Abelian Bunker

Voici some Abelian sandpile code golf:

ab=: -[:+/@(-"2((,-)=/~i.2)|.!.0])3&<

clrs =: ".(;._2) 0 : 0
  0   0   0
 54  85 134
109 171 255
163 255 255
)
sand =: ,"_1/^:2 ] 3 3&$"0 (ab^:_) 12000 (<50 50)} 101 101$0
echo '../images/abelian-bunker.png' linki sand { clrs

Abelian sandpiles sit on vertices of a graph and the piles begin to topple over whenever the grains of sand outnumber the edges leaving the vertex. Our graph is just a grid graph, where each square on the plane is connected to its four neighbors. So, all entries in the matrix with sand \(\ge 4\) are redistributed to the neighboring squares. Eventually this process settles; we let the sand fall off the edge, if it gets there.

sand =: ,"_1/^:2 ] 3 3&$"0 (ab^:_) 101 101 $ 4
echo '../images/abelian-bunker-0.png' linki sand { |. clrs

Journey to smallest golf:

ab=: +[:(_4&*+[:+/((,-)(,:|.)0 1)|.!.0])3&< NB. first try
mw=: +[:+/@(_4&*,((,-)2]\i:1)|.!.0])3&<     NB. tangenstorm version
ab=: -[:(4&*(-+/)((,-)2]\i:1)|.!.0])3&<     NB. subtraction trick 1
ab=: -[:+/@(-"2((,-)=/~i.2)|.!.0])3&<       NB. subtraction trick 2
ae=: {{y-+/(-"2 u|.!.0])y>:#u}}             NB. direct definition version

tangentstorm's much better neighborhood phrases:

nhood =: (,-)(,:|.)0 1 NB. my original
nhood =: (,-)=/~i.2    NB. Michal Wallace's improvements
nhood =: (,-)2]\i:1