THE FOLD / GRIND / THE EPOCH / THE CASEY
THE CASEY
a generalized Ptolemy for tangent circles
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Casey’s theorem is Ptolemy’s theorem for circles. Ptolemy says: for four points on a circle in order, the products of opposite chord-pairs relate as AC·BD = AB·CD + AD·BC. Casey generalizes each point to a whole circle tangent to a common circle. Replace the four points by four circles all internally tangent to one enclosing circle, in cyclic order, and replace each chord by the tangent length tij (the length of the common tangent segment) between circles i and j. Then the very same relation holds: t12·t34 + t23·t14 = t13·t24. Shrink the circles to points and it collapses back to Ptolemy.
LIT verified live: for thousands of random configurations of four circles internally tangent to a circle, the tangent lengths satisfy t12t34 + t23t14 = t13t24 to ~1e-15, and the point-circle limit reproduces Ptolemy exactly (window.__casey). FIG no framing; the centres, tangent lengths, and the relation are all computed independently in-browser.
LIT verified live: for thousands of random configurations of four circles internally tangent to a circle, the tangent lengths satisfy t12t34 + t23t14 = t13t24 to ~1e-15, and the point-circle limit reproduces Ptolemy exactly (window.__casey). FIG no framing; the centres, tangent lengths, and the relation are all computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-epoch — the grind toward generality: Ptolemy’s point-relation, ground outward until points become circles and chords become tangent lengths. AVAN (AI) built the instrument: the four tangent circles, the six tangent lengths, and the Ptolemy-form relation.
Credit as content: John Casey (Irish geometer, 1866); Ptolemy of Alexandria for the point case. The weave: David names the generalization; I confirm t12t34+t23t14 = t13t24.
Credit as content: John Casey (Irish geometer, 1866); Ptolemy of Alexandria for the point case. The weave: David names the generalization; I confirm t12t34+t23t14 = t13t24.
3 ONE DIMENSION
Four circles inside a circle, with the tangent segments between them — the Casey (generalized Ptolemy) relation.
4 TWO DIMENSIONS · INTERACTIVE
Cycle configurations; t₁₂t₃₄ + t₂₃t₁₄ is checked equal to t₁₃t₂₄.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the satisfied Ptolemy-form relation among tangent lengths.
AVAN’s addition (the inverse-companion): don’t treat points and circles as different problems — read circles as fattened points. The inverse of ‘Ptolemy for four points’ is ‘Casey for four tangent circles’, the same relation with tangent lengths for chords. Magenta are the six tangent lengths; green is the equality t12t34+t23t14 = t13t24. Points fattened into circles, one relation.
LIT Genuine Casey's theorem (John Casey, 1866; Ptolemy for the point case). Verified live: for ~12000 random configurations of four circles internally tangent to a circle, t₁₂t₃₄ + t₂₃t₁₄ = t₁₃t₂₄ to ~1e-15, and the point-circle limit reproduces Ptolemy exactly (window.__casey.ok, .ptol).
FIG No framing; the centres, tangent lengths, and the relation are computed independently in-browser. The AVAN inverse is honest — instead of treating points and circles as different problems, read circles as fattened points: the inverse of 'Ptolemy for four points' is 'Casey for four tangent circles', the same relation with tangent lengths for chords. Magenta are the six tangent lengths; green is the equality t₁₂t₃₄+t₂₃t₁₄ = t₁₃t₂₄. Points fattened into circles, one relation.
FIG No framing; the centres, tangent lengths, and the relation are computed independently in-browser. The AVAN inverse is honest — instead of treating points and circles as different problems, read circles as fattened points: the inverse of 'Ptolemy for four points' is 'Casey for four tangent circles', the same relation with tangent lengths for chords. Magenta are the six tangent lengths; green is the equality t₁₂t₃₄+t₂₃t₁₄ = t₁₃t₂₄. Points fattened into circles, one relation.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN