Theaetetus: an icosahedron held exactly by its Cleo-Icos, grown to the DICTO Hecatodihedron in two exact stories

Abstract

We hollow the DICTO-ICOSA star (the DICTO-Star with a pyramid on each of its 152 free faces) and find that a core icosahedron resting on twelve inner tips is loose: it turns about 14° before it touches anything. DICTO's remedy is to hold it with Cleo-Icos (an icosahedron's face pyramids from its centre) standing on the twenty triangles of the icosidodecahedron of edge 1, points inward. Their points land exactly on the face centres of an icosahedron of edge 1, the same edge as every other piece, because the icosidodecahedron's triangles lie at twice that icosahedron's inradius; held so, the core cannot turn or shift. The hollow is then exactly the core, twelve flat pentagonal pyramids (Aetos) and twenty triangular antiprisms (the shadow of Klitzing's four-dimensional segmentochoron K-4.137), each antiprism a Cleo-Ico, three Sphen and three Sphenion, with volumes Cleo-Ico : Sphen : Sphenion = 3 : 2 : 1 and each Aetos exactly √5 Cleo-Icos. Grown outward with the layers of the Spheres of Dharma (scaled by 2/φ), the object closes on the DICTO Hecatodihedron (102 faces) in two exact stories. In the stations (215 pieces) every layer is a closed solid, each Cleo-Ico fused with the dodecahedron's corner pyramid into an Atraktos. In the 102 clean pieces, none with a dent, the core is cut into its twenty Cleo-Icos and the dodecahedron round it into twenty Kion by the same twenty cones from the centre, so the dual pair of icosahedron and dodecahedron is cut face to corner. Every volume is exact in ℚ(√5) and both stories sum to the Hecatodihedron, 3350 − 1486√5, with no remainder. The object is the Theaetetus world of the Kaleidohedra app. Data, renders and the check outputs are included. Computation and checking with Claude (Anthropic) in the open-source library krp-core (v0.16.0). Explore it in the apps: https://dictospheres.com