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Tangency properties of a pentagonal tiling generated by a piecewise isometry
Marcello Trovati1, Peter Ashwin
1Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter Exeter EX4 4QF, United Kingdom.
Chaos (Woodbury, N.Y.)
|January 1, 2008
Summary
Piecewise isometries (PWIs) create geometric packings. A specific PWI generates a pentagonal cell packing whose tangency graph is a forest, unlike other PWIs that can form infinite cycles.
Area of Science:
- Dynamical systems
- Geometric topology
- Computational geometry
Background:
- Piecewise isometries (PWIs) exhibit complex dynamics.
- PWIs can generate intricate geometric structures and planar packings.
- The study of tangency graphs reveals connectivity properties of geometric packings.
Purpose of the Study:
- To analyze the geometric packing generated by a specific PWI introduced by Goetz.
- To determine the structure of the tangency graph for this pentagonal cell packing.
- To investigate whether the forest property of the tangency graph is general for PWIs.
Main Methods:
- Analysis of a specific piecewise isometry (PWI).
- Characterization of periodically coded cells as pentagons.
- Construction and examination of the tangency graph for the generated packing.
Main Results:
- The tangency graph associated with Goetz's PWI packing is a forest (contains no nontrivial cycles).
- This forest property is demonstrated through detailed analysis of the packing's connectivity.
- A counterexample is provided, showing a PWI that generates a packing with an infinite number of cycles in its tangency graph.
Conclusions:
- The specific PWI analyzed generates a planar packing with a tree-like connectivity structure.
- The findings highlight that not all PWIs produce packings with acyclic tangency graphs.
- This research contributes to understanding the diverse topological properties of geometric packings generated by PWIs.
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