Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice <i>A<sub>n</sub></i>
The orthogonal projections of the Voronoi and Delone cells of root lattice <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-08-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/9/1082 |
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| author | Nazife Ozdes Koca Abeer Al-Siyabi Mehmet Koca Ramazan Koc |
| author_facet | Nazife Ozdes Koca Abeer Al-Siyabi Mehmet Koca Ramazan Koc |
| author_sort | Nazife Ozdes Koca |
| collection | DOAJ |
| description | The orthogonal projections of the Voronoi and Delone cells of root lattice <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman−Beenker tiles, Robinson triangles, and Danzer triangles to name a few. We point out that the symmetries representing the dihedral subgroup of order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </semantics> </math> </inline-formula> involving the Coxeter element of order <inline-formula> <math display="inline"> <semantics> <mrow> <mi>h</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of the Coxeter−Weyl group <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> play a crucial role for <inline-formula> <math display="inline"> <semantics> <mi>h</mi> </semantics> </math> </inline-formula>-fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4-, 5-, 6-, 7-, 8-, and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mrow> <mn>3</mn> </mrow> </msub> </mrow> </semantics> </math> </inline-formula>, whose Wigner−Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an <inline-formula> <math display="inline"> <semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math> </inline-formula>-fold symmetry. |
| first_indexed | 2024-04-11T20:56:17Z |
| format | Article |
| id | doaj.art-2d4263c8f7bd4613a2148d98f42b93af |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T20:56:17Z |
| publishDate | 2019-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-2d4263c8f7bd4613a2148d98f42b93af2022-12-22T04:03:40ZengMDPI AGSymmetry2073-89942019-08-01119108210.3390/sym11091082sym11091082Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice <i>A<sub>n</sub></i>Nazife Ozdes Koca0Abeer Al-Siyabi1Mehmet Koca2Ramazan Koc3Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, 123 Muscat, OmanDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, 123 Muscat, OmanDepartment of Physics, Cukurova University, 1380 Adana, TurkeyDepartment of Physics, Gaziantep University, 27310 Gaziantep, TurkeyThe orthogonal projections of the Voronoi and Delone cells of root lattice <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman−Beenker tiles, Robinson triangles, and Danzer triangles to name a few. We point out that the symmetries representing the dihedral subgroup of order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </semantics> </math> </inline-formula> involving the Coxeter element of order <inline-formula> <math display="inline"> <semantics> <mrow> <mi>h</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of the Coxeter−Weyl group <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> play a crucial role for <inline-formula> <math display="inline"> <semantics> <mi>h</mi> </semantics> </math> </inline-formula>-fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4-, 5-, 6-, 7-, 8-, and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mrow> <mn>3</mn> </mrow> </msub> </mrow> </semantics> </math> </inline-formula>, whose Wigner−Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an <inline-formula> <math display="inline"> <semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math> </inline-formula>-fold symmetry.https://www.mdpi.com/2073-8994/11/9/1082LatticesCoxeter–Weyl groupsVoronoi and Delone cellstilings by rhombi and triangles |
| spellingShingle | Nazife Ozdes Koca Abeer Al-Siyabi Mehmet Koca Ramazan Koc Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice <i>A<sub>n</sub></i> Symmetry Lattices Coxeter–Weyl groups Voronoi and Delone cells tilings by rhombi and triangles |
| title | Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice <i>A<sub>n</sub></i> |
| title_full | Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice <i>A<sub>n</sub></i> |
| title_fullStr | Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice <i>A<sub>n</sub></i> |
| title_full_unstemmed | Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice <i>A<sub>n</sub></i> |
| title_short | Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice <i>A<sub>n</sub></i> |
| title_sort | prototiles and tilings from voronoi and delone cells of the root lattice i a sub n sub i |
| topic | Lattices Coxeter–Weyl groups Voronoi and Delone cells tilings by rhombi and triangles |
| url | https://www.mdpi.com/2073-8994/11/9/1082 |
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