and 10 faces (4-gon x2 and 5-gon x8). Antiprism type symmetry. The shape is obtained by cutting off two 4-deg vertices of a square trapezohedron (V10[3842]F8[48], the dual of a square antiprism V8[48]F10[3842]). A similar shape, a truncated triangular trapezohedron V12[312]F8[3256] is known as a Dürer’s solid.

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