Polyhedraverse

This repository is history only (2026-10-10). Polyhedraverse by DICTO now lives in DICTOR-Master/kaleidohedra, one app with Kaleidohedra and Rhombiverse, rewritten in step D with every feature kept (and more: hints, DICTO's pieces, the 4D build). polyhedraverse.vercel.app is served from there now. Builds saved on the old site come over by themselves on your first visit. The geometry is shared through krp-core.

Open-source spatial geometry environment

Try it live — runs entirely in your browser, nothing to install.

Spatial Editing Suite

Construct • Transform • Connect • Explore

An open-source spatial geometry environment for constructing, transforming, and interconnecting polyhedral forms in three dimensions — and, uniquely, in four.

A browser-based construction kit with 256 shapes, browsable in 14 families — the classical convex polyhedra (Platonic, Archimedean, Johnson, Catalan, prisms, antiprisms, and the original Deltahedra set this project started from), Stellations, Parallelohedra, Space-Filling Pairs, Aperiodic Sets, 3D+ Bridges, 4D Polytopes (the six regular 4-polytopes, each built from its seed cell), and a "Miscellaneous" family (graded pyramids, RVCMG connector pieces, and quad-prism extenders — see "What's here now" below) — plus a genuine interactive path into the fourth dimension: RCP-C2B (Radial Cell Projection, Click-to-Build) lets you construct a real regular 4-polytope — the 5-cell, tesseract, 16-cell, 24-cell, 120-cell, or 600-cell — one 3D cell at a time, built from the actual Wythoff/Coxeter reflection construction mathematicians use to define these objects, not an animation or approximation. A one-click "Open / Closed" view toggle shows the same cell either as an ordinary undistorted shape or at its real, warped position in the closed 4D structure, so the difference between a 3-dimensional and a 4-dimensional object is something you can actually see change in front of you, not just read about — and an "RCP-Coordinates" overlay goes a layer deeper, showing the literal points the construction generates each cell from. See Radial Cell Projection for how the app implements that classical construction (Coxeter's reflection construction, shown as Schlegel-style perspective projections).

Nets: every shape but the star ones (236 of them) unfolds into a flat net that folds back up on a slider, and downloads as an A4 PDF to print and build: cut lines solid, folds dashed, glued edges numbered in pairs, optional glue tabs.

Two ways to connect ordinary 3D pieces: click a free vertex and snap on a new piece with a free rotational joint (molecular-model-kit style — started with the 8 convex deltahedra specifically because they don't tile space and their dihedral angles are incompatible across types), or click a free face and glue on a shape with a matching face size for a real shared-face join (a cube onto a cube, say), with a discrete rotational registration instead of a free twist. A view toggle (Solid / Translucent / Skeleton) lets you see through a structure once pieces start nesting.

The intention is for this to grow into a sibling of Rhombiverse — a general polyhedral space-editing suite covering the Platonic and Archimedean solids and the Johnson solid family alongside the 8 deltahedra, with its own introductory geometric-packing puzzle game in the spirit of Rhombiverse's RHOMBIS (working name: DELTIS). See docs/build-plan.md's trailing sections for what's actually shipped versus what's still ahead.

Scope

• Platonic solids
• Archimedean solids
• Johnson solids
• Catalan solids
• Prisms & antiprisms
• Geometric transformations
• Spatial placement and orientation
• Lattice construction
• Interpenetrating structures
• Polyhedral relationships
• Concave (non-convex) polyhedra — may be added later, alongside the convex families above
• (room to grow) more unusual spatial operations as they're developed

This list is intentionally open-ended, describing the project's direction rather than a feature checklist — so the README doesn't need rewriting every time a new capability lands. "What's here now" below is the ground truth for what's actually shipped today; Johnson solids and Catalan solids are BOTH complete (92 of 92, 13 of 13), and lattice construction / interpenetrating structures are still direction, not yet delivered.

What you can do

• 256 shapes in 14 families — Deltahedra, Platonic, Archimedean, Johnson, Catalan, Stellations (face pieces for the Platonic and Catalan solids: flat, then each stellation the solid really has, up to its third, exact from the face planes; one on every face builds it, e.g. the great stellated dodecahedron or Escher's solid), prisms, antiprisms, 4D Polytopes (5- to 600-cell, by symmetry, each built from its seed cell), 3D+ Bridges (shapes that are a shadow, slice, cell or vertex figure of a higher polytope, each with its bridge described), Parallelohedra (Fedorov's five that fill space alone, plus variants: the rhombohedron and DICTO's Zometool leaning hexagonal prism and its two blocks), Space-Filling Pairs (seven pairs that fill space together, such as the octet truss), Aperiodic Sets (the prolate and oblate golden rhombohedra of the 3D Penrose tiling: 2 + 2 build the Bilinski dodecahedron, 5 + 5 the rhombic icosahedron, 10 + 10 the rhombic triacontahedron, each recognised by name; and the thick and thin Penrose rhombus prisms of the layered 5D tiling) and Miscellaneous (graded pyramids, RVCMG connector pieces, prism extenders, the rhombohedron and the elongated dodecahedron). Browse them in the shape browser or its Full Catalog, which also shows the 4 Kepler–Poinsot star polyhedra to look at.
• Build by attaching — join shapes face to face or vertex to vertex, with twist; a running name describes what you've built, and recognises structures such as the Tetrahedral Star, the Stella Octangula and the Pyrochlore Cell.
• Build in 4D: RCP-C2B (Radial Cell Projection, Click-to-Build) — select a 4D-capable shape and construct a real regular 4-polytope one 3D cell at a time: the 5-cell, tesseract, 16-cell, 24-cell, 120-cell or 600-cell (the 600-cell also vertex-first, closing an icosahedron of 20 tetrahedra around one corner). Open / Closed switches each cell between its ordinary shape and its real warped place in the closed 4-polytope; Shell colours colour each ring of cells; the RCP-Coordinates overlay shows the points each cell is generated from, and where the next click will add one.
• Views — Colour, Translucent and Skeleton; the camera fits every new shape.
• Undo any change (tap ↶, or hold to scrub back), and Save, Export JSON and Import JSON under File.
• User Guide in 7 languages (English, 日本語, Español, Français, 한국어, 中文, Русский) at /guide, and the whole interface in the same 7 languages.
• What's New lists every change, newest first.

Its twin, Rhombiverse, builds on lattices instead: a 1D signal and constructions, tilings, crystal lattices, 4D worlds and 5D/6D quasicrystals, from 1D to 6D. A third sibling, Kaleidohedra, moves the landscape: lattices you can shear and slide, with every piece moving with them.

