  We construct a combinatorial model that is described by the cube recurrence,
a nonlinear recurrence relation introduced by Propp, which generates families
of Laurent polynomials indexed by points in $\mathbb{Z}^3$. In the process, we
prove several conjectures of Propp and of Fomin and Zelevinsky, and we obtain a
combinatorial interpretation for the terms of Gale-Robinson sequences. We also
indicate how the model might be used to obtain some interesting results about
perfect matchings of certain bipartite planar graphs.
