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Complete classification of six-dimensional iso-edge domains.
Mathieu Dutour Sikirić1, Wessel van Woerden2
1Rudjer Bosković Institute, Bijenicka 54, 10000 Zagreb, Croatia.
Acta Crystallographica. Section A, Foundations and Advances
|November 7, 2024
Summary
This study fully classifies generic iso-edge subdivisions of six-dimensional translational lattices, yielding over 55 million distinct affine types. Canonical forms and database hashing techniques were employed for this comprehensive enumeration.
Area of Science:
- Discrete Geometry
- Computational Crystallography
- Combinatorial Theory
Background:
- Translational lattices are fundamental structures in various scientific fields.
- Iso-edge subdivisions offer a way to analyze lattice symmetries and properties.
- Previous classifications were limited in scope and dimensionality.
Purpose of the Study:
- To achieve a complete classification of generic iso-edge subdivisions for six-dimensional translational lattices.
- To enumerate all possible affine types of these subdivisions.
- To introduce efficient computational methods for lattice analysis.
Main Methods:
- Development and application of the method of canonical forms.
- Utilizing hashing techniques, commonly found in modern databases, for efficient data management.
- Systematic enumeration and classification of subdivisions based on affine types.
Main Results:
- A complete list of 55,083,357 distinct affine types of iso-edge subdivisions has been generated.
- The classification covers all generic subdivisions of six-dimensional translational lattices.
- The method of canonical forms proved effective for handling the large number of classifications.
Conclusions:
- The comprehensive classification provides a foundational dataset for further research in lattice theory.
- The employed computational methods demonstrate scalability for complex combinatorial problems.
- This work opens avenues for applying advanced database techniques to geometric and crystallographic problems.
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