Abelian Groups
Michael Creutz and Per Bak showed that the sandpile model forms an abelian group under the operation of addition. This means that the operation of sand addition to the system is commutative, and that the grid model for a given size grid has an identity state.
Self-Organized Criticality
Bak and others have concluded that that the sandpile model exhibits self-organized criticality. Their experiments involved creating a sandpile graph with random sand values and allowing the system to stablize. The effect of setting each cell in the graph (individually) to the threshold value was studied on 50 X 50 and smaller grids. The number of cells affected and the time to reach stabilization where shown to follow a power law behavior. The exponent for these grid sandpile models has always been found to be <= 2. It has been suggested that modifying the model in some way to generate different exponents may introduce further applications for these type of models.