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k-Isocoronal tilings
Eduard Taganap1, Ma Louise Antonette De Las Peñas1
1Department of Mathematics, Ateneo de Manila University, Quezon City, Metro Manila 1108, Philippines.
Acta Crystallographica. Section A, Foundations and Advances
|December 22, 2018
Summary
This study introduces a method to systematically derive k-isocoronal tilings from tile-transitive tilings. It classifies these tilings within the Euclidean plane, expanding geometric understanding.
Area of Science:
- Geometry
- Tiling Theory
- Symmetry Groups
Background:
- Isocoronal tilings are defined by the number of symmetry orbits of their vertex coronae.
- Understanding these tilings is crucial for classifying geometric patterns.
Purpose of the Study:
- To present a framework for systematically deriving planar edge-to-edge k-isocoronal tilings.
- To classify isocoronal tilings in the Euclidean plane.
Main Methods:
- Derivation of k-isocoronal tilings from tile-s-transitive tilings (s ≤ k).
- Analysis of vertex coronae and symmetry group actions.
- Classification based on transitivity classes.
Main Results:
- A systematic framework for generating k-isocoronal tilings is established.
- The relationship between k-isocoronal, vertex-k-transitive (k-isogonal), and k-uniform tilings is clarified.
- A classification of isocoronal tilings in the Euclidean plane is provided.
Conclusions:
- The presented framework enables the systematic construction and classification of k-isocoronal tilings.
- This work contributes to a deeper understanding of geometric symmetries and tiling patterns.
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