The DICTO Palintriacontahedron

Abstract

The DICTO Palintriacontahedron is the hull of the icosidodecahedron's central pyramids flipped out at full height: 32 corners and 60 triangles, each half a rhombic-dodecahedron rhombus. It is the member of the dodecahedron–icosahedron hull family with edge ratio exactly 3/√5, in closed form the hull of (0, ±1, ±φ) and ((5 + √5)/6)·{(±1, ±1, ±1), (0, ±φ, ±1/φ)} (cyclic): the rhombic triacontahedron turned inside out. From Vajrayana (DICTO DOI: 10.5281/zenodo.23429911). Exact in Q(√5) unless marked float. 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