Pentagons for the Kites: How One Idea for the DICTO Ogdoecontahedron Grew

The DICTO Ogdoecontahedron is a ball of 20 hexagons and 60 kites, the kites gathered in twelve five-pointed stars. DICTO proposed putting a pentagon in the middle of each star. One flat cut at each of the twelve star points does it, and what is left turns out to be hiding a famous shape: its corners sit exactly at the midpoints of the edges of a football, the shape of the carbon molecule C60. From there the idea kept growing. Abstract. DICTO proposed replacing the five-kite stars of the DICTO Ogdoecontahedron (OG) with pentagons. One plane cut at each of the twelve five-fold corners, through the side corners of its five kites, removes exactly the twelve corners that lie off the OG's sphere and nothing else. Theorem. The convex hull of the OG's 90 sphere corners is the rectified truncated icosahedron: the 90 corners are the edge midpoints of a regular truncated icosahedron. Following DICTO's further proposals we find: an all-equal-edge version (equilateral triangles, regular pentagons, hexagons with alternating angles); hexagons as truncated triangles, possible only in a chiral, left and right family; a nonzero Dehn invariant (exact) and a best lattice packing of 0.776244; a kit of four shapes and a left/right pair for the octet truss, built on a pentagonal connector across every octahedron edge; and DICTO's Bull-horns OG adaptor adapted to the cut solid, so that two shapes fill space; the gyrated octet, closed by three twin pieces; and, on DICTO's proposal, the all-equal-edge solid on a pyrochlore network, which turns into the body-centred cubic packing of truncated octahedra. The solid itself is known; the relation to the OG, the chiral family and the kits were not found in the sources checked.