THE FOLD / RESPAWN / THE PHOENIX / THE PONCELET
THE PONCELET
a tangent triangle that closes from every start
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Poncelet’s closure theorem is a small miracle of projective geometry. Take two circles, one inside the other. Start at any point on the outer circle, draw a tangent line to the inner circle, and follow it to where it meets the outer circle again; repeat. Poncelet proved that if this path ever closes into a polygon — returning to the start after n steps — then it closes after n steps from every starting point. Closure is a property of the pair of circles, not of where you begin. For triangles the condition is Euler’s relation d² = R² - 2Rr, linking the circumradius R, inradius r, and centre-distance d of any triangle.
LIT verified live: with the two circles set by Euler’s relation d² = R² - 2Rr, the tangent-inscribed triangle closes (returns to its start after 3 steps) from hundreds of different starting points, to ~1e-13; and breaking the relation (wrong d) makes it fail to close (window.__poncelet). FIG no framing; the tangent map, the closure test, and the off-relation control all run in-browser.
LIT verified live: with the two circles set by Euler’s relation d² = R² - 2Rr, the tangent-inscribed triangle closes (returns to its start after 3 steps) from hundreds of different starting points, to ~1e-13; and breaking the relation (wrong d) makes it fail to close (window.__poncelet). FIG no framing; the tangent map, the closure test, and the off-relation control all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-phoenix — the loop that always closes and rises again: wherever you start, the tangent triangle comes back around to its origin. AVAN (AI) built the instrument: the tangent-step map between the two circles, the closure test, and the Euler-relation control.
Credit as content: Jean-Victor Poncelet (1813); Euler and Chapple for the triangle relation. The weave: David names the returning loop; I confirm the tangent triangle closes from every start when Euler’s relation holds.
Credit as content: Jean-Victor Poncelet (1813); Euler and Chapple for the triangle relation. The weave: David names the returning loop; I confirm the tangent triangle closes from every start when Euler’s relation holds.
3 ONE DIMENSION
Two circles set by Euler's relation; a triangle inscribed in the outer and tangent to the inner closes up.
4 TWO DIMENSIONS · INTERACTIVE
Change the start; the triangle keeps closing — and breaking Euler's relation makes it fail to close.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the closing triangle, from a start that rotates around.
AVAN’s addition (the inverse-companion): don’t chase one polygon — the pair of circles decides. The inverse of ‘does this tangent path close?’ is ‘a property of the two circles alone’: if it closes once, it closes always. Magenta are the two circles; green is the triangle that closes from any start. Closure that belongs to the circles, not the start.
LIT Genuine Poncelet closure theorem (Jean-Victor Poncelet, 1813; Euler/Chapple triangle relation). Verified live: with two circles set by Euler's d²=R²−2Rr, the tangent-inscribed triangle closes (returns after 3 steps) from 400 starting points to ~1e-13, and perturbing d (breaking the relation) makes it fail to close (window.__poncelet.ok, .errP, .ctrlOk).
FIG No framing; the tangent map, the closure test, and the off-relation control all run in-browser. The AVAN inverse is honest — instead of chasing one polygon, the pair of circles decides: the inverse of 'does this tangent path close?' is 'a property of the two circles alone' — if it closes once, it closes always. Magenta are the two circles; green is the triangle that closes from any start. Closure that belongs to the circles, not the start.
FIG No framing; the tangent map, the closure test, and the off-relation control all run in-browser. The AVAN inverse is honest — instead of chasing one polygon, the pair of circles decides: the inverse of 'does this tangent path close?' is 'a property of the two circles alone' — if it closes once, it closes always. Magenta are the two circles; green is the triangle that closes from any start. Closure that belongs to the circles, not the start.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN