You can easily generalize for n dimensions by (albeit incredible slow!!!!):

ClearAll@Sandpile    
Sandpile[x_SparseArray] := Sandpile@Normal@x
Normal@Sandpile[s_List] ^:= s    
Sandpile /: Sandpile[a_List] + Sandpile[n_Integer] := Sandpile@a + Sandpile[ConstantArray[n, Dimensions@a]]
Sandpile /: Sandpile[a_List] + Sandpile[b_List] /; Dimensions@a === Dimensions@b := Module[
  {sum, temp, rest, sides},
  sum = a + b;      
  sides = Permutations@Append[ConstantArray[0, {ArrayDepth@a - 1}], 1];
  sides = Join[sides, -sides];
  While[Max@sum >= 4,
   sum = ArrayPad[sum, 1, 0];
   temp = Quotient[sum, 4];
   sum -= 4 temp;
   sum += Total[RotateLeft[temp, ##1] & /@ sides];
   sum = ArrayPad[sum, -1]
   ];
  Sandpile@sum
  ]

Sandpile$Colors = {0 -> Black, 1 -> Yellow, 2 -> Blue, 3 -> Red, _Integer -> Pink};
Format[Sandpile[t_List /; ArrayDepth@t <= 2]] := ArrayPlot[If[ArrayDepth@t == 1, {t}, t] /. Sandpile$Colors, 
  ImageSize -> 400, Frame -> False]
Format[Sandpile[t_List /; ArrayDepth@t == 3]] := ListDensityPlot3D[t, ImageSize -> 400, Axes -> False,
  ColorFunction -> (Blend[{{0, Black}, {1, Yellow}, {2, Blue}, {3, Red}}, #] &),
  ColorFunctionScaling -> False, SphericalRegion -> True, Boxed -> False]

For me it's just conservation of sand

As a physicist myself, I'm ashamed I haven't thought that...

Perhaps a better generalization would've be a graph, where a node topple its content to its neighbors if it has more sand than neighbors and the outer nodes (boundary) works as a dump site, where it can hold an infinite amount of sand without toppling.

I'm still a bit confused on the syntax

You can find more info at stackexchange. I just prefer TagSet over UpSet, since it makes more obvious what upvalues you are modifying.

You won the price :) If you're going to the tech event in Amsterdam I'll buy you a beer!

- Congratulations! This post is now a Staff Pick! Thank you for your wonderful contributions. Please, keep them coming!

I don't often use ^:= but this was a good point to use it. Though I guess one could also use TagSet, not sure though. Both I only use once a year or so haha. Also I thought it would be a good idea to show the Format[...] 'trick', most people don't know about it, but can be very useful to see what is going on...