DICTO ATOM-13×13: thirteen clusters of thirteen, twelve around one, six golden and seven periodic, with the pieces that fill between them

Abstract

Twelve equal spheres can touch a thirteenth and no more (Schütte and van der Waerden 1952), and twelve atoms round one make the icosahedral cluster of liquids and metals (Frank 1952). We set thirteen polyhedral thirteens side by side, each a centre and twelve around it. Six are golden, on the icosahedron's five-fold axes: the Icosa-13 (Lotus seed), DESHI, the Dodeca-13, UNITY with its closed form VENUS, the DICTO-Star and the Bodhi seed. Seven are periodic, a cell and twelve neighbours in a honeycomb: the rhombic dodecahedron (RD 13, with its rounded form the DICTO Spheroid-13 RD), our skewed RD, the Bain RD, the golden Bilinski dodecahedron, the regular-hexagon elongated dodecahedron, the truncated octahedron (4 + 5 + 4, the centre seen through two square windows) and the DICTO Jewel (with its stella octangulas). For each we give how the twelve touch the centre and one another, and the kits that fill between them: 24 kits from 13 to 213 pieces, every one checked piece by piece and buildable from the centre out. Two golden clusters close exactly into a φ²-times larger copy of their own form. Every periodic block fills space by translation alone, and all seven are the same tiling in different shapes: under a linear change of coordinates each neighbourhood is the face-centred cubic one, whose radius-one perfect code is known (Kovačević and Tan 2018; explicitly x₁ + 3x₂ + 9x₃ ≡ 0 mod 13, Kovačević 2026), its weights Singer's difference set {0, 1, 3, 9} mod 13 (Singer 1938). Credit: the clusters and their names (Lotus seed, DESHI, UNITY, VENUS, DICTO-Star, Bodhi seed, DICTO Spheroid-13 RD, the DICTO Jewel, the truncated octahedron's 4 + 5 + 4 with its two square windows, the piece names, DICTO ATOM-13×13), the choice of members and the gold centre are DICTO's. Computation and checking with Claude (Anthropic). Files: the paper (PDF), and a data archive with every cluster and kit as pieces, the 24 kit plans for the Polyhedraverse app, the facts table, renders and scripts. Version 2 (2026-10-14): the paper is written in the first person throughout; no other changes.