  We study crystal growth inside an infinite octant on a cubic lattice. The
growth proceeds through the deposition of elementary cubes into inner corners.
After re-scaling by the characteristic size, the interface becomes
progressively more deterministic in the long-time limit. Utilizing known
results for the crystal growth inside a two-dimensional corner, we propose a
hyperbolic partial differential equation for the evolution of the limiting
shape. This equation is interpreted as a Hamilton-Jacobi equation which helps
in finding an analytical solution. Simulations of the growth process are in
excellent agreement with analytical predictions. We then study the evolution of
the sub-leading correction to the volume of the crystal, the asymptotic growth
of the variance of the volume of the crystal, and the total number of inner and
outer corners. We also show how to generalize the results to arbitrary spatial
dimension.
