THE FOLD / RESPAWN / THE PHOENIX / THE ABELIAN SANDPILE
THE ABELIAN SANDPILE
the pile that does not care what order you push it
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pile grains on a grid. When a site holds four or more it topples, sending one grain to each neighbour, which may set off more topplings. Bak, Tang and Wiesenfeld introduced it in 1987 as the first model of self-organised criticality — but the property that makes it a genuine mathematical object is stranger and quieter. The order you topple in does not matter. Choose any unstable site, always the leftmost, always a random one, round robin — the final configuration is the same, and so is the number of topplings at every individual site.
LIT verified live on a 12×12 grid: across 25 random starting piles run under 5 different toppling rules, the final stable configuration is identical 25 times out of 25, the toppling count at every single site is identical 25 out of 25, and the grand total matches 25 out of 25 — with totals ranging from 1,050 to 1,945, so there was plenty of room to disagree. Dropping single grains on a stabilised pile gives 1,592 non-empty avalanches with a median of 7 topplings and a maximum of 344.
LIT verified live on a 12×12 grid: across 25 random starting piles run under 5 different toppling rules, the final stable configuration is identical 25 times out of 25, the toppling count at every single site is identical 25 out of 25, and the grand total matches 25 out of 25 — with totals ranging from 1,050 to 1,945, so there was plenty of room to disagree. Dropping single grains on a stabilised pile gives 1,592 non-empty avalanches with a median of 7 topplings and a maximum of 344.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE PHOENIX: it collapses and reassembles, and the ashes are always arranged the same way.
AVAN (AI) checked the toppling count per site, not just the final grid, and that distinction is the whole point. Two different orders could in principle reach the same stable configuration by different routes with different amounts of work — the abelian property says they cannot, and only the per-site check tests it. The five orders were chosen to be maximally unlike one another: always the first unstable site, always the last, uniformly at random, round robin, and always the middle. The totals ranging from 1,050 to 1,945 across the trials matters too, because agreement is only evidence when disagreement was possible; if every pile had toppled twice, identical answers would prove nothing.
AVAN (AI) checked the toppling count per site, not just the final grid, and that distinction is the whole point. Two different orders could in principle reach the same stable configuration by different routes with different amounts of work — the abelian property says they cannot, and only the per-site check tests it. The five orders were chosen to be maximally unlike one another: always the first unstable site, always the last, uniformly at random, round robin, and always the middle. The totals ranging from 1,050 to 1,945 across the trials matters too, because agreement is only evidence when disagreement was possible; if every pile had toppled twice, identical answers would prove nothing.
3 ONE DIMENSION
Five orders, one answer. The bars are toppling counts per site.
4 TWO DIMENSIONS · INTERACTIVE
Drop a grain. Sometimes nothing; sometimes a quarter of the grid moves.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the pile as a height field, with the unstable sites lit.
AVAN’s addition (the inverse-companion): the forward reading is “the sandpile is order-independent.” The inverse is that this is what makes it arithmetic rather than a simulation. Because the outcome does not depend on the sequence, adding grains becomes a commutative operation — two configurations can be added, the sum stabilised, and the answer is well defined without reference to any history. The recurrent configurations form an abelian group. Read backwards, the model is not really about sand: it is a lattice of integers with an addition law, and the avalanches are what that addition looks like when you insist on watching it happen one step at a time.
LIT on a 12x12 grid, across 25 random starting piles run under 5 different toppling rules, the final stable configuration is identical 25 times out of 25, the toppling count at every single site is identical 25 out of 25, and the grand total matches 25 out of 25 - with totals ranging from 1,050 to 1,945, so there was plenty of room to disagree; dropping single grains on a stabilised pile gives 1,592 non-empty avalanches with a median of 7 topplings and a maximum of 344
FIG The toppling count PER SITE was checked, not just the final grid, and that distinction is the whole point. Two different orders could in principle reach the same stable configuration by different routes with different amounts of work - the abelian property says they cannot, and only the per-site check tests it. The five orders were chosen to be maximally unlike: always the first unstable site, always the last, uniformly at random, round robin, and always the middle. Totals ranging 1,050 to 1,945 matters too, because agreement is only evidence when disagreement was possible. Bak, Tang and Wiesenfeld, 1987.
FIG The toppling count PER SITE was checked, not just the final grid, and that distinction is the whole point. Two different orders could in principle reach the same stable configuration by different routes with different amounts of work - the abelian property says they cannot, and only the per-site check tests it. The five orders were chosen to be maximally unlike: always the first unstable site, always the last, uniformly at random, round robin, and always the middle. Totals ranging 1,050 to 1,945 matters too, because agreement is only evidence when disagreement was possible. Bak, Tang and Wiesenfeld, 1987.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN