The Euclid–Kepler–Pacioli Cell: all five Platonic solids on 13 points in one periodic cell, and a corner-sharing network of Kepler stars and icosahedra

Abstract

Place Euclid's six roofs on a cube of edge 2 to obtain the regular dodecahedron, then reflect the twelve roof vertices back through the cube faces: they land exactly on a regular icosahedron inside the cube. Repeating the cube by translations of 2 gives a periodic structure, which we call the Euclid–Kepler–Pacioli (EKP) cell, with 13 nodes per cell, edge lengths in the chain 2 : 2/φ : 2/φ² (cube : dodecahedron : icosahedron), and space group Pm-3 (No. 200), nodes on Wyckoff positions 1b and 12j (0, y, z) with y = 1/(2φ), z = 1/(2φ²). Each roof vertex is a vertex of a neighbouring cell's icosahedron, and the ridges of the six neighbouring cells meet inside every cube as Pacioli's three interlocking golden rectangles. The cell nests all five Platonic solids (icosahedron ⊂ octahedron ⊂ stella octangula ⊂ cube ⊂ dodecahedron) with Kepler's great stellated dodecahedron around the icosahedron. From the same node set, great stellated dodecahedra on the even cells and icosahedra on the odd cells share only corners: the EKP network. All statements are checked by 48 exact or sampled tests. The individual nestings are classical; the periodic cell as a whole and the EKP network were not found in a literature search (2026-10-06, extended 2026-10-15) and we present them as candidates, not as priority claims. 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