The DICTO Jewel

Put Euclid's dodecahedron on a cube, as roofs on its six faces, and carve away the stella octangulas of the six neighbouring cubes: what remains is the DICTO Jewel, with 38 corners and 60 faces, twelve of them thick Penrose rhombi (the windows) and 48 triangles, and volume exactly 3/16 at the scale where a 2 × 2 × 2 block of cells (the Hexa's cube, §4) has edge 1, or 12 on cells of edge 2. Jewels and stella octangulas on the other cells fill space in the rock-salt pattern, 3 : 1 by volume [2]; we prove this exactly: each stella's cube is the stella and six carved roofs of exactly 1/96 each. DICTO's clusters [2, 10], named here as single solids: the DICTO Jewel-Tetra (4 Jewels, 228 faces, 3/4), whose copies share out all Jewels, and the DICTO Jewel-Octa (6 Jewels and the stella that exactly fills their hole, 19/16), which tiles space with loose stellas (stellas inside no piece) on exactly 12 lattices of translations, all one tiling up to its symmetry. Together they sit in the pattern of the octet truss, eight Jewel-Tetras round each Jewel-Octa, but as overlapping solids, not as a tiling: without overlap at most one stella in six can be inside a Jewel-Octa, and the 12 lattices reach that bound. The third, the DICTO Jewel-Rhombi, is eight Jewels on the corners of the face-centred cubic cell, meeting on 18 windows, with the stella sealed inside (396 faces, 25/16). Each of the three tiles space with loose stellas (on 4, 12 and 25 lattices of translations; 2, 1 and 9 up to symmetry); none can without stellas, since Jewels are only three quarters of space. The DICTO Hexa and DICTO Hexa-Key [12] (5/4 and 3/4, filling space 5 : 3, two windows between neighbouring Hexas), checked numerically in [12], are proved exactly. As a solid the Jewel is pyritohedral, so it twins like the Ogdoecontahedron [5]; its packing does not twin by a (111) mirror, because that mirror does not keep the cube lattice. All claims are exact in ℚ(√5) except where marked float. Builds on Kagome Hulls (DICTO DOI 10.5281/zenodo.23376317) and The DICTO Hexa (DICTO DOI 10.5281/zenodo.23376320). Finding #41 on dictospheres.com. Scripts, exact checks and data: https://github.com/DICTOR-Master/kaleidohedra/tree/d475b59/studies/2026-10-20-dicto-jewel, also attached here (dicto-jewel-study-d475b59.zip).