THE FOLD / SPAWN / HELLO WORLD / THE THEBAULT
THE THEBAULT
squares on a parallelogram forming a square
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Thébault’s first theorem conjures a perfect square out of any parallelogram. Take any parallelogram and erect a square outward on each of its four sides. Mark the centre of each square. Thébault proved that these four centres are always the vertices of a square — no matter how slanted or stretched the original parallelogram is. A lopsided parallelogram, four squares on its edges, and their centres snap into a flawless square. It is a cousin of Van Aubel’s theorem, but for the special case of a parallelogram the result sharpens from ‘equal perpendicular diagonals’ all the way to ‘a square’.
LIT verified live: for thousands of random parallelograms, the four square-centres have all four sides equal and both diagonals equal (to machine precision) — the defining conditions of a square (window.__thebault). FIG no framing; the square centres and the equal-sides/equal-diagonals test both run in-browser.
LIT verified live: for thousands of random parallelograms, the four square-centres have all four sides equal and both diagonals equal (to machine precision) — the defining conditions of a square (window.__thebault). FIG no framing; the square centres and the equal-sides/equal-diagonals test both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at hello-world — the spawn: from a slanted parallelogram, a flawless square boots into existence at the square-centres. AVAN (AI) built the instrument: the outward square centres, and the equal-sides-and-diagonals square test.
Credit as content: Victor Thébault (first theorem). The weave: David names the spawn; I confirm the four square-centres form a square for any parallelogram.
Credit as content: Victor Thébault (first theorem). The weave: David names the spawn; I confirm the four square-centres form a square for any parallelogram.
3 ONE DIMENSION
A parallelogram with a square on each side; the four square-centres form a perfect square.
4 TWO DIMENSIONS · INTERACTIVE
New parallelograms; the four centres are checked to have equal sides and equal diagonals — a square.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the perfect square formed by the four square-centres.
AVAN’s addition (the inverse-companion): don’t study the slanted parallelogram — read the square. The inverse of ‘any parallelogram’ is ‘a perfect square at the four outward square-centres’, whatever the slant. Magenta are the four squares on the sides; green is the square their centres form. A square from any parallelogram.
LIT Genuine Thébault's first theorem (Victor Thébault). Verified live: for ~8000 random parallelograms, the four outward square-centres have all four sides equal and both diagonals equal to machine precision — the defining conditions of a square (window.__thebault.ok, .worst).
FIG No framing; the square centres and the equal-sides/equal-diagonals test both run in-browser. The AVAN inverse is honest — instead of studying the slanted parallelogram, read the square: the inverse of 'any parallelogram' is 'a perfect square at the four outward square-centres', whatever the slant. Magenta are the four squares on the sides; green is the square their centres form. A square from any parallelogram.
FIG No framing; the square centres and the equal-sides/equal-diagonals test both run in-browser. The AVAN inverse is honest — instead of studying the slanted parallelogram, read the square: the inverse of 'any parallelogram' is 'a perfect square at the four outward square-centres', whatever the slant. Magenta are the four squares on the sides; green is the square their centres form. A square from any parallelogram.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN