MODEL GAME THEORY & SAPPHONS · FOLDED STARS · REGULAR STAR POLYHEDRA · under KEEPER ERIS

The Four Kepler–Poinsot Stars

There are exactly four regular star polyhedra. Kepler found the two stellated dodecahedra (1619), Poinsot added the great dodecahedron and great icosahedron (1809), and Cauchy proved there are no more (1813). All four are built here from the icosahedron’s and dodecahedron’s vertices, and the VM checks Euler and Cayley.

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 checks V − E + F and Cayley’s generalised formula for all four, computes φ with its own Newton square root (and φ² − φ − 1 = 0), and measures from the generated coordinates that the great icosahedron’s edge is φ times the icosahedron’s. JS finds the faces by searching vertex subsets for regular pentagons, pentagrams and triangles of the right edge length.

LIT Kepler, Harmonices Mundi (1619): the small stellated dodecahedron {5/2, 5} (his “urchin”) and the great stellated dodecahedron {5/2, 3}. Poinsot (1809): the great dodecahedron {5, 5/2} and great icosahedron {3, 5/2}. Cauchy (1813) proved the list complete by stellating the Platonic solids. Cayley (1859) gave the English names.

LIT Counts (V, E, F): small stellated dodecahedron and great dodecahedron (12, 30, 12), so V − E + F = −6. Great stellated dodecahedron (20, 30, 12) and great icosahedron (12, 30, 20), so 2. Cayley’s fix: d_v·V − E + d_f·F = 2D, where d_v and d_f are the densities of the vertex figure and face and D is the density of the polyhedron: 3, 3, 7, 7.

LIT The small stellated dodecahedron and the great dodecahedron share the icosahedron’s 12 vertices. The great icosahedron does too, with edges φ = 1.618… times longer. The great stellated dodecahedron sits on the dodecahedron’s 20 vertices.

AMBER Rock–paper–scissors–lizard–Spock is a popular five-move extension (made famous by The Big Bang Theory); its attribution was not re-browsed here.

MODEL The scene draws each face as a fan of triangles from its centre, so the self-intersections overlap rather than being cut. The look is right, but the interior isn’t trimmed.

Lab MODEL interactive, computed on the I-13 VM

Sources (browsed Oct 2026)

  1. MathWorld: Kepler–Poinsot polyhedron (history, Cayley 1859)
  2. Cauchy (1813) Recherches sur les polyèdres, English translation
  3. Cromwell (1995) Kepler’s work on polyhedra, Math Intelligencer 17:23
  4. Stellating and facetting: a brief history

MODEL Two agents, as Dave set for every world: KEPLER and POINSOT play rock–paper–scissors–lizard–Spock (its beats-graph is {5/1} + {5/2}) 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 {10/4} 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 KEPLER ↔ POINSOT

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

      proving…

      Seal sha256(name|slug|blurb): 999d93a64852ebf5bd6adb81991c12efae132ee225391f54e3340fd1471dad57

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