Symmetry and Algorithmic Complexity of Polyominoes and Polyhedral Graphs

TL;DR

This paper introduces a new way to measure geometric symmetry using algorithmic complexity, applying it to polyominoes and polyhedral graphs to reveal their structural properties.

Contribution

It develops an algorithmic symmetry definition and applies an AP-based complexity approximation to analyze geometric and topological features of polyominoes and polyhedral networks.

Findings

Algorithmic complexity correlates with geometric symmetry.

AP-based methods effectively characterize polyhedral properties.

Compression-based approximations can profile complex geometric structures.

Abstract

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.