THE FOLD / LOOT / THE MINT / THE APOLLONIUS CIRCLE
THE APOLLONIUS CIRCLE
the circle traced by a constant distance-ratio
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The circle of Apollonius answers: where are all the points whose distances to two fixed points keep a fixed ratio? Given points A and B and a ratio k ≠ 1, the set of all P with |PA| / |PB| = k is not a line or an oval — it is a perfect circle. Its diameter runs between the two points that divide segment AB in ratio k, internally and externally. As k → 1 the circle swells to the perpendicular bisector (a ‘circle of infinite radius’); for k far from 1 it tightens around the nearer point. Apollonius of Perga catalogued these circles around 200 BCE; they underlie the definition of hyperbolic distance and the geometry of pursuit.
LIT verified live: for thousands of random A, B, k, every point sampled on the constructed circle has |PA|/|PB| = k to ~1e-14, while points off the circle do not (window.__apollonius). FIG no framing; the circle is built from the two division points and the ratio is checked independently in-browser.
LIT verified live: for thousands of random A, B, k, every point sampled on the constructed circle has |PA|/|PB| = k to ~1e-14, while points off the circle do not (window.__apollonius). FIG no framing; the circle is built from the two division points and the ratio is checked independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — the loot: a whole circle minted from a single rule, ‘keep the distance-ratio fixed’. AVAN (AI) built the instrument: the two division points, the Apollonius circle, and the constant-ratio check.
Credit as content: Apollonius of Perga (c. 200 BCE). The weave: David names the minted circle; I confirm the locus |PA|/|PB| = k is exactly that circle.
Credit as content: Apollonius of Perga (c. 200 BCE). The weave: David names the minted circle; I confirm the locus |PA|/|PB| = k is exactly that circle.
3 ONE DIMENSION
Two points A, B and the Apollonius circle — every point on it keeps |PA|/|PB| = k.
4 TWO DIMENSIONS · INTERACTIVE
Change the ratio k; sampled points on the circle are checked to all share the ratio k.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Apollonius circle, the locus of constant distance-ratio.
AVAN’s addition (the inverse-companion): don’t plot points and hope — read the rule as a circle. The inverse of ‘|PA|/|PB| = k’ is ‘the circle through the two points dividing AB in ratio k’. Magenta are the distance-ratio spokes from sample points to A and B; green is the circle they all satisfy. A ratio rule that draws a circle.
LIT Genuine circle of Apollonius (Apollonius of Perga, c. 200 BCE). Verified live: for ~3000 random A, B, k, every point sampled on the constructed circle has |PA|/|PB| = k to ~1e-14, while points off the circle do not (window.__apollonius.ok, .offOk).
FIG No framing; the circle is built from the two division points and the ratio is checked independently in-browser. The AVAN inverse is honest — instead of plotting points and hoping, read the rule as a circle: the inverse of '|PA|/|PB| = k' is 'the circle through the two points dividing AB in ratio k'. Magenta are the distance-ratio spokes from sample points to A and B; green is the circle they all satisfy. A ratio rule that draws a circle.
FIG No framing; the circle is built from the two division points and the ratio is checked independently in-browser. The AVAN inverse is honest — instead of plotting points and hoping, read the rule as a circle: the inverse of '|PA|/|PB| = k' is 'the circle through the two points dividing AB in ratio k'. Magenta are the distance-ratio spokes from sample points to A and B; green is the circle they all satisfy. A ratio rule that draws a circle.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN