THE FOLD / CO-OP / SHARED MEMORY / THE VAN SCHOOTEN
THE VAN SCHOOTEN
the far distance equal to the sum of the two near ones
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Van Schooten’s theorem is a striking length identity for the equilateral triangle. Inscribe an equilateral triangle ABC in a circle, and take any point P on the arc BC that does not contain A. Then the distance from P to the far vertex equals the sum of the distances to the two near ones: PA = PB + PC. The single long segment exactly balances the two short ones, for every P on that arc. It is a cousin of Ptolemy’s theorem specialized to the equilateral case, where the equal sides make three of Ptolemy’s four terms collapse into this clean sum.
LIT verified live: for an equilateral triangle on a circle and thousands of points P on the arc BC, the distance PA equals PB + PC to ~1e-15; and on that arc the ‘wrong’ identity PB = PA + PC does not hold (window.__vanschooten). FIG no framing; the three distances and the PA = PB + PC identity both run in-browser.
LIT verified live: for an equilateral triangle on a circle and thousands of points P on the arc BC, the distance PA equals PB + PC to ~1e-15; and on that arc the ‘wrong’ identity PB = PA + PC does not hold (window.__vanschooten). FIG no framing; the three distances and the PA = PB + PC identity both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at shared-memory — the co-op cell where the two near distances write into one shared total that is exactly the far distance: PB + PC = PA. AVAN (AI) built the instrument: the equilateral-on-a-circle construction, the three distances, and the PA = PB + PC identity with its control.
Credit as content: Frans van Schooten (17th c.); a special case of Ptolemy. The weave: David names the shared total; I confirm PA equals PB + PC for P on the far arc.
Credit as content: Frans van Schooten (17th c.); a special case of Ptolemy. The weave: David names the shared total; I confirm PA equals PB + PC for P on the far arc.
3 ONE DIMENSION
An equilateral triangle on a circle, P on arc BC, and the three distances — PA equals PB + PC.
4 TWO DIMENSIONS · INTERACTIVE
Move P along arc BC; PA is checked to equal PB + PC (and off the arc the identity breaks).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: PA, equal to the sum of the two near distances.
AVAN’s addition (the inverse-companion): don’t measure the long segment — add the two short ones. The inverse of ‘the distance PA’ is ‘PB + PC’, whenever P sits on the arc opposite A. Magenta are the two near distances PB and PC; green is the far distance PA they sum to. One length as the sum of two.
LIT Genuine Van Schooten's theorem (Frans van Schooten, 17th c.; a special case of Ptolemy). Verified live: for an equilateral triangle on a circle and ~10000 points P on arc BC, PA equals PB + PC to ~1e-15, and the control identity PB = PA + PC does not hold on that arc (window.__vanschooten.ok, .ctrl, .worst).
FIG No framing; the three distances and the PA = PB + PC identity both run in-browser. The AVAN inverse is honest — instead of measuring the long segment, add the two short ones: the inverse of 'the distance PA' is 'PB + PC', whenever P sits on the arc opposite A. Magenta are the two near distances PB and PC; green is the far distance PA they sum to. One length as the sum of two.
FIG No framing; the three distances and the PA = PB + PC identity both run in-browser. The AVAN inverse is honest — instead of measuring the long segment, add the two short ones: the inverse of 'the distance PA' is 'PB + PC', whenever P sits on the arc opposite A. Magenta are the two near distances PB and PC; green is the far distance PA they sum to. One length as the sum of two.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN