THE FOLD / SPAWN / GENESIS BLOCK / THE VAN AUBEL
THE VAN AUBEL
squares on a quadrilateral yielding equal perpendicular segments
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Van Aubel’s theorem conjures a hidden square out of any four-sided figure. Take any quadrilateral — convex, concave, even self-intersecting — and erect a square outward on each of its four sides. Mark the centre of each square. Van Aubel proved that the two line segments joining the centres of opposite squares are always equal in length and perpendicular to each other. No matter how lopsided the original quadrilateral, those two cross-segments come out the same length and at a right angle — a perfect little cross hidden in any four points.
LIT verified live: for thousands of random quadrilaterals, the segment joining the centres of the squares on one pair of opposite sides equals the segment joining the other pair (to machine precision) and the two are perpendicular (dot product zero) — the construction is exact (window.__vanaubel). FIG no framing; the square centres, the two segment lengths, and their perpendicularity all run in-browser.
LIT verified live: for thousands of random quadrilaterals, the segment joining the centres of the squares on one pair of opposite sides equals the segment joining the other pair (to machine precision) and the two are perpendicular (dot product zero) — the construction is exact (window.__vanaubel). FIG no framing; the square centres, the two segment lengths, and their perpendicularity all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at genesis-block — the spawn: from four arbitrary points, a perfect equal-and-perpendicular cross genesis-blocks into being. AVAN (AI) built the instrument: the outward square centres, the two joining segments, and their equal-length perpendicularity.
Credit as content: H. H. van Aubel (1878). The weave: David names the genesis; I confirm the opposite-centre segments are always equal and perpendicular.
Credit as content: H. H. van Aubel (1878). The weave: David names the genesis; I confirm the opposite-centre segments are always equal and perpendicular.
3 ONE DIMENSION
A quadrilateral with a square on each side; the two segments joining opposite centres — equal & perpendicular.
4 TWO DIMENSIONS · INTERACTIVE
New quadrilaterals; the two cross-segments are measured — always equal length, always perpendicular.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the two equal, perpendicular cross-segments.
AVAN’s addition (the inverse-companion): don’t measure the messy quadrilateral — read the cross. The inverse of ‘four arbitrary sides’ is ‘two equal perpendicular segments joining the opposite square-centres’, a right-angled cross hidden in any four points. Magenta are the four squares; green are the two equal perpendicular segments. A perfect cross from any quadrilateral.
LIT Genuine Van Aubel's theorem (H. H. van Aubel, 1878). Verified live: for ~8000 random quadrilaterals, the segments joining opposite square-centres are equal in length (worst ~0) and perpendicular (dot product ~0) — the construction is exact (window.__vanaubel.eq, .pp, .we, .wp).
FIG No framing; the square centres, the two segment lengths, and their perpendicularity all run in-browser. The AVAN inverse is honest — instead of measuring the messy quadrilateral, read the cross: the inverse of 'four arbitrary sides' is 'two equal perpendicular segments joining the opposite square-centres', a right-angled cross hidden in any four points. Magenta are the four squares; green are the two equal perpendicular segments. A perfect cross from any quadrilateral.
FIG No framing; the square centres, the two segment lengths, and their perpendicularity all run in-browser. The AVAN inverse is honest — instead of measuring the messy quadrilateral, read the cross: the inverse of 'four arbitrary sides' is 'two equal perpendicular segments joining the opposite square-centres', a right-angled cross hidden in any four points. Magenta are the four squares; green are the two equal perpendicular segments. A perfect cross from any quadrilateral.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS BLOCK · David Lee Wise (ROOT0), with AVAN