Hyperbolic crystallography of two-periodic surfaces and associated structures

Research output: Contribution to journalJournal articlepeer-review

  • Martin Cramer Pedersen
  • Stephen T. Hyde

This paper describes the families of the simplest, two-periodic constant mean curvature surfaces, the genus-two HCB and SQL surfaces, and their isometries. All the discrete groups that contain the translations of the genus-two surfaces embedded in Euclidean three-space modulo the translation lattice are derived and enumerated. Using this information, the subgroup lattice graphs are constructed, which contain all of the group-subgroup relations of the aforementioned quotient groups. The resulting groups represent the two-dimensional representations of subperiodic layer groups with square and hexagonal supergroups, allowing exhaustive enumeration of tilings and associated patterns on these surfaces. Two examples are given: a two-periodic [3,7]-tiling with hyperbolic orbifold symbol and a surface decoration.The intrinsic, hyperbolic crystallography of the two-periodic, genus-two HCB and SQL surfaces is presented. All discrete groups containing the translations of the Euclidean embeddings of these surfaces are derived and examples of applications are given.

Original languageEnglish
JournalActa Crystallographica Section A: Foundations and Advances
Volume73
Issue number2
Pages (from-to)124-134
Number of pages11
ISSN0108-7673
DOIs
Publication statusPublished - 1 Mar 2017
Externally publishedYes

    Research areas

  • constant mean curvature surfaces, hyperbolic crystallography, hyperbolic geometry

ID: 229370446