  We describe the statistical mechanics of a melting crystal in three
dimensions and its relation to a diverse range of models arising in
combinatorics, algebraic geometry, integrable systems, low-dimensional gauge
theories, topological string theory and quantum gravity. Its partition function
can be computed by enumerating the contributions from noncommutative instantons
to a six-dimensional cohomological gauge theory, which yields a dynamical
realization of the crystal as a discretization of spacetime at the Planck
scale. We describe analogous relations between a melting crystal model in two
dimensions and N=4 supersymmetric Yang-Mills theory in four dimensions. We
elaborate on some mathematical details of the construction of the quantum
geometry which combines methods from toric geometry, isospectral deformation
theory and noncommutative geometry in braided monoidal categories. In
particular, we relate the construction of noncommutative instantons to deformed
ADHM data, torsion-free modules and a noncommutative twistor correspondence.
