  We establish quantitative spherical shape theorems for rotor-router
aggregation and abelian sandpile growth on the graphical Sierpinski gasket
($SG$) when particles are launched from the corner vertex.
  In particular, the abelian sandpile growth problem is exactly solved via a
recursive construction of self-similar sandpile tiles. We show that sandpile
growth and patterns exhibit a $(2\cdot 3^n)$-periodicity as a function of the
initial mass. Moreover, the cluster explodes---increments by more than 1 in
radius---at periodic intervals, a phenomenon not seen on $\mathbb{Z}^d$ or
trees. We explicitly characterize all the radial jumps, and use the renewal
theorem to prove the scaling limit of the cluster radius, which satisfies a
power law modulated by log-periodic oscillations. In the course of our proofs
we also establish structural identities of the sandpile groups of subgraphs of
$SG$ with two different boundary conditions, notably the corresponding identity
elements conjectured by Fairchild, Haim, Setra, Strichartz, and Westura.
  Our main theorems, in conjunction with recent results of Chen, Huss,
Sava-Huss, and Teplyaev, establish $SG$ as a positive example of a state space
which exhibits "limit shape universality," in the sense of Levine and Peres,
among the four Laplacian growth models: divisible sandpiles, abelian sandpiles,
rotor-router aggregation, and internal diffusion-limited aggregation (IDLA). We
conclude the paper with conjectures about radial fluctuations in IDLA on $SG$,
possible extensions of limit shape universality to other state spaces, and
related open problems.