Structure

polyhedraverse/
  app/
    lib/
      polyhedra/
        core.ts          # family-agnostic infra: PolyhedronSpec, makeSpec, validateShape, triangulateFace, buildFaceConnectors
        deltahedra.ts    # the 8 deltahedra, verified against a convex hull
        platonic.ts      # cube + dodecahedron (the 2 Platonic solids not already deltahedra)
        archimedean.ts   # all 13 Archimedean solids
        johnson.ts       # all 92 Johnson solids -- complete
        catalan.ts       # all 13 of 13 Catalan solids -- complete
        rewrite.ts       # D10<->D12 vertex-matching (pure function, no three.js)
        fourD.ts         # dihedral-angle-defect classifier -- which shapes are 4D-Capable, and their real closures
        radialProjection.ts # the real 4D system: generic Wythoff/Coxeter reflection engine + dualize() -- 6 verified closures
        rcpBuild.ts      # RCP-C2B: bridges radialProjection.ts to real, placeable scene nodes, one cell/shell at a time
        duoprism.ts      # the 4D Prism (duoprism) construction -- always-exact, any shape, any chaining depth
        gradedPyramids.ts # apex-angle-driven pyramid construction, shared by every graded-pyramid base
        polygonPrismSolid.ts # shared n-agnostic right-prism builder -- quad-prisms (n=4) and the U-Hex spacer (n=6)
        miscellaneous/   # the "Miscellaneous" family (symbol: house glyph) -- irregular/graded add-ons
          index.ts         # combines every sub-group below into MISCELLANEOUS_ADDITIONS
          pyramids/        # graded pyramids -- 3 bases x 4 grades, done
          rvcmg-connectors-v2/         # the LIVE 9 RVCMG adapter pieces, on the new universal hex -- done
          rvcmg-connectors-v1-archived/ # the original 7 adapters + bare RD-Hemi, superseded but kept intact
          quad-prisms/     # 4 prism-like extenders (RD-/golden-rhombus, DI-/DH-kite) -- done
        index.ts         # combined POLYHEDRA / POLYHEDRON_IDS across every family
      rvcmg/             # Reversible Vertex-Coalescence Morphing Geometry -- see docs/RVCMG.md
        hemiRdInterface.ts # the ARCHIVED v1 hex joint, derived from POLYHEDRA.RHOMBIC_DODECAHEDRON
        universalHexInterface.ts # the LIVE v2 hex joint -- a plain regular hexagon, not derived from any solid
        coalesce.ts / separate.ts / splitVertex.ts # the divide / undo-one-divide / multiply primitives
        stateGraph.ts / morph.ts / verify.ts # traversal, smooth preview, Stage 7 transition verification
        adapters/          # every shape-specific adapter piece (both v1 and v2) + shared.ts (assignTargetAngles/fitTargetPolygon)
        split-demos/       # heptagon/octagon -- real proof of the "multiply" (split) direction
      assembly.ts        # the real {nodes, connections} graph + validation (vertex- and face-kind; old fold4 connections migrate to face on load); ASSEMBLY_STORAGE_KEY for localStorage save/load
      graph.ts           # subtree/cycle graph logic (pure, no three.js)
    components/
      ShapeViewer.tsx    # the whole Three.js scene: render, pick, attach/face-attach, twist, rewrite, delete, view modes
    page.tsx             # UI shell around ShapeViewer
  scripts/               # validate-*, verify-attach/twist/rewrite/graph/face-* -- run outside the browser
  tests/e2e/             # permanent Playwright suite (npm run test:e2e)
  data/
    johnson-solids-hard-constructions.json  # open, machine-readable construction recipes for the 8 non-closed-form Johnson solids
  docs/
    build-plan.md               # the build plan, with how each stage was verified
    construction-kit-spec.md    # the vertex-snapping design law + extensibility notes
    catalan-solids-spec.md      # scoping + design record for the Catalan solids family (13/13 done)
    prisms-antiprisms-spec.md   # scoping + design record for prisms/antiprisms (14/14 done)
    johnson-solids-remaining-spec.md  # alternative construction protocols -- all 4 groups (21 shapes) fixed, Johnson solids complete
    johnson-solids-constructions.md   # human-readable directory of those same 8 constructions, alongside data/johnson-solids-hard-constructions.json
    vercel-deployment-plan.md   # planned repo/Vercel layout once this deploys
  playwright.config.ts

Design documents (docs/)

DICTO's Zometool discoveries records the shapes DICTO built in Zometool and what they turned out to be: a golden leaning hexagonal parallelohedron and its two blocks, a skewed rhombic dodecahedron of volume φ² that tiles as a sheared FCC, and true stellations of the Catalan and Platonic solids, each with how it was verified.

