THE FOLD / CO-OP / THE PUSH / THE CAYLEY-MENGER
THE CAYLEY-MENGER
a simplex volume from its edge lengths alone
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Cayley–Menger determinant computes the volume of a simplex from its edge lengths alone — no coordinates needed. Heron’s formula gives a triangle’s area from its three sides; Cayley and Menger generalized it to every dimension. Arrange the squared pairwise distances into a bordered matrix (a row and column of 1’s, a 0 corner), and its determinant yields the squared volume: 16·Area² = -det(CM) for a triangle, 288·Vol² = det(CM) for a tetrahedron. Distances in, volume out — the metric fully determines the shape’s size, and a negative or zero determinant flags points that cannot be embedded at all.
LIT verified live: for thousands of random triangles and tetrahedra, the volume computed from the Cayley–Menger determinant (using only pairwise squared distances) matches the volume computed the ordinary way from coordinates, to ~1e-7 (window.__cayleymenger). FIG no framing; the distance-only determinant and the coordinate volume are computed by different routes and agree.
LIT verified live: for thousands of random triangles and tetrahedra, the volume computed from the Cayley–Menger determinant (using only pairwise squared distances) matches the volume computed the ordinary way from coordinates, to ~1e-7 (window.__cayleymenger). FIG no framing; the distance-only determinant and the coordinate volume are computed by different routes and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-push — every pair of vertices pushes in its one distance, and together the pairwise pushes fix the whole simplex’s volume without a single coordinate. AVAN (AI) built the instrument: the Cayley–Menger bordered determinant, the coordinate volume, and their agreement.
Credit as content: Arthur Cayley (1841), Karl Menger (1928); Heron of Alexandria for the triangle. The weave: David names the collective push; I confirm the determinant of distances equals the volume.
Credit as content: Arthur Cayley (1841), Karl Menger (1928); Heron of Alexandria for the triangle. The weave: David names the collective push; I confirm the determinant of distances equals the volume.
3 ONE DIMENSION
A triangle with its three edge lengths; the Cayley–Menger determinant turns those distances into its area.
4 TWO DIMENSIONS · INTERACTIVE
New shapes; the volume from the distance-only determinant is compared to the coordinate volume.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the tetrahedron's volume, from its six edge lengths.
AVAN’s addition (the inverse-companion): don’t place the points — measure between them. The inverse of ‘the volume of a simplex’ is ‘the bordered determinant of its pairwise squared distances’, so the metric alone fixes the size and reveals when points can’t be embedded. Magenta are the six edge lengths; green is the volume they determine. Shape from distance, no coordinates.
LIT Genuine Cayley–Menger determinant (Arthur Cayley 1841, Karl Menger 1928; Heron for the triangle). Verified live: for ~4000 random triangles (n=2) and tetrahedra (n=3), the volume from the bordered determinant of pairwise squared distances matches the coordinate volume to ~1e-7 (window.__cayleymenger.ok2, .ok3, .worst).
FIG No framing; the distance-only determinant and the coordinate volume are computed by different routes and agree. The AVAN inverse is honest — instead of placing the points, measure between them: the inverse of 'the volume of a simplex' is 'the bordered determinant of its pairwise squared distances', so the metric alone fixes the size and reveals when points can't be embedded. Magenta are the six edge lengths; green is the volume they determine. Shape from distance, no coordinates.
FIG No framing; the distance-only determinant and the coordinate volume are computed by different routes and agree. The AVAN inverse is honest — instead of placing the points, measure between them: the inverse of 'the volume of a simplex' is 'the bordered determinant of its pairwise squared distances', so the metric alone fixes the size and reveals when points can't be embedded. Magenta are the six edge lengths; green is the volume they determine. Shape from distance, no coordinates.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN