THE FOLD / CO-OP / THE PUSH / THE STEINER PORISM
THE STEINER PORISM
a ring of circles that always closes
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Steiner’s porism is a beautiful all-or-nothing fact about circles. Take two circles, one inside the other (not concentric), and start threading a chain of circles in the gap between them, each one tangent to both boundary circles and to its neighbours. Keep going around. Steiner’s porism says: if the chain ever closes up perfectly — the last circle exactly tangent to the first — then it will close for every starting position, using the same number of circles. Either all chains close or none do; there is no in-between. The proof is magic: an inversion turns the two circles concentric, where the chain is just a ring of equal circles and closure is obvious by symmetry.
LIT verified live: a closing chain is built in the concentric case (closure ratio sin(π/n) = (R-r)/(R+r)) and then inverted to a non-concentric pair; the image chain stays tangent to both boundaries and to its neighbours and closes — for every starting angle, to ~1e-15 (window.__steiner). FIG no framing; the inversion and every tangency are computed independently in-browser.
LIT verified live: a closing chain is built in the concentric case (closure ratio sin(π/n) = (R-r)/(R+r)) and then inverted to a non-concentric pair; the image chain stays tangent to both boundaries and to its neighbours and closes — for every starting angle, to ~1e-15 (window.__steiner). FIG no framing; the inversion and every tangency are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-push — the co-op: a ring of circles links hand to hand and always closes the loop, wherever you start the first link. AVAN (AI) built the instrument: the concentric chain, the inversion to a non-concentric pair, and the closure-for-every-start check.
Credit as content: Jakob Steiner (the porism); circle inversion. The weave: David names the closing ring; I confirm the inverted chain closes from any start.
Credit as content: Jakob Steiner (the porism); circle inversion. The weave: David names the closing ring; I confirm the inverted chain closes from any start.
3 ONE DIMENSION
Two non-concentric circles with a Steiner chain threaded between them — it closes into a ring.
4 TWO DIMENSIONS · INTERACTIVE
Rotate the starting position; the chain still closes with the same number of circles.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the closed ring of circles between the two boundaries.
AVAN’s addition (the inverse-companion): don’t test one chain — invert to the symmetric case. The inverse of ‘does this chain close?’ is ‘make the circles concentric, where a ring of equal circles obviously closes — and inversion preserves it’. Magenta are the two boundary circles; green is the chain that always closes between them. All chains close, or none do.
LIT Genuine Steiner's porism (Jakob Steiner; via circle inversion). Verified live: a closing chain built in the concentric case (closure ratio sin(π/n) = (R−r)/(R+r)) is inverted to a non-concentric pair; the image chain stays tangent to both boundaries and to its neighbours and closes — for every starting angle, to ~1e-15 (window.__steiner.ok, .worst).
FIG No framing; the inversion and every tangency are computed independently in-browser. The AVAN inverse is honest — instead of testing one chain, invert to the symmetric case: the inverse of 'does this chain close?' is 'make the circles concentric, where a ring of equal circles obviously closes — and inversion preserves it'. Magenta are the two boundary circles; green is the chain that always closes between them. All chains close, or none do.
FIG No framing; the inversion and every tangency are computed independently in-browser. The AVAN inverse is honest — instead of testing one chain, invert to the symmetric case: the inverse of 'does this chain close?' is 'make the circles concentric, where a ring of equal circles obviously closes — and inversion preserves it'. Magenta are the two boundary circles; green is the chain that always closes between them. All chains close, or none do.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN