  We introduce a definition of algorithmic symmetry able to capture essential
aspects of geometric symmetry. We review, study and apply a method for
approximating the algorithmic complexity (also known as Kolmogorov-Chaitin
complexity) of graphs and networks based on the concept of Algorithmic
Probability (AP). AP is a concept (and method) capable of recursively
enumeration all properties of computable (causal) nature beyond statistical
regularities. We explore the connections of algorithmic complexity---both
theoretical and numerical---with geometric properties mainly symmetry and
topology from an (algorithmic) information-theoretic perspective. We show that
approximations to algorithmic complexity by lossless compression and an
Algorithmic Probability-based method can characterize properties of
polyominoes, polytopes, regular and quasi-regular polyhedra as well as
polyhedral networks, thereby demonstrating its profiling capabilities.
