The DICTO Dodeca-13: the 13-dodecahedron cluster made solid, its gaps as two exact pieces, and separable finned units

Abstract

A regular dodecahedron with a regular dodecahedron on each of its 12 faces is unique (each outer one is the centre's mirror image across the shared face) and is the first shell of the 120-cell laid flat. The 12 outer dodecahedra never overlap; they touch only along the centre's 30 edges, where three dodecahedra leave the well-known gap of 360° − 3 × 116.565° = 10.3048°. We show that the gaps are filled exactly by two kinds of congruent piece: 30 wedges (volume φ³/20 at edge 1), one on each edge of the centre, and 20 needles (volume (45 − 19√5)/600), one at each corner. Together the 63 pieces form one closed solid with no inner gap, the DICTO Dodeca-13 (volume 106.058472), whose outside is 72 regular pentagons, 60 thin trapezoids and 20 small triangles. Because each wedge is mirror-symmetric and each needle three-fold, the filled solid splits into a bare centre and 12 identical finned units that lift straight out. The cluster's convex hull, its densest translation packing (body-centred cubic, density 0.7154) and its relation to the DICTO-Star are also given. The 10.3° gap is known; the gap as two exact solid pieces, and the separable units, were not found in a literature search (2026-10-09). 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