Kaleidohedra by DICTO: shearable lattices, new space-fillers and the Euclid–Kepler–Pacioli (EKP) cell
Year: 2026
Type: software
Abstract: Kaleidohedra by DICTO: working notes, verification scripts and a browser app for lattices and the polyhedra on them. This snapshot records finding #8 of DISCOVERIES.md, first recorded 2026-10-06. All Kaleidohedra findings (DISCOVERIES.md; each checked by a script in CI; credit DICTO unless noted): DICTO skewed rhombic dodecahedron: a rhombic-dodecahedron-type space-filler whose four equal edge directions meet at 60° three times and 72° three times (60° and 72° rhombi), tiling as a sheared FCC. Not found in the literature: candidate. Equal-edge rule for sheared FCC (Kaleidohedra): the equal-edge cell of every sheared FCC lattice, by a circumcentre shift. General idea known; this form not found. Bain disphenoids become regular tetrahedra (known, Bain 1924). Bain rhombic dodecahedron: 4 squares and 8 rhombi of 60° (known). Regular-hexagon elongated dodecahedron: 4 regular hexagons, 4 squares and 4 rhombi of 60°, all edges equal, filling space by translation; Kaleidohedra's symbol. Known (the truncated octahedron with one zone removed; Grünbaum 2010, Fig. 2b); DICTO reached it independently by a new route, the Bain stretch. Exactly nine "most regular" space-fillers (Kaleidohedra two-way search): every equal-edge space-filling zonohedron whose faces are only squares, regular hexagons and 60° rhombi. DICTO skewed elongated dodecahedra (two forms), extending #1. Euclid–Kepler–Pacioli (EKP) cell, and 8b, the Euclid–Kepler–Pacioli network (below). Also TARGETS.md and geometry-targets.json: the 160 equal-edge space-filling cells whose edge directions meet only at special angles, found by an exact search both ways. Euclid–Kepler–Pacioli cell. Euclid's roofs on a cube of edge 2 make the regular dodecahedron; reflecting the 12 roof vertices back through the cube faces makes the regular icosahedron. Cube, dodecahedron and icosahedron edges are φ² : φ : 1, and every roof vertex is a vertex of a neighbouring cell's icosahedron. Repeated on a simple cubic lattice: 13 nodes per cell, space group Pm-3 (No. 200), nodes on Wyckoff 1b and 12j with y = 1/(2φ), z = 1/(2φ²). Every cell carries the icosahedron's 31 rotation axes, parallel throughout the crystal; 7 of them are crystal symmetries, and m-3 is exactly the cube's symmetries that are also the icosahedron's. The cell also contains Kepler's great stellated dodecahedron (the icosahedron with 20 golden-triangle spikes reaching the dodecahedron's vertices). All five Platonic solids in one cell. The six neighbours' roof ridges meet inside each cube as Pacioli's three interlocking golden rectangles, whose corners are the icosahedron. The icosahedron's vertices lie on the edges of the octahedron on the cube-face centres at the golden section; that octahedron is the overlap of Kepler's stella octangula (two regular tetrahedra on alternate cube corners). So icosahedron ⊂ octahedron ⊂ tetrahedra ⊂ cube ⊂ dodecahedron nest exactly in one cell. Euclid–Kepler–Pacioli network. Great stellated dodecahedra on the even cells (an FCC lattice) and icosahedra on the odd cells share only corners: each star's 12 roof tips are vertices of the six neighbouring icosahedra, with no overlap (exact separating-axis test). Dodecahedra on the even cells alone form the known optimal lattice packing of the regular dodecahedron, density (5+√5)/8. Every claim is checked numerically by scripts/verify-roof-fold.mjs on each push. A web-level literature search (2026-10-06, recorded in DISCOVERIES.md) did not find the cell or the network; the individual solids, Euclid's construction and the optimal packing are classical. Names chosen by DICTO.
Author: DICTO
Language: English
Subfield: Materials Chemistry
Field: Materials Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220421428