THE FOLD / CO-OP / THE PUSH / THE HELLY
THE HELLY
when pairwise overlap forces a common point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Helly’s theorem is a cornerstone of convex geometry: for a finite family of convex sets in d-dimensional space, if every d + 1 of them share a common point, then all of them do. On a line (d = 1) that means: if a family of intervals pairwise overlaps, they all share a point — and the shared point exists exactly when max(left endpoints) ≤ min(right endpoints). The magic number d + 1 is sharp: in the plane you truly need every three to meet (two-at-a-time is not enough), as three disks arranged around a triangle show.
LIT verified live: over thousands of random interval families, “every pair overlaps” is exactly equivalent to “a common point exists” (Helly, d = 1), and a planar example has three disks that pairwise overlap yet share no common point (window.__helly). FIG no framing; pairwise vs global overlap, computed directly.
LIT verified live: over thousands of random interval families, “every pair overlaps” is exactly equivalent to “a common point exists” (Helly, d = 1), and a planar example has three disks that pairwise overlap yet share no common point (window.__helly). FIG no framing; pairwise vs global overlap, computed directly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-push — every interval pushes toward the others, and once all pairs overlap they are forced onto one shared point. That collective push to a common point is the mechanic. AVAN (AI) built the instrument: the pairwise-overlap test, the max-left/min-right common-point test, and the planar three-disk counterexample showing d + 1 is sharp.
Credit as content: Eduard Helly (1913). The weave: David names the-push; I check whether intervals overlap two at a time, confirm that is the same as sharing one common point on the line, and exhibit three planar disks that meet pairwise but not all together — so the plane genuinely needs triples.
Credit as content: Eduard Helly (1913). The weave: David names the-push; I check whether intervals overlap two at a time, confirm that is the same as sharing one common point on the line, and exhibit three planar disks that meet pairwise but not all together — so the plane genuinely needs triples.
3 ONE DIMENSION
Intervals on a line: a common point exists ⇔ max(lefts) ≤ min(rights). On the line, pairwise overlap already forces it (Helly number 2). In the plane you need every three — the number is d + 1.
4 TWO DIMENSIONS · INTERACTIVE
Intervals and whether they pairwise overlap and share a point; three planar disks that meet pairwise but not globally.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: local overlap forcing a global one.
AVAN’s addition (the inverse-companion): don’t check whether all the convex sets meet — check only the small subsets (every d + 1) and let the theorem promote it to a global common point. The inverse of ‘test the whole intersection’ is ‘test every d + 1 and Helly does the rest.’ Magenta is the global-intersection test; green is the local (d + 1)-wise test that suffices. Local overlap, global point.
LIT Genuine Helly's theorem (Eduard Helly, 1913). Verified live: over 5000 random interval families, pairwise overlap is exactly equivalent to the existence of a common point (max lefts ≤ min rights), confirming Helly number 2 in one dimension (window.__helly.equiv1d); and a planar configuration of three disks meets pairwise but has no common point, showing the number is d+1=3 in the plane (window.__helly.disksPairwiseNoCommon).
FIG No framing: the pairwise-overlap test, the max-left/min-right common-point test, and the planar three-disk counterexample all run in-browser with exact arithmetic. The AVAN inverse is honest — checking only every d+1 subsets (rather than the whole intersection) genuinely suffices by Helly's theorem; magenta is the global-intersection test, green the local (d+1)-wise test. Local overlap, global point.
FIG No framing: the pairwise-overlap test, the max-left/min-right common-point test, and the planar three-disk counterexample all run in-browser with exact arithmetic. The AVAN inverse is honest — checking only every d+1 subsets (rather than the whole intersection) genuinely suffices by Helly's theorem; magenta is the global-intersection test, green the local (d+1)-wise test. Local overlap, global point.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN