The DICTO Hexa: a cube of 8 DICTO Hexa Jewels, the Hexa-Key, and their 5 : 3 checkerboard filling of space

Abstract

The DICTO Jewel is the regular dodecahedron of the Euclid–Kepler–Pacioli cell carved by the stella octangulas of its six neighbouring cells (volume 12 at cube edge 2); with stella octangulas it fills space. Eight DICTO Hexa Jewels (exact DICTO Jewels on a 2 × 2 × 2 block of cells, unsheared and passing through each other inside) form one solid, the DICTO Hexa: a cube of edge 4 with 24 outward roofs on its faces, volume 80 (6⅔ Jewels). No shear makes eight Jewels meet face to face as a cube; the Hexa exists only as overlapping exact Jewels. Hexas mate only window to window; at the FCC pattern of spacing 4 each has 12 neighbours, and the space between them is one solid, the DICTO Hexa-Key (8 stella octangulas and 12 double roofs, volume 48). Hexas and Hexa-Keys fill space as a checkerboard, 5 : 3 by volume, the Stella–Jewel Lattice one level up. The Hexa splits exactly into 8 cubes and 24 roofs, or into 4 whole and 4 trimmed Jewels. Clusters of eight Hexas, a diamond network of Hexas and stellas, and the absence of any turning join are also given. Not found in a web search (2026-10-09) or in our reference library (2026-10-15). 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