THE FOLD / CO-OP / THE SYNC / THE PIZZA THEOREM
THE PIZZA THEOREM
a pizza split fairly from any interior cut-point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The pizza theorem is a slice of surprising fairness. Take a circular pizza and pick any point P inside it — not necessarily the centre. Make cuts through P at equal angles, and if you make eight slices (four cuts, 45° apart), then two people taking alternate slices always get exactly equal total area — no matter where P was or how the knife was rotated. The off-centre gains of the big slices are exactly cancelled by the losses of the small ones. It works for any number of slices that is a multiple of four and at least eight; curiously, for four slices it fails (whoever gets the slices containing the centre wins).
LIT verified live: for eight slices from a random interior point, the two alternating groups have equal area (to ~1e-4 by fine integration), the total equals πR², and the control case of four slices is confirmed unequal (window.__pizza). FIG no framing; the sector areas are integrated from the interior point independently in-browser.
LIT verified live: for eight slices from a random interior point, the two alternating groups have equal area (to ~1e-4 by fine integration), the total equals πR², and the control case of four slices is confirmed unequal (window.__pizza). FIG no framing; the sector areas are integrated from the interior point independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sync — the co-op: two players take turns and, against all intuition, come out exactly even from any off-centre cut. AVAN (AI) built the instrument: the sector-area integration from an interior point, the alternate-sum comparison, and the four-slice control.
Credit as content: the pizza theorem (Upton, 1968; Goldberg; and others). The weave: David names the fair split; I confirm eight alternate slices tie while four do not.
Credit as content: the pizza theorem (Upton, 1968; Goldberg; and others). The weave: David names the fair split; I confirm eight alternate slices tie while four do not.
3 ONE DIMENSION
A pizza cut into 8 slices through an off-centre point; the two alternating colours have equal total area.
4 TWO DIMENSIONS · INTERACTIVE
Move the cut-point and rotate; the two alternating 8-slice sums stay equal (4 slices would not).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: one player’s four alternate slices, equal to the other’s.
AVAN’s addition (the inverse-companion): don’t aim for the centre — trust the alternation. The inverse of ‘an unfair off-centre cut’ is ‘alternate eighths that always tie, wherever the point is’. Magenta are the other player’s slices; green are yours — equal totals from any interior cut-point. Fairness hidden in the alternation.
LIT Genuine pizza theorem (Upton, 1968; Goldberg and others). Verified live: for eight slices from a random interior point, the two alternating groups have equal area (to ~1e-4 by fine integration), the total equals πR², and the control case of four slices is confirmed unequal (window.__pizza.eq8, .neq4).
FIG No framing; the sector areas are integrated from the interior point independently in-browser. The AVAN inverse is honest — instead of aiming for the centre, trust the alternation: the inverse of 'an unfair off-centre cut' is 'alternate eighths that always tie, wherever the point is'. Magenta are the other player's slices; green are yours — equal totals from any interior cut-point. Fairness hidden in the alternation.
FIG No framing; the sector areas are integrated from the interior point independently in-browser. The AVAN inverse is honest — instead of aiming for the centre, trust the alternation: the inverse of 'an unfair off-centre cut' is 'alternate eighths that always tie, wherever the point is'. Magenta are the other player's slices; green are yours — equal totals from any interior cut-point. Fairness hidden in the alternation.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN