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3-Periodic weavings: piecewise linear embeddings of symmetric structures
Michael O'Keeffe1, Michael M J Treacy2
1School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287, USA.
Summary
Researchers explored symmetric 3-periodic weavings using symmetry and coordinates. A novel dia-w graph generates infinite supersymmetric thread weaves, a property unique to this and the 2-periodic sql-w fabric weaves.
Area of Science:
- Materials Science
- Crystallography
- Topology
Background:
- Symmetry and coordinates are fundamental to understanding complex structures.
- Periodic weavings, particularly 3-periodic ones, present unique geometric challenges.
- Previous work has explored 2-periodic fabric weaves, like the sql-w family.
Purpose of the Study:
- To systematically explore symmetric (vertex- and edge-transitive) 3-periodic weavings of piecewise linear threads.
- To identify and characterize new families of weavings with unique symmetry properties.
- To relate these weavings to known crystallographic structures and graph families.
Main Methods:
- Utilizing symmetry principles and coordinate systems for structural analysis.
- Deriving the dia-w graph from the diamond structure.
- Applying graph theory to classify and generate weavings.
Main Results:
- An infinite family of supersymmetric 3-periodic thread weaves was generated from the dia-w graph.
- This family exhibits a unique property shared only with the 2-periodic sql-w fabric weaves.
- Symmetric 3-periodic, 3-coordinated weavings related to the srs graph were described.
- A supersymmetric Borromean weave of hcb graphs was also presented.
Conclusions:
- The dia-w graph is a powerful tool for generating complex, highly symmetric weavings.
- The discovered weavings expand the known catalog of 3-periodic structures.
- These findings offer new insights into the interplay of symmetry, topology, and geometry in materials and abstract structures.
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