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Symmetry groups associated with tilings on a flat torus
Mark L Loyola1, Ma Louise Antonette N De Las Peñas1, Grace M Estrada1
1Mathematics Department, Ateneo de Manila University, Katipunan Avenue, Loyola Heights, Quezon City, Metro Manila 1108, Philippines.
Acta Crystallographica. Section A, Foundations and Advances
|December 25, 2014
Summary
This study examines symmetry in Kepler, Heesch, and Laves tilings on flat and round tori. Researchers determined the symmetry groups of these 3D geometric models, relevant for nanotube geometric modeling.
Area of Science:
- Geometric topology
- Crystallography
- Group theory
Background:
- Tiling theory explores patterns and symmetries.
- Flat and round tori are fundamental geometric objects.
- Understanding symmetry is crucial for materials science.
Purpose of the Study:
- Investigate symmetry and color symmetry of Kepler, Heesch, and Laves tilings.
- Analyze their geometric realization on a round torus in 3D space.
- Determine the symmetry group of these 3D tilings.
Main Methods:
- Embedding planar tilings onto a round torus.
- Analyzing planar symmetry transformations.
- Identifying axial symmetries in 3D realizations.
Main Results:
- Characterization of symmetry and color symmetry for specific tiling types.
- Determination of the symmetry group for tilings on a round torus.
- Established a link between planar and 3D symmetries.
Conclusions:
- The study provides a framework for understanding tiling symmetries in 3D.
- Results are applicable to geometric modeling of nanotori.
- Symmetry group determination is key for nanotube characterization.
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