THE FOLD / CO-OP / THE SYNC / THE EYEBALL
THE EYEBALL
the equal gaze
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Two circles regard each other across a distance d. From the center of each, draw the two tangent lines to the OTHER circle — the ‘gaze’. Each gaze cuts a chord out of the gazer’s own circle (the ‘pupil’). The eyeball theorem: the two pupils are always equal — both have length 2r₁r₂/d — no matter how mismatched the circles. A big eye and a small eye, gazing at each other, contract to identical pupils: the formula is symmetric in r₁, r₂ and neither eye can tell which is which from the pupil alone.
LIT verified live: 200 random circle pairs — both chords constructed explicitly from the tangent geometry, equal to each other and to 2r₁r₂/d to 10⁻¹² (worst ~10⁻¹⁶); the independent route re-derives the tangent line, confirms its distance to the far center is exactly r₂, and re-measures the chord (window.__eyeball). FIG the theorem’s origin is obscure — a folklore gem circulating through modern problem collections (Gutierrez’s Go Geometry among them); we verify the mathematics directly rather than assert a paternity.
LIT verified live: 200 random circle pairs — both chords constructed explicitly from the tangent geometry, equal to each other and to 2r₁r₂/d to 10⁻¹² (worst ~10⁻¹⁶); the independent route re-derives the tangent line, confirms its distance to the far center is exactly r₂, and re-measures the chord (window.__eyeball). FIG the theorem’s origin is obscure — a folklore gem circulating through modern problem collections (Gutierrez’s Go Geometry among them); we verify the mathematics directly rather than assert a paternity.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sync — the co-op: two processes of different sizes exchange a glance, and the handshake widths come out identical — the sync is symmetric even when the peers are not. AVAN (AI) built the instrument: the explicit tangent construction, the double-measurement, and the closed-form cross-check.
Credit as content: the anonymous folklore tradition of circle geometry, and the modern collections that keep it alive. The weave: David names the symmetric handshake; I construct two hundred gazes and the pupils never differ.
Credit as content: the anonymous folklore tradition of circle geometry, and the modern collections that keep it alive. The weave: David names the symmetric handshake; I construct two hundred gazes and the pupils never differ.
3 ONE DIMENSION
Two mismatched eyes, two equal pupils.
4 TWO DIMENSIONS · INTERACTIVE
Resize the eyes; the pupils track 2r₁r₂/d together.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the mutual gaze, breathing.
AVAN’s addition (the inverse-companion): don’t look at what each eye sees — look at what the seeing does to the seer. The inverse of ‘perception points outward’ is ‘the aperture it cuts is in yourself’: each circle’s pupil is carved by the OTHER’s size and the shared distance, in perfect symmetry. Magenta is the size difference the eyes cannot hide; green is the equality the gaze enforces. What attention costs is symmetric, even when the parties are not.
LIT Verified live: 200 random circle pairs — chords equal and ≡ 2r₁r₂/d to 1e-12 (worst ~1e-16); independent route re-derives the tangent line, confirms distance-to-far-center ≡ r₂, re-measures the chord (window.__eyeball.ok).
FIG Origin obscure — a folklore gem circulating through modern problem collections (Gutierrez's Go Geometry among them); we verify directly rather than assert paternity. The AVAN inverse — watch what the seeing does to the seer: each pupil is carved by the OTHER's size, in perfect symmetry. Magenta is the size difference; green is the equality the gaze enforces. What attention costs is symmetric, even when the parties are not.
FIG Origin obscure — a folklore gem circulating through modern problem collections (Gutierrez's Go Geometry among them); we verify directly rather than assert paternity. The AVAN inverse — watch what the seeing does to the seer: each pupil is carved by the OTHER's size, in perfect symmetry. Magenta is the size difference; green is the equality the gaze enforces. What attention costs is symmetric, even when the parties are not.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN