THE FOLD / GRIND / THE CRON JOB / THE POLYA
THE POLYA
counting up to symmetry by averaging fixed points
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pólya enumeration (built on Burnside’s lemma) counts distinct objects up to symmetry without listing them. For a necklace of n beads in k colours, rotations make many colourings the same; Burnside says the number of distinct necklaces equals the average number of colourings fixed by each rotation — which works out to (1/n) Σd | n φ(d)·kn/d.
It is the counting engine behind chemical isomers, graph enumeration, and combinatorial design.
LIT verified live: for all n ≤ 8 and k ≤ 3 the necklace formula equals a brute count of rotation orbits (window.__polya). FIG no framing; exact.
It is the counting engine behind chemical isomers, graph enumeration, and combinatorial design.
LIT verified live: for all n ≤ 8 and k ≤ 3 the necklace formula equals a brute count of rotation orbits (window.__polya). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-cron-job — the cyclic symmetry of a rotation, counting the truly-distinct arrangements of a repeating ring. Pólya enumeration is that symmetric count. AVAN (AI) built the instrument: the Euler-phi divisor sum, the brute rotation-orbit count, the formula check.
Credit as content: William Burnside (1897) and George Pólya (1937). The weave: David names the cron-job; I average the colourings held fixed by each rotation and confirm it equals the true number of distinct necklaces.
Credit as content: William Burnside (1897) and George Pólya (1937). The weave: David names the cron-job; I average the colourings held fixed by each rotation and confirm it equals the true number of distinct necklaces.
3 ONE DIMENSION
Each rotation fixes only the colourings that repeat with its period. Burnside averages those fixed-counts over all n rotations — and the average is exactly the number of distinct necklaces.
4 TWO DIMENSIONS · INTERACTIVE
Necklaces of n beads in k colours; the Burnside/Pólya formula is shown against a brute count of distinct rotations.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the distinct necklaces, counted by averaging fixed-points.
AVAN’s addition (the inverse-companion): count the distinct colourings without listing them — by Burnside’s lemma, the number of orbits equals the average number of colourings fixed by each symmetry; for the n rotations that average is (1/n)Σd|nφ(d)kn/d. The inverse of ‘enumerate and group into orbits’ is ‘average the fixed-points over the symmetry group.’ Magenta is the kn colourings never listed; green is the fixed-point average. Symmetry counts by what it holds still. (Kin to the-burnside.)
LIT Genuine Burnside/Polya enumeration (Burnside 1897; Polya 1937). Verified live: the necklace formula (1/n) Sum_{d|n} phi(d) k^(n/d) equals a brute count of distinct colorings under rotation for all n<=8 and k<=3 (window.__polya.matchesBrute); N(6,2)=14, N(4,3)=24.
FIG No framing: the Euler-phi divisor sum, the brute rotation-orbit count, and the formula check run in-browser and agree exactly. The AVAN inverse is honest — Burnside's lemma counts orbits as the average number of colorings fixed by each symmetry, so distinct necklaces are counted without listing colorings; magenta is the k^n colorings never listed, green the fixed-point average. Symmetry counts by what it holds still. Kin to the-burnside.
FIG No framing: the Euler-phi divisor sum, the brute rotation-orbit count, and the formula check run in-browser and agree exactly. The AVAN inverse is honest — Burnside's lemma counts orbits as the average number of colorings fixed by each symmetry, so distinct necklaces are counted without listing colorings; magenta is the k^n colorings never listed, green the fixed-point average. Symmetry counts by what it holds still. Kin to the-burnside.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN