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In this paper, we use a branch of polyhedral geometry, Ehrhart theory, toexpand our combinatorial understanding of congruences for partition functions.Ehrhart theory allows us to give a new decomposition of partitions, which inturn allows us to define statistics called {\it supercranks} thatcombinatorially witness every instance of divisibility of $p(n,3)$ by any prime$m \equiv -1 \pmod 6$, where $p(n,3)$ is the number of partitions of $n$ intothree parts. A rearrangement of lattice points allows us to demonstrate withexplicit bijections how to divide these sets of partitions into $m$equinumerous classes. The behavior for primes $m' \equiv 1 \pmod 6$ is alsodiscussed.

Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts