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We present limiting shape results for a non-abelian variant of the abeliansandpile growth model (ASGM), some of which have no parallel in the ASGM. Oneof our limiting shapes is an octagon. In our model, mass spreads from theorigin by the toppling rule in Zhang's sandpile model. Previously, severallimiting shape results have been obtained for the ASGM using abelianness andmonotonicity as main tools. As both properties fail for our model, we use a newproof technique: in our main proof, we introduce several cellular automata tomimic our growth model.

Limiting shapes for a non-abelian sandpile growth model and related cellular automata