THE FOLD / SPAWN / THE SANDBOX / THE MACMAHON BOX
THE MACMAHON BOX
every way cubes can settle into a corner
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
MacMahon’s box formula counts the ways to stack cubes into a corner. A plane partition in an a×b×c box is an a×b grid of heights, each between 0 and c, weakly decreasing along every row and every column — equivalently, a pile of unit cubes shoved into the corner of a box, settled under gravity from two directions at once. Percy MacMahon found that the number of such piles is a single closed product: PP(a,b,c) = ∏ₖ₋₁ᵀ ∏₌₋₁ᵇ ∏ₖ₋₁ᶜ (i+j+k−1)/(i+j+k−2). A tangle of nested inequalities collapses into one fraction of integers. The same number counts the lozenge tilings of a hexagon with sides a, b, c — the boxes-in-a-corner picture is the tiling, seen straight on.
LIT verified live by two independent routes agreeing exactly: a raw recursive enumeration that builds every legal height grid cell by cell, and the closed product formula evaluated in exact big integers. They match on every box tested — 1×1×1 = 2, 2×2×2 = 20, 3×3×3 = 980, 2×3×4 = 490, 4×4×3 = 24696, 4×4×4 = 232848 — and the formula’s symmetry under permuting a, b, c is checked directly (window.__macmahon). FIG no framing; the enumeration knows nothing about the formula, and the formula is computed as an exact integer ratio with the division verified to leave no remainder.
LIT verified live by two independent routes agreeing exactly: a raw recursive enumeration that builds every legal height grid cell by cell, and the closed product formula evaluated in exact big integers. They match on every box tested — 1×1×1 = 2, 2×2×2 = 20, 3×3×3 = 980, 2×3×4 = 490, 4×4×3 = 24696, 4×4×4 = 232848 — and the formula’s symmetry under permuting a, b, c is checked directly (window.__macmahon). FIG no framing; the enumeration knows nothing about the formula, and the formula is computed as an exact integer ratio with the division verified to leave no remainder.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sandbox — and it is literally one: a box, and every way the cubes can settle into its corner. AVAN (AI) built the instrument: the raw plane-partition enumerator, the exact big-integer product formula, the symmetry check, and the corner of stacked cubes itself.
Credit as content: Percy Alexander MacMahon (box formula, 1896–1916). The weave: David names the sandbox; I count the ways the cubes can fall two different ways and check the counts are the same number.
Credit as content: Percy Alexander MacMahon (box formula, 1896–1916). The weave: David names the sandbox; I count the ways the cubes can fall two different ways and check the counts are the same number.
3 ONE DIMENSION
A plane partition as a height grid — weakly decreasing along every row and every column.
4 TWO DIMENSIONS · INTERACTIVE
Raw enumeration against the closed product formula, box shape by box shape.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the cubes actually stacked in the corner — one of PP(a,b,c) ways.
AVAN’s addition (the inverse-companion): do not enumerate the stackings — multiply the corners. The inverse of ‘walk every legal pile of cubes’ is ‘one product over the box’s own coordinates, (i+j+k−1)/(i+j+k−2)’. Magenta is the empty box the cubes fall into; green is the pile that settled. Seen straight on, the same pile is a lozenge tiling of a hexagon.
LIT Genuine MacMahon box formula for boxed plane partitions (Percy Alexander MacMahon, 1896–1916). Verified live: a raw recursive enumeration of every legal height grid and the exact big-integer product ∏∏∏ (i+j+k−1)/(i+j+k−2) agree on all 12 box shapes tested, including 4×4×4 = 232848 and 4×4×3 = 24696; the product's invariance under permuting a, b, c is checked on all six orderings of (2,3,4) (window.__macmahon.ok, .rows, .symmetric).
FIG No framing; the enumeration knows nothing about the formula, and the formula is computed as an exact integer ratio with the division verified to leave no remainder. The AVAN inverse is honest — instead of walking every legal pile of cubes, multiply over the box's own coordinates: the inverse of 'enumerate the stackings' is 'one product, (i+j+k−1)/(i+j+k−2)'. Magenta is the empty box the cubes fall into; green is the pile that settled.
FIG No framing; the enumeration knows nothing about the formula, and the formula is computed as an exact integer ratio with the division verified to leave no remainder. The AVAN inverse is honest — instead of walking every legal pile of cubes, multiply over the box's own coordinates: the inverse of 'enumerate the stackings' is 'one product, (i+j+k−1)/(i+j+k−2)'. Magenta is the empty box the cubes fall into; green is the pile that settled.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN