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

Folded Paper Stars · Lucky Stars & Froebel

Tie a paper strip in an overhand knot, flatten it, and the knot is a regular pentagon. Wrap the rest of the strip around it and pinch the edges, and it puffs into a lucky star. Four woven strips make a 16-pointed Froebel star. The I-13 VM computes the pentagon and the fold angles.

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"

LIT Lucky stars: start with a strip of about 1:13, tie a loose overhand knot, pull it snug and flatten it into a neat pentagon. Fold the short tail onto it, wrap the long tail around the pentagon edge by edge, tuck the end, then hold the pentagon by its edges and press each side in with a fingernail so it puffs into a five-pointed star.

MODEL Geometry of the knot (derived here): the strip runs along a side and the parallel diagonal of the pentagon, which are s·sin 72° apart, so a strip of width w makes a pentagon of side s = w / sin 72° ≈ 1.0515 w. The number of wraps is estimated as (13w − 2s)/s.

LIT Froebel star (Fröbelstern): woven and folded from four identical strips of about 1:25 to 1:30, in about forty steps. It ends with eight flat prongs and eight cone tips. Named after Friedrich Fröbel, founder of the kindergarten, who promoted paper folding, but he probably didn’t invent it. Instructions date to at least the 19th century (an 1885 Dorcas Magazine article), and the German name has been common since the 1960s.

MODEL Every page in this batch shows a folded star {n/k}: a flat star outline with an apex above and below, 2n + 2 vertices and 4n triangles (the OBJ export). Its fold height rises as the agents get closer to equilibrium.

MODEL Two agents, as Dave set for every world: FOLDER A and FOLDER B play two folders choosing valley or mountain creases (pure coordination) 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 {5/2} 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.

⏸ Pause Reseed Export training + workspace log (.json) Export star + trajectory (.svg) Export folded star (.obj)

Gallery of kept moments (WebGL snapshots):

Shared board MODEL FOLDER A ↔ FOLDER B

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

proving…

Seal sha256(name|slug|blurb): a55571831e11da41fe5002806867ff4a73064071eab40b09ffdebcf4f07e3d40

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