  Building on earlier work of Diaconis and Fulton (1991) and Lawler, Bramson,
and Griffeath (1992), Propp in 2001 defined a deterministic analogue of
internal diffusion-limited aggregation. This growth model is a "convergent
game" of the sort studied by Eriksson (1996). In one dimension, we show that
the model is equivalent to a simple dynamical system with three integer-valued
parameters; an invariant of this dynamical system yields links between the
behavior of the system and diophantine approximation of quadratic irrationals.
In two dimensions, we give constraints on the shape of the occupied region, as
well as evidence for the proposition that the asymptotic shape of the boundary,
suitably rescaled as time goes to infinity, is a circle. While the rotor-router
model displays many of the same intriguing features as the Bak-Tang-Weisenfeld
abelian sandpile model, we suggest that the new model is likely to be easier to
analyze rigorously.
