The DICTO Metaxyhexecontahedron

Abstract

The DICTO Metaxyhexecontahedron is the convex hull of a regular icosahedron and a dodecahedron placed over its faces, at the one size where the icosahedron edge is exactly 9/φ⁴ times the dodecahedron edge: in closed form the hull of (0, ±1, ±φ) and (φ⁵/9)·{(±1, ±1, ±1), (0, ±φ, ±1/φ)} (cyclic). 32 corners, 90 edges, 60 congruent isosceles triangles, an insphere and no midsphere. In the twelve-station golden cycle of Vajrayana (DICTO DOI: 10.5281/zenodo.23429911) it closes the last step, touching the rhombic triacontahedron and the next icosahedron, each of its two layers one kind of piece. Exact in Q(√5). A short introduction to one solid, with its exact data and checks included, and a record of the catalogues, software libraries and reference-library sources searched. Computation and checking with Claude (Anthropic). Explore DICTO's worlds in the apps: https://dictospheres.com Version 2: the cut figures re-rendered with solid cut faces (each cut solid filled in its own colour; glass shells hollow), following DICTO's standing practice for cut views; text and results unchanged.