Hexagonal and trigonal quasiperiodic tilings
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866912505981304832 |
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| author | Coates, Sam Koga, Akihisa Matsubara, Toranosuke Tamura, Ryuji Sharma, Hem Raj McGrath, Ronan Lifshitz, Ron |
| author_facet | Coates, Sam Koga, Akihisa Matsubara, Toranosuke Tamura, Ryuji Sharma, Hem Raj McGrath, Ronan Lifshitz, Ron |
| contents | Exploring nonminimal-rank quasicrystals, which have symmetries that can be found in both periodic and aperiodic crystals, often provides new insight into the physical nature of aperiodic long-range order in models that are easier to treat. Motivated by the prevalence of experimental systems exhibiting aperiodic long-range order with hexagonal and trigonal symmetry, we introduce a generic two-parameter family of 2-dimensional quasiperiodic tilings with such symmetries. We focus on the special case of trigonal and hexagonal Fibonacci, or golden-mean, tilings, analogous to the well studied square Fibonacci tiling. We first generate the tilings using a generalized version of de Bruijn's dual grid method. We then discuss their interpretation in terms of projections of a hypercubic lattice from six dimensional superspace. We conclude by concentrating on two of the hexagonal members of the family, and examining a few of their properties more closely, while providing a set of substitution rules for their generation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_11848 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hexagonal and trigonal quasiperiodic tilings Coates, Sam Koga, Akihisa Matsubara, Toranosuke Tamura, Ryuji Sharma, Hem Raj McGrath, Ronan Lifshitz, Ron Soft Condensed Matter Exploring nonminimal-rank quasicrystals, which have symmetries that can be found in both periodic and aperiodic crystals, often provides new insight into the physical nature of aperiodic long-range order in models that are easier to treat. Motivated by the prevalence of experimental systems exhibiting aperiodic long-range order with hexagonal and trigonal symmetry, we introduce a generic two-parameter family of 2-dimensional quasiperiodic tilings with such symmetries. We focus on the special case of trigonal and hexagonal Fibonacci, or golden-mean, tilings, analogous to the well studied square Fibonacci tiling. We first generate the tilings using a generalized version of de Bruijn's dual grid method. We then discuss their interpretation in terms of projections of a hypercubic lattice from six dimensional superspace. We conclude by concentrating on two of the hexagonal members of the family, and examining a few of their properties more closely, while providing a set of substitution rules for their generation. |
| title | Hexagonal and trigonal quasiperiodic tilings |
| topic | Soft Condensed Matter |
| url | https://arxiv.org/abs/2201.11848 |