THE FOLD / BOSS / THE WALL / THE DESARGUES
THE DESARGUES
perspective from a point equals perspective from a line
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Desargues’ theorem is a cornerstone of projective geometry, linking two kinds of ‘perspective’. Two triangles ABC and A′B′C′ are perspective from a point if the three lines AA′, BB′, CC′ meet at one center O. They are perspective from a line if the three intersection points of corresponding sides — AB∩A′B′, BC∩B′C′, CA∩C′A′ — are collinear. Desargues proved these are equivalent: a common center forces a common axis, and vice versa. It is self-dual (swap ‘point’ and ‘line’ and it still holds) and it is exactly the condition a projective plane needs to come from a field.
LIT verified live: for tens of thousands of triangle pairs placed in perspective from a random center, the three corresponding-side intersections are always collinear, and pushing a single vertex off its center-ray breaks both the perspectivity and the collinearity together (window.__desargues). FIG no framing; the perspective construction, the side intersections, and the collinearity test all run in-browser.
LIT verified live: for tens of thousands of triangle pairs placed in perspective from a random center, the three corresponding-side intersections are always collinear, and pushing a single vertex off its center-ray breaks both the perspectivity and the collinearity together (window.__desargues). FIG no framing; the perspective construction, the side intersections, and the collinearity test all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-wall — the axis of perspectivity is a single straight wall, and Desargues says two triangles share a center point exactly when their sides meet along that one wall. AVAN (AI) built the instrument: the point-perspective construction, the three side intersections, the collinearity check, and the off-ray control.
Credit as content: Girard Desargues (1639). The weave: David names the wall; I confirm perspective-from-a-point forces the three side-meetings onto a single line.
Credit as content: Girard Desargues (1639). The weave: David names the wall; I confirm perspective-from-a-point forces the three side-meetings onto a single line.
3 ONE DIMENSION
Two triangles perspective from a center O; their corresponding sides meet at three points P, Q, R on one line (the axis).
4 TWO DIMENSIONS · INTERACTIVE
New configurations; the three side-intersections are checked for collinearity, and an off-ray push breaks it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the axis of perspectivity, the line through the three side-meetings.
AVAN’s addition (the inverse-companion): don’t look for the center — look for the axis. The inverse of ‘the two triangles share a center point O’ is ‘their corresponding sides meet on a single line’, and Desargues makes the two conditions identical (and self-dual). Magenta is the center O; green is the axis line the sides meet on. Point and line, two faces of one perspective.
LIT Genuine Desargues' theorem (Girard Desargues, 1639). Verified live: for 20000 triangle pairs perspective from a random center O, the three corresponding-side intersections P=AB∩A′B′, Q=BC∩B′C′, R=CA∩C′A′ are always collinear, and pushing one vertex off its center-ray breaks the collinearity (window.__desargues.persp, .ctrl).
FIG No framing; the perspective construction, the side intersections, and the collinearity test all run in-browser. The AVAN inverse is honest — instead of looking for the center, look for the axis: perspective-from-a-point equals perspective-from-a-line, and the theorem is self-dual. Magenta is the center O; green is the axis line the sides meet on. Point and line, two faces of one perspective.
FIG No framing; the perspective construction, the side intersections, and the collinearity test all run in-browser. The AVAN inverse is honest — instead of looking for the center, look for the axis: perspective-from-a-point equals perspective-from-a-line, and the theorem is self-dual. Magenta is the center O; green is the axis line the sides meet on. Point and line, two faces of one perspective.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN