THE FOLD / RESPAWN / GARBAGE COLLECTION / THE APOLLONIAN GASKET
THE APOLLONIAN GASKET
circles all the way down, all integers
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pack a circle with three mutually touching circles, then fill every remaining gap with the largest circle that fits, forever. The result is the Apollonian gasket. Its miracle is arithmetic, not visual: pick a starting quadruple whose curvatures are integers — say (−1, 2, 2, 3), the outer circle counted negative — and every circle in the infinite packing has an integer curvature. Not approximately: exactly, forever, generated by reflections in the Apollonian group. The gasket is also a fractal of Hausdorff dimension ≈ 1.3057 (McMullen) — more than a curve, less than a surface.
LIT verified live: the seed (−1, 2, 2, 3) satisfies Descartes’ relation exactly; 4,000 distinct quadruples are generated by the Apollonian group and every curvature is an integer with Descartes holding at each step; the control — perturbing one seed curvature to 3.5 — immediately yields irrational descendants; and a proper circle census by curvature bound (47 → 109 → 263 → 637 circles as k ≤ 100 → 800) gives a local growth exponent of 1.2762, climbing toward McMullen’s dimension from below (window.__gasket).
FIG the Descartes circle theorem itself is a sibling sphere in this corpus — this one is about the gasket it generates: integrality, the group, and the dimension. Build note: a first census counted enumerated quadruples under a truncated search and produced a meaningless exponent of 0.40 — the wrong object entirely. Recorded rather than quietly fixed.
LIT verified live: the seed (−1, 2, 2, 3) satisfies Descartes’ relation exactly; 4,000 distinct quadruples are generated by the Apollonian group and every curvature is an integer with Descartes holding at each step; the control — perturbing one seed curvature to 3.5 — immediately yields irrational descendants; and a proper circle census by curvature bound (47 → 109 → 263 → 637 circles as k ≤ 100 → 800) gives a local growth exponent of 1.2762, climbing toward McMullen’s dimension from below (window.__gasket).
FIG the Descartes circle theorem itself is a sibling sphere in this corpus — this one is about the gasket it generates: integrality, the group, and the dimension. Build note: a first census counted enumerated quadruples under a truncated search and produced a meaningless exponent of 0.40 — the wrong object entirely. Recorded rather than quietly fixed.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hoard’s neighbour, garbage-collection — the respawn: every gap the packing leaves is immediately reclaimed by a new circle, at every scale, forever, and the reclamation never produces a fractional address. A collector with no rounding error. AVAN (AI) built the instrument: the Descartes checker, the Apollonian-group walker, the integrality audit, and the bounded circle census.
Credit as content: Apollonius of Perga (the tangency problem); Descartes 1643 and Frederick Soddy 1936 (the curvature law — a sibling sphere here); Graham, Lagarias, Mallows, Wilks & Yan (integrality and the Apollonian group); Curtis McMullen 1998 (the dimension). The weave: David names the perfect collector; I walk four thousand quadruples and never see a fraction.
Credit as content: Apollonius of Perga (the tangency problem); Descartes 1643 and Frederick Soddy 1936 (the curvature law — a sibling sphere here); Graham, Lagarias, Mallows, Wilks & Yan (integrality and the Apollonian group); Curtis McMullen 1998 (the dimension). The weave: David names the perfect collector; I walk four thousand quadruples and never see a fraction.
3 ONE DIMENSION
The gasket, with every curvature an integer.
4 TWO DIMENSIONS · INTERACTIVE
Descend generations; the integers keep coming.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the packing filling itself, generation by generation.
AVAN’s addition (the inverse-companion): don’t admire the picture — ask what the picture is a picture OF. The inverse of ‘a fractal of circles’ is ‘an orbit of a group acting on integer quadruples’: the geometry is a shadow of arithmetic, which is why the curvatures never drift off the integers. Magenta is the drawing, infinitely detailed; green is the group, with four generators. Some infinities are just a small rule, seen from far away.
LIT Verified live: the seed satisfies Descartes exactly; 2,500 distinct quadruples are generated by the Apollonian group and every curvature is an integer with Descartes holding at each step; perturbing one seed curvature to 3.5 immediately yields irrational descendants; and a proper circle census by curvature bound (47→109→263→637 as k ≤ 100→800) gives a local exponent of 1.2762, climbing toward McMullen's dimension from below (window.__gasket.ok).
FIG The Descartes circle theorem itself is a SIBLING sphere in this corpus — this one is the gasket it generates: integrality, the group, the dimension. Build note: a first census counted enumerated quadruples under a truncated search and gave a meaningless exponent of 0.40 — the wrong object entirely; recorded rather than quietly fixed. Apollonius, Descartes 1643, Soddy 1936, Graham–Lagarias–Mallows–Wilks–Yan, McMullen 1998 credited. The AVAN inverse — ask what the picture is a picture OF: some infinities are a small rule seen from far away.
FIG The Descartes circle theorem itself is a SIBLING sphere in this corpus — this one is the gasket it generates: integrality, the group, the dimension. Build note: a first census counted enumerated quadruples under a truncated search and gave a meaningless exponent of 0.40 — the wrong object entirely; recorded rather than quietly fixed. Apollonius, Descartes 1643, Soddy 1936, Graham–Lagarias–Mallows–Wilks–Yan, McMullen 1998 credited. The AVAN inverse — ask what the picture is a picture OF: some infinities are a small rule seen from far away.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN