  A fundamental characteristic of matter is its ability to form ordered
structures under the right thermodynamic conditions. Predicting these
structures - and their properties - from the attributes of a material's
building blocks is the holy grail of materials science. Here, we investigate
the self-assembly of 145 hard convex polyhedra whose thermodynamic behavior
arises solely from their anisotropic shape. Our results extend previous works
on entropy-driven crystallization by demonstrating a remarkably high propensity
for self-assembly and an unprecedented structural diversity, including some of
the most complex crystalline phases yet observed in a non-atomic system. In
addition to 22 Bravais and non-Bravais crystals, we report 66 plastic crystals
(both Bravais and topologically close-packed), 21 liquid crystals (nematic,
smectic, and columnar), and 44 glasses. We show that from simple measures of
particle shape and local order in the disordered fluid, the class of ordered
structure can be predicted.
