Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings

@article{Hyde2003SomeNT,
  title={Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings},
  author={Stephen T. Hyde and Stuart Ramsden},
  journal={The European Physical Journal B - Condensed Matter and Complex Systems},
  year={2003},
  volume={31},
  pages={273-284}
}
  • S. HydeS. Ramsden
  • Published 2003
  • Computer Science
  • The European Physical Journal B - Condensed Matter and Complex Systems
Abstract:We demonstrate the usefulness of two-dimensional hyperbolic geometry as a tool to generate three-dimensional Euclidean (E3) networks. The technique involves projection of edges of tilings of the hyperbolic plane (H2) onto three-periodic minimal surfaces, embedded in E3. Given the extraordinary wealth of symmetries commensurate with H2, we can generate networks in E3 that are difficult to construct otherwise. In particular, we form four-, five- and seven-connected (E3) nets containing… 

Periodic entanglement I: networks from hyperbolic reticulations

This paper presents a construction technique and limited catalogue of entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces.

2D hyperbolic groups induce three-periodic Euclidean reticulations

Abstract.Many crystalline networks can be viewed as decorations of triply periodic minimal surfaces. Such surfaces are covered by the hyperbolic plane in the same way that the Euclidean plane covers

Three-dimensional entanglement: knots, knits and nets

Three-dimensional entanglement, including knots, periodic arrays of woven filaments (weavings) and periodic arrays of interpenetrating networks (nets), forms an integral part of the analysis of

Enumerating tilings of triply-periodic minimal surfaces with rotational symmetries

We present a technique for the enumeration of all isotopically distinct ways of tiling, with disks, a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This

Isotopic tiling theory for hyperbolic surfaces

In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More

Enumerating Isotopy Classes of Tilings Guided by the Symmetry of Triply Periodic Minimal Surfaces

We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a

Enumeration of index 3 and 4 subgroups of hyperbolic triangle symmetry groups

Abstract This paper explores the area of crystallography on the hyperbolic plane, in particular the study of the subgroup structure of hyperbolic symmetry groups. In this work, the index 3 and 4

References

SHOWING 1-5 OF 5 REFERENCES

Small worlds

This paper considers some particular instances of small world models, and rigorously investigates the distribution of their inter‐point network distances, framed in terms of approximations, whose accuracy increases with the size of the network.

Geometry and the Imagination

The simplest curves and surfaces Regular systems of points Projective configurations Differential geometry Kinematics Topology Index.

Geometry of surfaces

This text intends to provide the student with the knowledge of a geometry of greater scope thatn the classical geometry taught today, which is no longer an adequate basis for mathematics of physics,

Mineralogical Society of America; Washington D.C.

  • Nature,
  • 1996