The DICTO-Star: an icosidodecahedron, 12 regular dodecahedra and 20 tridiminished icosahedra closing exactly in ordinary space

Abstract

On each of the 12 pentagons of an icosidodecahedron place a regular dodecahedron, and on each of its 20 triangles a tridiminished icosahedron (Johnson solid J63), attached by its one triangle that borders only pentagons. The 33 regular-faced pieces (edge 1) meet only in whole faces and do not overlap. At every one of the icosidodecahedron's 60 edges the three dihedral angles add to exactly 360° (142.6226° + 116.5651° + 100.8123°), and the dodecahedron's face there is exactly a pentagon of the J63, so every edge and every corner is closed without a gap or filler. The volume is exactly (195 + 89√5)/3 ≈ 131.3367. With whole icosahedra in place of the J63s (a known virtual model) the pieces overlap; J63 is the icosahedron with exactly the three overlapping pyramids removed, which is what lets the arrangement exist exactly in flat space. The star does not continue into a space-filling by regular pieces. It was not found in a literature search (2026-10-09), and we present it as a candidate, not as a priority claim. First released inside Kaleidohedra v2026.10.11-octet (DICTO DOI: 10.5281/zenodo.23284390); this is its own record. 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