Kagome Hulls: six corner-sharing networks of solid clusters in two space-filling lattices, one rule, and its icosahedral exception

Abstract

Two space-filling lattices share the same rock-salt arrangement of cubic cells: the Stella–Jewel Lattice (DICTO Jewels on the even cells, stella octangulas on the odd) and the Sunstar Lattice (regular dodecahedra on the even cells, Dogstars on the odd). Grouping even cells into solid clusters shaped like a convex hull (rhombohedron, cuboctahedron hollow or solid, cube, rhombic dodecahedron, octahedron), we find in both lattices a network in which any two clusters share at most one piece and every shared piece lies in exactly two clusters, corner to corner as in the Kagome and pyrochlore nets. The rhombohedral hull gives a perfect Kagome net, with every piece in exactly two clusters. Across the six hulls one rule holds: a hull with n corners sits on a network where each cluster has n neighbours. The 13-dodecahedron cluster is the exception: its hull has 12 corners, but at most 3 clusters can share corners with it and no lattice exists. The rule breaks where icosahedral symmetry meets periodic space. Corner-sharing octahedra and tetrahedra are known (perovskite, pyrochlore); the clusters of DICTO Jewels and dodecahedra, the rule across six hulls and the exception were 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