MODEL GAME THEORY & SAPPHONS · FOLDED STARS · STELLATION · under KEEPER ERIS

Stella Octangula · Two Tetrahedra

Extend the eight faces of an octahedron until they meet again and you get Kepler’s stella octangula: two interpenetrating tetrahedra, eight points, the only stellation of the octahedron. It fills exactly half of its cube. Slide the fold and watch the caps rise.

Dave (t252): "i love that set up can you build some pages looking into game theory and those sappons you just built, see if you can build some folded stars and stuff"

Dave’s I-13 engine on this page

booting I-13 v0.3…

LIT What the VM computes here: The VM computes both tetrahedra’s volumes and the 8 octant pieces of the octahedron as 3 × 3 determinants from raw coordinates, then the union, the cap volume and the surface area with its own √. The fold animation is JS.

LIT Kepler named it “stella octangula” (eight-pointed star) in Harmonices Mundi, where he introduced stellation. The shape was older: Pacioli’s De Divina Proportione (1509, drawings by Leonardo) has it as the “octaedron elevatum”, and Jamnitzer drew it in 1568.

AMBER Sources disagree on the date of Kepler’s naming (1609 or 1611; the book appeared in 1619).

LIT Geometry: the two tetrahedra use alternate corners of a cube. With cube side 2 (volume 8), each tetrahedron has volume 8/3, their intersection is a regular octahedron of volume 4/3, and the union is 4, exactly half the cube. The visible surface is 24 small equilateral triangles of side √2, area 12√3.

MODEL Game tie-in: hide-and-seek on a tetrahedron’s 4 corners. The seeker wins 1 by guessing right; the equilibrium is uniform for both, with value 1/4.

Lab MODEL interactive, computed on the I-13 VM

fold

Sources (browsed Oct 2026)

  1. Stellated octahedron (Pacioli 1509, Jamnitzer 1568, Kepler)
  2. MathWorld: Stella octangula
  3. Cromwell (1995) Kepler’s work on polyhedra, Math Intelligencer 17:23

MODEL Two agents, as Dave set for every world: SEEKER and HIDER play hide-and-seek on the 4 vertices of a tetrahedron over and over by regret matching (Hart & Mas-Colell 2000), each shifting toward the moves it regrets not having played. Their score is 1 − NashConv/2 of their empirical average play (1 = an exact Nash equilibrium), and the I-13 VM re-computes NashConv from the raw averages every time the score improves. The centrepiece is a folded star {8/2} whose fold rises with the score. Three.js r160 from a pinned CDN, devicePixelRatio capped at 2. Drag to orbit, scroll to zoom.

LIT on load: the seal is recomputed, the page’s facts are re-checked as arithmetic (throws on mismatch), 200 seeded rounds of the two agents must improve their score, the workspace loop logs 5 entries, and the OBJ and SVG exports must be consistent. Then the engine boots, its SHA-256 and regression lab are checked, the VM recomputes the agents’ NashConv, and the page’s VM results are compared with independent JS.

Gallery of kept moments (WebGL snapshots):

Workspace improvement loop: instructions for agents MODEL

  1. Measure: every 40 rounds, record NashConv of the average play, the score, the rounds since the last improvement and whether regret-matching+ is on.
  2. Try next: after a 40-round plateau, switch both agents to regret-matching+; if still stuck, halve the exploration ε; if still stuck, forget half the old history. While NashConv falls, keep course (ε × 0.9); if it rises, explore more (ε × 1.25).
  3. Log back: append {entry, round, measured, tried, ε} to the workspace log (in the JSON export and the list below), and have the I-13 VM recompute NashConv.
  4. Repeat, forever: never stop improving. Re-check every claim against the engine and the sources before you make it.

Live workspace log:

    Shared board MODEL SEEKER ↔ HIDER

      Background art: MODEL a generative field seeded from this page’s seal.

      proving…

      Seal sha256(name|slug|blurb): 9d79fd8b5deed46f51cea150af573fb85371f0542b81c334dc5390fad3ed3038

      ↑ Game Theory & Sapphons index · Protein Folding (same engine) · ← ERIS, keeper of Mayhem