Computing the bridge length: the key ingredient in a continuous isometry classification of periodic point sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911345549508608 |
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| author | McManus, Jonathan Kurlin, Vitaliy |
| author_facet | McManus, Jonathan Kurlin, Vitaliy |
| contents | The fundamental model of any periodic crystal is a periodic set of points at all atomic centres. Since crystal structures are determined in a rigid form, their strongest equivalence is rigid motion (composition of translations and rotations) or isometry (also including reflections). The recent classification of periodic point sets under rigid motion used a complete invariant isoset whose size essentially depends on the bridge length, defined as the minimum `jump' that suffices to connect any points in the given set.
We propose a practical algorithm to compute the bridge length of any periodic point set given by a motif of points in a periodically translated unit cell. The algorithm has been tested on a large crystal dataset and is required for an efficient continuous classification of all periodic crystals. The exact computation of the bridge length is a key step to realising the inverse design of materials from new invariant values. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_23288 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing the bridge length: the key ingredient in a continuous isometry classification of periodic point sets McManus, Jonathan Kurlin, Vitaliy Computational Geometry Materials Science 68U05, 74E15, 51M15 The fundamental model of any periodic crystal is a periodic set of points at all atomic centres. Since crystal structures are determined in a rigid form, their strongest equivalence is rigid motion (composition of translations and rotations) or isometry (also including reflections). The recent classification of periodic point sets under rigid motion used a complete invariant isoset whose size essentially depends on the bridge length, defined as the minimum `jump' that suffices to connect any points in the given set. We propose a practical algorithm to compute the bridge length of any periodic point set given by a motif of points in a periodically translated unit cell. The algorithm has been tested on a large crystal dataset and is required for an efficient continuous classification of all periodic crystals. The exact computation of the bridge length is a key step to realising the inverse design of materials from new invariant values. |
| title | Computing the bridge length: the key ingredient in a continuous isometry classification of periodic point sets |
| topic | Computational Geometry Materials Science 68U05, 74E15, 51M15 |
| url | https://arxiv.org/abs/2410.23288 |