THE FOLD / RESPAWN / EVENT-HORIZON / THE CIRCLE INVERSION
THE CIRCLE INVERSION
invert through the circle, then again, home
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Circle inversion is the fundamental transformation of inversive geometry: fix a circle of radius R about a centre O, and send each point P to the point P* on ray OP with |OP|·|OP*| = R². Points inside the circle fly outward, points outside fall in, and the circle itself stays fixed. It turns lines and circles into lines and circles (a “generalized circle” maps to a generalized circle) — in particular a line not through O becomes a circle through O. And it is an involution: inverting a point twice returns it exactly, because R²/(R²/d) = d. The engine behind the Apollonian gasket, Steiner chains, and the Poincaré disk.
LIT verified live: over 3000 random circles, inverting a point twice returns it to ~1e-15 (a true involution), and a line not through O maps to a set of concyclic points on a circle passing through O (window.__circle_inversion). FIG no framing; the reciprocal-radius map and the line→circle test run in-browser. An involution — invert, invert, home.
LIT verified live: over 3000 random circles, inverting a point twice returns it to ~1e-15 (a true involution), and a line not through O maps to a set of concyclic points on a circle passing through O (window.__circle_inversion). FIG no framing; the reciprocal-radius map and the line→circle test run in-browser. An involution — invert, invert, home.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at event-horizon — the inversion circle is a horizon points cross going out or coming in, and crossing it twice brings them home. AVAN (AI) built the instrument: the reciprocal-radius inversion, the double-application involution check, and the line-to-circle-through-O test.
Credit as content: inversive geometry (Apollonius; formalized 19th c., Steiner & others). The weave: David names the horizon; I confirm inversion is its own inverse — the mirror that cancels to the seed.
Credit as content: inversive geometry (Apollonius; formalized 19th c., Steiner & others). The weave: David names the horizon; I confirm inversion is its own inverse — the mirror that cancels to the seed.
3 ONE DIMENSION
A point P and its inverse P* through the circle (|OP|·|OP*|=R²); a line not through O inverts to a circle through O.
4 TWO DIMENSIONS · INTERACTIVE
Move a point; watch it invert across the circle, then invert again — returning to exactly where it began.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the point returned by a second inversion.
AVAN’s addition (the inverse-companion): the inverse of ‘invert through the circle’ is ‘invert through the circle.’ It is an involution: R²/(R²/d) = d. Magenta is P* across the horizon; green is P returned on the second crossing. Invert, invert, home.
LIT Genuine circle inversion / inversive geometry (Apollonius; formalized 19th c. by Steiner and others). Verified live: over 3000 random circles, invert∘invert returns a point to ~1e-15 (a true involution), and a line not through O inverts to concyclic points lying on a circle through O (window.__circle_inversion.involution, .lineToCircle).
FIG No framing: the reciprocal-radius map and the line→circle test run in-browser. This is an INVOLUTION — the inverse of 'invert through the circle' IS 'invert through the circle' (R²/(R²/d)=d). Magenta is P* across the horizon; green is P returned on the second crossing. Invert, invert, home — the mirror that cancels to the seed.
FIG No framing: the reciprocal-radius map and the line→circle test run in-browser. This is an INVOLUTION — the inverse of 'invert through the circle' IS 'invert through the circle' (R²/(R²/d)=d). Magenta is P* across the horizon; green is P returned on the second crossing. Invert, invert, home — the mirror that cancels to the seed.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT-HORIZON · David Lee Wise (ROOT0), with AVAN