KALEIDOHEDRA by DICTO
In development — live at kaleidohedra.dictospheres.com, rhombiverse.dictospheres.com and polyhedraverse.dictospheres.com. Since 2026-10-08 this repository is one app with two front doors: Kaleidohedra (orange), lattices you can shear and slide continuously, every piece moving with the lattice, and Rhombiverse (cyan), lattices of space-filling shapes from 1D to 6D. Each door has its own address, welcome, colours, guide and saved worlds; behind them the worlds, typography and controls are shared. The DICTO menu (top left) opens on the app you came in by, with the others a button away. Polyhedraverse (green), the portrait gallery of 333 polyhedra and the six regular 4D polytopes, now lives here too (step D, 2026-10-09): its space, shape browser, nets, 4D builds and hints are in both doors' DICTO menu, and since 2026-10-10 it has its own third door here, polyhedraverse.dictospheres.com (SITE=polyhedraverse); the old Polyhedraverse repository is archived (KRP, DICTO's plan for the three). The geometry all three share is krp-core.
- Per-app content (head, changelog, icons):
site/<app>/; the one DICTO User Guide (shared sections, a chapter per app, 7 languages): site/guide/. The apps' table (names, colours, which tools each has): src/app/site.js; one type scale for all: src/app/tokens.css.
- Building a front door:
SITE=kaleidohedra npm run build or SITE=rhombiverse npm run build (each Vercel project sets its own). Locally, index.html?site=rhombiverse opens Rhombiverse's door.
- DICTO's newest worlds (DICTO → 3D+, each from his ideas and papers, with its DICTO DOI in the Almanac): Every Mix of the Platonic Solids (#47), The Reuleaux Platonic Solids (#45), Aristotle's Pyramid (#44), The OG on the Octet Truss (#37) and Universal PiTet (#38) with its cycle film and The Kepler–Poinsot–UP Cycle; in Studies, the DICTO stretched solids (#49, The Polyhedral Stretching Theorem), opened directly by kaleidohedra.dictospheres.com/?open=studies:stretch. Polyhedraverse's newest family is the Mosseri–Sadoc tiles (1982): the six golden tetrahedra of Mosseri and Sadoc, among them DICTO's Hound Tooth, Tricap, Cleo and AXE.
Polyhedraverse's families include DICTO's Reuleaux Platonic Solids (paper #45, DICTO DOI 10.5281/zenodo.23529499), the five curved bodies of Kaleidohedra's Reuleaux world, drawn smooth and fitted by their corners (one body data in krp-core serves both). On all three doors the Soft view (the puffed-square button in the tools column) shows any build with every flat face bulged into a soft cap, in a Plain, Metallic or Ceramic finish; it is a view only, the real pieces stay underneath for tapping and snapping (krp-core puffed.js, src/app/soft-finish.js).
It began with two shapes DICTO found by building with their hands: a golden leaning hexagonal prism, and a skewed rhombic dodecahedron that fills space as a sheared FCC lattice (first notes in Polyhedraverse's docs/). Everything since grew from asking what else such shapes can do.
DISCOVERIES.md — what has been found so far, how each is verified, and whether it is known.TARGETS.md — the target list: the 160 equal-edge space-filling cells whose edges meet only at special angles, searched both ways (discover.py, discover_run.py, output data/geometry-targets.json). All 160 can be seen in the app's Targets world (DICTO → 3D+), alone, with their face neighbours or as a 3×3×3 block; scripts/verify-targets.mjs rebuilds each from its data and checks it tiles space.enumerate.py — the earlier, smaller list (targets.json), all included in the new one.assets/brand/ — the logo, with the name (kaleidohedra-logo.jpg) and without (kaleidohedra-mark.jpg), plus the site icons and preview image, drawn as an orange wireframe of the same shape. The logo shows the regular-hexagon elongated dodecahedron (DISCOVERIES.md #5), which is also the app's live symbol: it turns as a wireframe on Kaleidohedra's welcome screen.
Findings and attribution
Geometry found by DICTO in this project, each checked numerically by a script in CI and dated by
the commit that first recorded it (full list, verification and literature status in DISCOVERIES.md):
| Finding | First recorded | Literature status |
|---|
| DICTO skewed rhombic dodecahedron (60° and 72° rhombi, tiles as a sheared FCC) | 2026-10-01 | not found — candidate |
| Regular-hexagon elongated dodecahedron (4 regular hexagons, 4 squares, 4 60° rhombi), reached independently by a new route (the Bain stretch) | 2026-10-01 | known — the truncated octahedron with one zone removed (Grünbaum 2010) |
| DICTO skewed elongated dodecahedra (two forms) | 2026-10-01 | not yet searched |
| Euclid–Kepler–Pacioli cell: cube, dodecahedron, icosahedron, great stellated dodecahedron, octahedron, stella octangula and Pacioli's golden rectangles in one cell, 13 nodes, Pm-3 | 2026-10-06 | not found — candidate |
| Euclid–Kepler–Pacioli network: Kepler's great stellated dodecahedra and icosahedra sharing only corners | 2026-10-06 | not found — candidate |
| EKP windows: carving the six neighbouring stella octangulas out of the dodecahedron leaves 12 Penrose thick rhombi (72°) at its own face angles | 2026-10-08 | not found — candidate |
| Stretched dodecahedron: 8 regular pentagons, 4 hexagons and 2 rectangles at any stretch, squares at one edge | 2026-10-08 | not found — candidate |
| Study of the windows (10a): pushed out by √(7 − 4φ), a convex 74-face solid of 12 Penrose thick rhombi, 6 golden rhombi, 8 equilateral triangles and 48 triangles; buildable from a flat net | 2026-10-08 | study of the windows, not a separate claim |
"Not found" means a web-level search turned up nothing, not proof of novelty. Cite as
DICTO, Kaleidohedra discoveries, #N (first recorded date), https://doi.org/10.5281/zenodo.23173809. Each release is archived on Zenodo under that DOI; the latest, v2026.10.09-octet, adds #14 DICTO-Star, #15 the DICTO Jewel clusters and #16 the Sunstar clusters; v2026.10.09-cluster added #13, the 13-dodecahedron cluster made solid; v2026.10.08-star-chain #12, the Star Chain Reaction, and study 12a, the Jewel chain (archived as the Dragon chain); v2026.10.08-checkerboard added study 10b; v2026.10.08-convex the windows made convex (now study 10a); v2026.10.08 #10 and #11.
Shared geometry
The shared geometry (lattices, cells, quasicrystals, rhombic dodecahedron pieces) lives in
krp-core, shared with Rhombiverse and pinned here as a git
submodule at src/krp-core. Clone with it:
git clone --recurse-submodules https://github.com/DICTOR-Master/kaleidohedra
# or, in an existing clone: git submodule update --init
