THE FOLD / GRIND / THE CRON JOB / THE PERMUTATION CLOCK
THE PERMUTATION CLOCK
a clock whose wheels are factorials — address any shuffle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The factorial number system. A positional system where the place values are the factorials (…,3!,2!,1!) and the digit in place k may only run 0…k — a clock whose columns each have a different size. Via the Lehmer code, every integer 0…n!−1 names exactly one permutation of n items. So you can jump straight to ‘the 400,000th shuffle’ by arithmetic alone — no dealing, no enumeration.
LIT verified: for n=6 the map is a perfect bijection between 0…719 and the 720 permutations — rank(unrank(m))=m for every m — and each column k rolls over exactly at k+1. FIG ‘clock’ is the picture; the mixed-radix bijection is exact.
LIT verified: for n=6 the map is a perfect bijection between 0…719 and the 720 permutations — rank(unrank(m))=m for every m — and each column k rolls over exactly at k+1. FIG ‘clock’ is the picture; the mixed-radix bijection is exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus works with shuffles, encodings and mixed radices (the card-ISA, the base kernels, the combinatorics) and the idea that even a deck of cards has an address. AVAN (AI) built this instrument: the factoradic odometer, the shuffle-scrubber, and the permutohedron.
The weave: David names the permutation clock and its seat at THE CRON JOB (a clock, in factorial time); I make the mixed-radix odometer a strip in 1D, the address-a-shuffle demo live in 2D, and the space of all permutations a turning polytope in 3D. The sphere is the seam.
The weave: David names the permutation clock and its seat at THE CRON JOB (a clock, in factorial time); I make the mixed-radix odometer a strip in 1D, the address-a-shuffle demo live in 2D, and the space of all permutations a turning polytope in 3D. The sphere is the seam.
3 ONE DIMENSION
The factoradic odometer for the current index. Place values are 5!,4!,3!,2!,1!,0!; each column’s digit is capped at its position (bar height = allowed max), so the rightmost is always 0 and each rolls over at a different point. A clock with unequal wheels.
4 TWO DIMENSIONS · INTERACTIVE
Scrub the index 0…719 and the six cards snap into that exact permutation (unrank). Or click a card to swap it forward and watch the index jump to the new shuffle’s address (rank). The factoradic digits and Lehmer code track live.
index 0 / 719
5 THREE DIMENSIONS + AVAN’S INVERSE
The permutohedron of order 4, turning: all 24 permutations of four items as the corners of a polytope, edges joining shuffles that differ by one adjacent swap. Green is the Steinhaus–Johnson–Trotter tour — a single-swap path that visits every shuffle once (a Gray code for permutations, kin to THE GRAY).
AVAN’s addition (the inverse-companion): the magenta edges join each permutation to its inverse (the shuffle that undoes it). It is an involution — a fold of the polytope onto itself, with the self-inverse shuffles as its fixed points. The clock counts every arrangement; the inverse map pairs each with its undo.
LIT A genuine factoradic + Lehmer code. Verified live: for n=6 the map is a perfect bijection between 0..719 and the 720 permutations (rank(unrank(m))=m for all m), and each column rolls over at k+1. The permutohedron (24 shuffles, single-swap edges), its SJT Gray-code tour, and the inverse involution are the real group structure (verifiable: window.__factoradic.bijection720===true).
FIG 'Clock' is the picture; the mixed-radix bijection and the permutohedron structure are exact. Addressing 'the millionth shuffle' is a literal, verified capability, not a metaphor.
FIG 'Clock' is the picture; the mixed-radix bijection and the permutohedron structure are exact. Addressing 'the millionth shuffle' is a literal, verified capability, not a metaphor.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN