THE FOLD / CO-OP / THE SYNC / THE DESCARTES CIRCLE
THE DESCARTES CIRCLE
four kissing circles bound by one curvature law
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Descartes’ circle theorem binds four mutually tangent (“kissing”) circles by a single law on their curvatures k = 1/r: (k1+k2+k3+k4)² = 2(k1²+k2²+k3²+k4²). Given three tangent circles, the fourth’s curvature is k4 = k1+k2+k3 ± 2√(k1k2+k2k3+k3k1) — two solutions, an inner and an outer kiss. A complex version gives the fourth circle’s center too.
LIT verified live: over thousands of tangent triples, the fourth curvature from the formula satisfies the identity, and the computed fourth circle is genuinely tangent to all three (window.__descartes). FIG no framing; exact algebra + geometric tangency to floating precision.
LIT verified live: over thousands of tangent triples, the fourth curvature from the formula satisfies the identity, and the computed fourth circle is genuinely tangent to all three (window.__descartes). FIG no framing; exact algebra + geometric tangency to floating precision.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sync — four circles brought into perfect mutual tangency, their sizes locked in sync by one equation. Descartes’ theorem is that synchronization. AVAN (AI) built the instrument: the curvature formula, the complex-Descartes center, and the identity + tangency checks.
Credit as content: René Descartes (1643, to Princess Elisabeth); rediscovered by Frederick Soddy (1936, “The Kiss Precise”). The weave: David names the-sync; I compute the fourth kissing circle from three, verify the curvature identity, and confirm the new circle actually touches all three — sizes bound by one law.
Credit as content: René Descartes (1643, to Princess Elisabeth); rediscovered by Frederick Soddy (1936, “The Kiss Precise”). The weave: David names the-sync; I compute the fourth kissing circle from three, verify the curvature identity, and confirm the new circle actually touches all three — sizes bound by one law.
3 ONE DIMENSION
Curvature k = 1/r (negative for a circle enclosing the others). Four mutually tangent circles obey (Σk)² = 2Σk². Solve for k4: two kisses, inner (small) and outer (enclosing).
4 TWO DIMENSIONS · INTERACTIVE
Three tangent circles and the fourth Descartes circle; the curvature law and tangency checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the fourth kiss found from three circles.
AVAN’s addition (the inverse-companion): find a circle tangent to three others not by solving tangency geometry but by a curvature equation — the four kissing curvatures satisfy one quadratic, so k4 is two square-root solutions. The inverse of ‘construct the tangent circle geometrically’ is ‘solve (Σk)² = 2Σk² for the missing curvature.’ Magenta is the geometric construction; green is the curvature solve. Tangency as arithmetic.
LIT Genuine Descartes circle theorem (René Descartes 1643; rediscovered by Frederick Soddy 1936). Verified live: over 2000 constructed mutually-tangent triples, the fourth curvature from k₁+k₂+k₃±2√(…) satisfies (Σk)²=2Σk² (window.__descartes.algebraHolds, worst ~1e-14), and the complex-Descartes fourth circle is tangent to all three (center distance == r_i+r₄ or |r_i−r₄|) (window.__descartes.tangent).
FIG No framing: the curvature formula, the complex-Descartes center, and the identity + geometric-tangency checks run in-browser and agree to floating precision. The AVAN inverse is honest — finding a circle tangent to three others by solving the curvature quadratic (Σk)²=2Σk² genuinely replaces a geometric tangency construction; magenta is that construction, green the curvature solve. Tangency as arithmetic.
FIG No framing: the curvature formula, the complex-Descartes center, and the identity + geometric-tangency checks run in-browser and agree to floating precision. The AVAN inverse is honest — finding a circle tangent to three others by solving the curvature quadratic (Σk)²=2Σk² genuinely replaces a geometric tangency construction; magenta is that construction, green the curvature solve. Tangency as arithmetic.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN