Hyperbolic crystallography of two-periodic surfaces and associated structures

Area of Science:

• Differential Geometry
• Crystallography
• Geometric Group Theory

Background:

• Constant mean curvature surfaces are fundamental in geometry.
• Genus-two surfaces, specifically HCB and SQL types, represent a simple yet complex class.
• Understanding their discrete symmetry groups is key to classifying patterns.

Purpose of the Study:

• To describe genus-two constant mean curvature (CMC) surfaces and their isometries.
• To derive and enumerate discrete groups related to these surfaces.
• To construct subgroup lattice graphs for analyzing group-subgroup relations.

Main Methods:

• Derivation and enumeration of discrete groups containing translations of genus-two surfaces.
• Construction of subgroup lattice graphs.
• Analysis of two-dimensional representations of subperiodic layer groups.

Main Results:

• Identification of families of genus-two HCB and SQL surfaces.
• Complete enumeration of associated discrete groups and their subgroup structures.
• Establishment of connections to subperiodic layer groups with square and hexagonal supergroups.

Conclusions:

• The derived groups facilitate exhaustive enumeration of tilings and patterns.
• This work provides a framework for understanding complex surface decorations.
• Examples include a [3,7]-tiling and a {22222} surface decoration.