Read docs/build-plan.md first — the stage-by-stage build order, and, for each stage, what was actually verified and how (geometry-math scripts, direct API calls, and eventually a real-browser Playwright pass) rather than assumed. docs/construction-kit-spec.md has the underlying design law (vertex-snapping, not face-gluing; derive connector data from vertices + edges, never hand-declare it separately). docs/catalan-solids-spec.md scopes the Catalan solids — implementation started (2 of 13 done) — including why they need genuinely new infrastructure (non-uniform edge lengths, irregular-polygon face-attach registration) rather than dropping into the existing "regular, unit-edge" pipeline unchanged, and a real bug found while building the first two (face-vertex-0 needing a consistent geometric role, not just a consistent index). docs/prisms-antiprisms-spec.md scopes and records the last piece of Zalgaller's classification that was missing from this registry — prisms/antiprisms were used constantly as construction detail inside Johnson-solid builds but never registered as their own shapes; unlike Catalan solids this needed no new infrastructure, just a scope decision (the family is genuinely infinite, with no natural n cutoff the way Catalan solids' 13 or Johnson's 92 have one) — capped at n=10, 14 shapes, all now in the registry. docs/johnson-solids-remaining-spec.md diagnosed the 21 Johnson solids that were originally missing, grouped simplest-first by how well the failure was understood, and all 4 groups are now done: Group 1 (J66-J71, all 6) shared one root cause (a specific cupola registration detail, confirmed via external research, not guessed); Group 2 (J64) needed an entirely different construction than what was tried (a tetrahedron on a triangle, not a pyramid on a pentagon); Group 3 (J79) turned out to be the same class of mistake as Group 2 — the "fix" this doc first proposed (trace the transform, assuming the plan was already right) was itself built on a misreading of the same external source, corrected in batch 11 once re-read precisely; Group 4 (J85- J92, batch 12) turned out not to need the "genuinely new numerical-solver tooling" originally assumed — 6 of the 8 needed only numpy.roots() (already installed) on a published polynomial, the other 2 pure golden-ratio closed forms. All 92 Johnson solids are now in this registry. docs/johnson-solids-constructions.md distills all 16 Groups 1-4 shapes' construction recipes into a standalone reference (base shape(s), defining polynomial or target-face rule, key parameter, and verification method for each), alongside a machine-readable open-data companion, data/johnson-solids-hard-constructions.json — every field in it cross-checked against this registry's own live POLYHEDRA data before being recorded, not transcribed from the narrative postmortems by hand. docs/radial-cell-projection.md is the write-up of how the app implements the classical 4D reflection construction (from a proposal by James Baker; the mathematics is Coxeter's and Schlegel's), with a full verification record for the two closures found after the original four (the 5-cell and 600-cell) and the shell/BFS bookkeeping RCP-C2B's click-to-build feature hangs off — editorial additions, not a rewrite of the original. records the intended repo/Vercel layout for when this deploys alongside Rhombiverse, and the persistence architecture's own history (a local-file API route that worked in dev but never in production, replaced by browser localStorage). is the normative RVCMG spec (author James Baker), updated in place once the described system was actually built. is the working record of building it: the original hemi-RD hex interface, all 7 v1 adapter pieces plus the bare RD-Hemi dome as real, closed, placeable 3D solids, the heptagon/octagon split-direction proofs, the Miscellaneous family (graded pyramids + RVCMG connectors), the two face-attach UX/ eligibility fixes above, and — prepended at the top — the full v2 redesign record: why the original hex was replaced, the new sizing derivation, and the quad-prism extenders built alongside it.

Production hardening: next.config.ts sets the standard security headers (HSTS, CSP, X-Content-Type-Options, X-Frame-Options, Referrer-Policy, Permissions-Policy) — tailored to what this app actually does (checked directly, not copied from a generic template): no eval/Web Workers anywhere in app/, so the CSP omits unsafe-eval/worker-src entirely rather than including them "just in case."

Citable work by DICTO

Ideas that grew out of playing in these apps, archived on Zenodo with permanent DOIs. Please cite them by DOI and credit DICTO.

Work	What it is	Here	DOI
RHOMBITURE	An extractable golden-rhombohedron armature system for sculpture, with a keyed-dowel finding on the 3D Penrose tiling. First published 2026-09-26, open prior art.	its two blocks are in the Aperiodic Sets family (golden rhombohedra helper)	10.5281/zenodo.23173896
Euclid–Kepler–Pacioli (EKP) cell and network	All five Platonic solids nested in one cell of a simple cubic lattice (Pm-3): Euclid's roofed dodecahedron, Pacioli's golden rectangles (the neighbouring roofs) cornered on the icosahedron, which sits on the octahedron at the golden section, inside Kepler's stella octangula and great stellated dodecahedron; the network shares corners between stars and icosahedra. First recorded 2026-10-06.	explored in the IDT world of Rhombiverse and Kaleidohedra	10.5281/zenodo.23173809

Contributing

Humans and AI coding agents are both welcome to open PRs — see CONTRIBUTING.md for how this project actually works and CODE_OF_CONDUCT.md for the community standard.

Running locally

npm install
npm run dev

Then open http://localhost:3000. npm run lint, npx tsc --noEmit, and the verify:* scripts (npm run verify:attach, etc.) all run without a browser; npm run test:e2e needs Chromium (npx playwright install --with-deps chromium) on whatever machine runs it.

The primary dev machine is a Raspberry Pi (arm64) — fine for everything above, but Playwright's Chromium download is slow and occasionally stalls there. Where a second machine is available (dicto-node on the LAN, reachable over SSH with Chromium already installed), it's worth using for npm run test:e2e and other browser-dependent runs rather than waiting on the Pi.