# Available student project - Three-dimensional crystalline structures from two-dimensional hyperbolic tilings

## Research fields

- Topological and Structural Science
- Materials Science and Engineering

## Project details

Crystalline frameworks (nets) are a standard way to describe three-dimensional (3D) periodic structures in solid-state science. To achieve directed logical design of new materials, we need to know what framework structures are possible and which are most likely to form from a given set of building blocks.

The variety of 3D networks that can be realised in euclidean space is far from completely understood. We have developed a method for creating nets by mapping tilings of the 2D hyperbolic plane onto periodic minimal surfaces in 3D space. The techniques involve discrete symmetry groups, combinatorial tiling theory, computational and differential geometry, and advanced computer visualisation.

Some of the thousands of structures generated in this way are catalogued in the online database EPINET, which also includes a comprehensive overview of our techniques.

Recent applications of these ideas have included a model for the structure of keratin fibres in the outer layers of skin,and the self assembly of three-armed star copolymers into complex striped interwoven domains.

Open problems include:

- Systematic enumeration of multi-component interpenetrating nets.
- Characterisation of ambient isotopy classes of interwoven structures (periodic knot theory!).
- Data-mining physical properties (such as percolation thresholds or elasticity) of previously enumerated nets.
- Online interactive generation and caching of network structures.

## Required background

Willingness to engage with mathematical, physical, and computational disciplines.

Some prior knowledge of graph theory, group theory, topology, or differential geometry would be beneficial.

## Project suitability

- 3rd year special project
- PhB (2nd or 3rd year)
- Honours project
- Phd or Masters