An Infinite Class of Deltahedra

Abstract

The appropriate designation deltahedra (A-hedra) has been proposed by Cundy [1] for polyhedra all faces of which are equilateral triangles. Three of the regular (Platonic) solids the tetrahedron, octahedron, and icosahedron are convex deltahedra with 4, 8, and 20 equilateral triangular faces, respectively. Models of deltahedra can be made conveniently by using No. 10 rubber bands to join the 3-inch flanged equilateral triangle cardboard invented by Seattle architect, Fred Bassetti [2]. Kits of these Poly-O panels can be purchased from Book-Lab, Inc. [3]. Or, home-made can be cut from railroad board in the manner clearly described by Stewart [4], Pritchett [5], and Woolaver [6]. Three convex deltahedra can be formed by placing together the bases of two congruent pyramids: the 6-faced triangular dipyramid or ditetrahedron, the 8-faced square dipyramid or octahedron, and the 10-faced pentagonal dipyramid. Six equilateral triangles can be assembled into a regular hexagon. Two such hexagons joined around their perimeters form a degenerate dipyramid. If this is separated into two parts along two coincident long diagonals, the two parts can be opened up by pressing inward on the ends of the diagonals. After one of the opened parts is rotated through 900, the two parts can be rejoined to form a 12-faced Siamese dodecahedron which is convex. There are only eight convex deltahedra [7]. Completing the set are the 14-faced triaugmented triangular prism, formed by attaching square pyramids to the square lateral faces of a triangular prism; and the 16-faced dicapped square antiprism, formed by attaching square pyramids to the two bases of a square antiprism. Thus the eight convex deltahedra are formed from 4, 6, 8, 10, 12, 14, 16 and 20 equilateral triangles. There are no convex deltahedra with 18 faces, nor any with more than 20 faces. Once convexity is abandoned, the possibilities of forming deltahedra are endless. Members of one particularly interesting and attractive type are made by attaching regular tetrahedra to some or all of the faces of other deltahedra. The deltahedra thus made by augmentation are pseudo-stellated, since they are not the same as the solids produced by extending the planes of the faces of the basic polyhedron, except in the case of the augmented octahedron which is Kepler's famous stella octangula. A particularly attractive model is the pseudo-stellated icosahedron. Models of an infinite subclass of deltahedra, the spiral (twisted) type, can be constructed from strips of equilateral triangles (FIGURE 1) flexing about the common sides. Any desired lengths of these strips can be formed easily by the cardboard panel-rubber band method. Models of spiral deltahedra can be started by joining the sides of the left-most angles of three strips to form a trihedral angle. The construction proceeds by joining sides of triangles of adjacent strips that have an end in common. (Differently colored strips emphasize the spiral nature of the deltahedron.) The three strips may be chosen from the Hand L-patterns in four ways, namely: HHH, LLL, HHL, and LLH. The HHH form appears to twist to the right (see FIGURE 2), and the LLL form appears to twist to the left. However, in general, each form consists of a pile of regular octahedra capped top and bottom with a