  In this work we propose a generalization of Winfree's abstract Tile Assembly
Model (aTAM) in which tile types are assigned rigid shapes, or geometries,
along each tile face. We examine the number of distinct tile types needed to
assemble shapes within this model, the temperature required for efficient
assembly, and the problem of designing compact geometric faces to meet given
compatibility specifications. Our results show a dramatic decrease in the
number of tile types needed to assemble $n \times n$ squares to
$\Theta(\sqrt{\log n})$ at temperature 1 for the most simple model which meets
a lower bound from Kolmogorov complexity, and $O(\log\log n)$ in a model in
which tile aggregates must move together through obstacle free paths within the
plane. This stands in contrast to the $\Theta(\log n / \log\log n)$ tile types
at temperature 2 needed in the basic aTAM. We also provide a general method for
simulating a large and computationally universal class of temperature 2 aTAM
systems with geometric tiles at temperature 1. Finally, we consider the problem
of computing a set of compact geometric faces for a tile system to implement a
given set of compatibility specifications. We show a number of bounds on the
complexity of geometry size needed for various classes of compatibility
specifications, many of which we directly apply to our tile assembly results to
achieve non-trivial reductions in geometry size.
