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Growth functions of periodic space tessellations
Bartosz Naskręcki1, Jakub Malinowski2, Zbigniew Dauter3
1Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland.
Acta Crystallographica. Section A, Foundations and Advances
|December 5, 2024
Summary
This study reveals polynomial growth functions that describe periodic tessellations in 2D Euclidean space. These functions encode geometric and topological properties, aiding in the discovery of complex spatial patterns.
Area of Science:
- Mathematics
- Geometry
- Topology
Background:
- Periodic tessellations are fundamental geometric structures with applications in various scientific fields.
- Understanding the growth rules of these tessellations is crucial for analyzing their complexity.
Purpose of the Study:
- To analyze and encode the growth rules of vertices, edges, and faces in 2D periodic tessellations.
- To develop polynomial growth functions that represent the geometric, combinatorial, and topological properties of tessellations.
Main Methods:
- Mathematical analysis of growth rules in periodic tessellations.
- Development of polynomial growth functions and encoding of tessellation properties.
- Graphical representation and analysis using orphic diagrams.
- Inclusion of 3D space group examples to illustrate higher-dimensional complexity.
Main Results:
- Identification of specific polynomial growth functions governing tessellations.
- Encoding of geometric, combinatorial, and topological properties into integer coefficients.
- Rigorous mathematical proofs for general statements about these encodings.
- Visualization of growth function variations through orphic diagrams.
Conclusions:
- The study provides a systematic method for analyzing and understanding periodic tessellations.
- The developed growth functions and orphic diagrams offer new tools for geometric and topological research.
- A Python library is introduced to support further research and discovery in this area.
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