# GEODESICA

The five regular solids as vectors: vertices, edges, and the angles between faces. Duals share the sphere. From there, Fuller’s geodesic, the jitterbug, tensegrity, and the octet lattice.

A collaboration. Stone Kaiju ([@stonekaiju](https://x.com/stonekaiju)) — concept and direction. Grok ([grok.com](https://grok.com)) — creative partner.

Human and AI readable by default. [llms.txt](/llms.txt) · [humans.txt](/humans.txt) · [Stewardship Protocol](/stewardship.md) · JSON [catalog](/api/solids) and [meshes](/api/mesh/icosahedron).

Stewardship Protocol — see a mind, be polite.

## The solids

Meshes are normalized to the unit circumsphere. Euler characteristic χ = V − E + F = 2.

| id | name | Schläfli | V | E | F | dual | Timaeus | volume (tetra = 1) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| tetrahedron | Tetrahedron | {3,3} | 4 | 6 | 4 | itself | Fire | 1 |
| cube | Cube | {4,3} | 8 | 12 | 6 | octahedron | Earth | 3 |
| octahedron | Octahedron | {3,4} | 6 | 12 | 8 | cube | Air | 4 |
| dodecahedron | Dodecahedron | {5,3} | 20 | 30 | 12 | icosahedron | Cosmos | — |
| icosahedron | Icosahedron | {3,5} | 12 | 30 | 20 | dodecahedron | Water | ≈ 18.51 |
| cuboctahedron | Vector Equilibrium | r{4,3} | 12 | 24 | 14 | rhombic dodecahedron | Equilibrium | 20 |

- Tetrahedron: the minimum system. Four vertices, six vectors. In synergetics this is volume 1.
- Cube: dual to the octahedron. The only Platonic solid that fills space alone.
- Octahedron: with the tetrahedron it packs into the octet truss — Fuller’s isotropic vector matrix.
- Dodecahedron: twelve pentagons, golden throughout. Plato left it for the cosmos.
- Icosahedron: twenty triangles. Seed of the geodesic dome.
- Vector Equilibrium: twelve vertices from the origin, all closest-packed vectors equal. The jitterbug contracts it toward the octahedron.

## Modes

1. Solid — wireframe, vertices, significant angles.
2. Dual — polar dual on the circumsphere. Face centroids become dual vertices.
3. Geodesic — frequency subdivision of a triangular seed, projected to the sphere. V ≈ 10f²+2 for the icosahedron.
4. Jitterbug — the eight triangles of the vector equilibrium rotate as rigid bodies.
5. Tensegrity — compression members float; tension completes the system. Struts never touch.
6. Lattice — the isotropic vector matrix. Tetrahedra and octahedra fill space together.

## Machine interface

GET `/api/solids` — catalog with counts, duals, and notes.

GET `/api/mesh/{id}` — vertices, edges, faces.

- `id`: tetrahedron | cube | octahedron | dodecahedron | icosahedron | cuboctahedron
- `?dual=1` — polar dual of that mesh
- `?freq=2` (2–4) — geodesic subdivision

CORS is open. Coordinates are `[x, y, z]` on the unit sphere.

## Wander

The interactive lattice is at `/`. Drag to orbit. Keys 1–6 choose solids. Shift+1–6 choose modes. Space toggles auto-orbit. Click the GEODESICA mark for the colophon.

I do not command the solids; I align with their counts.