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THE RADON
four points that always split into two overlapping halves
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Radon’s theorem says that any d + 2 points in d-dimensional space can be split into two groups whose convex hulls overlap. In the plane (d = 2), that means any four points can be partitioned into two sets sharing a common point — the Radon point. Either one point sits inside the triangle of the other three, or the four form a quadrilateral whose two diagonals cross. The proof is pure linear algebra: four points in the plane always have an affine dependence ΣλiPi = 0 with Σλi = 0; grouping by the sign of λ gives the two overlapping sets.
LIT verified live: for thousands of random 4-point sets, solving the affine dependence and splitting by sign yields two sets whose weighted barycentres coincide — a shared Radon point inside both hulls (window.__radon). FIG no framing; the affine dependence solved and the shared point confirmed.
LIT verified live: for thousands of random 4-point sets, solving the affine dependence and splitting by sign yields two sets whose weighted barycentres coincide — a shared Radon point inside both hulls (window.__radon). FIG no framing; the affine dependence solved and the shared point confirmed.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-pull-request — four points submit a split into two teams, and the theorem guarantees the two teams’ territories always overlap at one shared point. That guaranteed merge-point is the review. AVAN (AI) built the instrument: the affine-dependence solver, the sign-based partition, and the check that both barycentres land on the same Radon point.
Credit as content: Johann Radon (1921). The weave: David names the-pull-request; I find the affine dependence of the four points, split them by the sign of the coefficients, and confirm the two groups’ convex hulls meet at a single common point — the split always overlaps.
Credit as content: Johann Radon (1921). The weave: David names the-pull-request; I find the affine dependence of the four points, split them by the sign of the coefficients, and confirm the two groups’ convex hulls meet at a single common point — the split always overlaps.
3 ONE DIMENSION
Four points, two cases: one inside the triangle of the other three (split {inner} vs {outer three}), or a crossing quadrilateral (split by diagonals). Either way the hulls share the Radon point.
4 TWO DIMENSIONS · INTERACTIVE
Four points, the Radon partition (two colours), the two hulls, and the shared Radon point; verified over many sets.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a split that always overlaps.
AVAN’s addition (the inverse-companion): given too many points to be independent (d + 2 in dimension d), don’t ask if they separate — find the partition that must overlap, read straight off the signs of their affine dependence. The inverse of ‘separate points into disjoint groups’ is ‘d + 2 points always split into two that intersect.’ Magenta is the four points; green is the shared Radon point of the two hulls. Enough points force an overlap.
LIT Genuine Radon's theorem (Johann Radon, 1921). Verified live: for thousands of random 4-point sets in the plane, solving the affine dependence ΣλᵢPᵢ=0 (Σλᵢ=0) and splitting by the sign of λ yields two sets whose weighted barycentres coincide — a shared Radon point in both hulls (window.__radon.matches, over window.__radon.tested sets).
FIG No framing: the affine-dependence solver, the sign-based partition, and the check that both barycentres land on the same point all run in-browser. The AVAN inverse is honest — given d+2 points (too many to be affinely independent), reading the overlapping partition straight off the signs of their affine dependence is the actual proof; magenta is the four points, green the shared Radon point. Enough points force an overlap.
FIG No framing: the affine-dependence solver, the sign-based partition, and the check that both barycentres land on the same point all run in-browser. The AVAN inverse is honest — given d+2 points (too many to be affinely independent), reading the overlapping partition straight off the signs of their affine dependence is the actual proof; magenta is the four points, green the shared Radon point. Enough points force an overlap.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN