TITLE:
Geometric and Topological Approaches to Crystallography—Connections between Symmetry, Lattices, Crystalline Structures
AUTHORS:
Ellie Richwine, Lucian M. Ionescu
KEYWORDS:
Topological Crystallography, Graph Cohomology, Bravais Lattices, Abel-Jacobi Maps, Materials Science, Crystal Networks
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.8,
August
31,
2026
ABSTRACT: This article explores the mathematical structures underpinning crystalline materials, bridging the gap between pure mathematics and materials science. To clarify the scope of this work, the paper is divided into a tutorial exposition and an original theoretical contribution. The tutorial components provide a rigorous yet accessible introduction to the geometric and topological modeling of crystals, building upon Toshikazu Sunada’s breakthrough framework of topological crystallography and subsequent formalizations by John C. Baez. We examine polyhedral geometry, duality, and lattice arrangements such as the Eisenstein and triangular lattices, framing them within the context of covering maps and Abel-Jacobi maps. Our original mathematical contribution is presented in Section 11, where we advance this foundation by introducing a simplified formulation of Graph Cohomology. The precise novelty of this section lies in its use of short exact sequences of graphs to categorify the connectivity of crystalline networks. By applying integer cohomology to quotient maps, this approach provides a unifying architectural template capable of rigorously tracking lattice defects as non-trivial cohomology classes. We conclude by discussing the broader heuristic applications of these tools in molecular biology, theoretical physics, and fault-tolerant quantum engineering.