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Three-periodic nets and tilings: regular and quasiregular nets
Olaf Delgado Friedrichs1, Michael O'Keeffe, Omar M Yaghi
1Department of Chemistry, Arizona State University, Tempe 85287, USA.
Acta Crystallographica. Section A, Foundations of Crystallography
|December 24, 2002
Summary
This study defines regular nets based on symmetry, identifying five regular 3-periodic nets and one quasiregular net. These nets exhibit unique tiling properties with consistent vertex, edge, ring, and tile characteristics.
Area of Science:
- Crystallography
- Materials Science
- Geometry
Background:
- Nets are fundamental structures in various scientific fields.
- Understanding the symmetry and tiling properties of nets is crucial for materials design and theoretical studies.
Purpose of the Study:
- To define and classify regular nets based on their coordination figures.
- To investigate the natural tiling properties and transitivity of these regular and quasiregular nets.
Main Methods:
- Definition of regular nets based on regular polygonal or polyhedral coordination figures.
- Analysis of natural tilings and essential rings associated with nets.
- Classification of nets based on their symmetry and transitivity properties.
Main Results:
- Identified five distinct regular 3-periodic nets.
- Characterized one quasiregular net with a quasiregular coordination figure.
- Demonstrated that regular nets exhibit transitivity 1111 (one type of vertex, edge, ring, and tile).
- Showed that the quasiregular net has transitivity 1112 (two types of natural tiles).
Conclusions:
- The proposed definition of regular nets accurately classifies specific 3-periodic nets.
- The study reveals distinct tiling characteristics for regular and quasiregular nets, offering insights into their structural complexity.