#
Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice A_{n}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Root Lattice ${A}_{n}$ and Its Coxeter–Weyl Group

## 3. Projections of the Faces of the Voronoi and Delone Cells

## 4. Examples of Prototiles and Patches of Tilings

#### 4.1. Projection of the Voronoi Cell of ${A}_{3}$

#### 4.2. Projection of the Delone Cells of the Root Lattice ${A}_{3}$

#### 4.3. Projection of the Voronoi Cell of the Root Lattice ${A}_{4}$

#### 4.4. Projection of the Root Lattice ${A}_{4}$ by Delone Cells

#### 4.5. Projection of the Voronoi Cell of the Root Lattice ${A}_{5}$

#### 4.6. Projection of the Delone Cells of the Root Lattice ${A}_{5}$

#### 4.7. Prototiles from the Projection of the Voronoi Cell $V\left(0\right)$ of the Root Lattice ${A}_{6}$

#### 4.8. Projection of the Delone Cells of the Root Lattice ${A}_{6}$

#### 4.9. Prototiles from the Projection of the Voronoi CELL $V\left(0\right)$ of the Root Lattice ${A}_{7}$

#### 4.10. Projection of the Delone Cells of the Root Lattice ${A}_{7}$

#### 4.11. Prototiles from Projection of the Voronoi Cell of the Root Lattice ${A}_{11}$

#### 4.12. Prototiles from the Projection of the Delone Cells of the Root Lattice ${A}_{11}$

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Three-Dimensional Cube and the Voronoi Cell $V\left(0\right)$ of ${A}_{2}$

#### Appendix A.2. Five-Dimensional Cube and the Voronoi Cell $V\left(0\right)$ of ${A}_{4}$

## References

- Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett.
**1984**, 53, 1951–1953. [Google Scholar] [CrossRef] - Di Vincenzo, D.; Steinhardt, P.J. Quasicrystals: The State of the Art; World Scientific Publishers: Singapore, 1991. [Google Scholar]
- Janot, C. Quasicrystals: A Primer, 2rd ed.; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Senechal, M. Quasicrystals and Geometry; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Baake, M.; Grimm, U. Aperiodic Order, Volume 1: A Mathematical Invitation; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Grünbaum, B.; Shephard, G.C. Tilings and Patterns; Freeman: New York, NY, USA, 1987. [Google Scholar]
- Penrose, R. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl.
**1974**, 10, 266–271. [Google Scholar] - Penrose, R. Roger Penrose’s Pentaplexity article on aperiodic tiling. Eureka
**1978**, 39, 16–22. [Google Scholar] - de Bruijn, N.G. Algebraic theory of Penrose’s non-periodic tilings of the plane. Nederl. Akad. Wetensch. Proc. Ser.
**1981**, 84, 38–66. [Google Scholar] [CrossRef] - Duneau, M.; Katz, A. Quasiperiodic patterns. Phys. Rev. Lett.
**1985**, 54, 2688–2691. [Google Scholar] [CrossRef] [PubMed] - Baake, M.; Joseph, D.; Kramer, P.; Schlottmann, M. Root lattices and quasicrystals. J. Phys. A Math. Gen.
**1990**, 23, L1037–L1041. [Google Scholar] [CrossRef] [Green Version] - Chen, L.; Moody, R.V.; Patera, J. Fields Institute for Research in Mathematical Sciences Monographs Series 10; AMS: Providence, RI, USA, 1998; pp. 135–178. [Google Scholar]
- Whittaker, E.J.W.; Whittaker, R.M. Some generalized Penrose patterns from projections of n-dimensional lattices. Acta Crystallogr. Sect. A Found. Adv.
**1988**, 44, 105–112. [Google Scholar] - Koca, M.; Koca, N.; Koc, R. Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices B
_{n}. Acta Crystallogr. Sect. A Found. Adv.**2015**, 71, 175–185. [Google Scholar] - Boyle, L.; Steinhardt, P.J. Coxeter pairs, Ammann patterns and Penrose-like tilings. arXiv
**2016**, arXiv:1608.08215. [Google Scholar] - Koca, N.O.; Koca, M.; Al-Siyabi, A. SU (5) grand unified theory, its polytopes and 5-fold symmetric aperiodic tiling. Int. J. Geom. Methods Mod. Phys.
**2018**, 5, 1850058. [Google Scholar] [CrossRef] - Masáková, Z.; Patera, J.; Pelantová, E. Inflation centres of the cut and project quasicrystals. J. Phys. A Math. Gen.
**1998**, 31, 1443. [Google Scholar] [CrossRef] - Meyer, Y. Algebraic Numbers and Harmonic Analysis, 1st ed.; North-Holland Pub. Co.: Amsterdam, The Netherlands, 1972. [Google Scholar]
- Lagarias, J.C. Meyer’s concept of quasicrystal and quasiregular sets. Commun. Math. Phys.
**1996**, 179, 365–376. [Google Scholar] [CrossRef] - Moody, R.V. Meyer Sets and Their Duals, in The Mathematics of Long-Range Aperiodic Order, NATO ASI Series C; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1997; Volume 489, pp. 403–441. [Google Scholar]
- Voronoi, G. Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. J. Für Die Reine Und Angew. Math.
**1908**, 134, 198–287. [Google Scholar] [CrossRef] - Voronoi, G. Part II of Voronoi (1908). J. Für Die Reine Und Angew. Math.
**1909**, 136, 67–181. [Google Scholar] [CrossRef] - Delaunay, N.B. Izv. Sur la partition régulière de l’espace à 4 dimensions. Première partie. Akad. Nauk SSSR Otdel. Fiz. Mat. Nauk.
**1929**, 1, 79–110. [Google Scholar] - Delaunay, N.B. Geometry of positive quadratic forms. Usp. Mat. Nauk.
**1938**, 3, 16–62. [Google Scholar] - Delaunay, N.B. Geometry of positive quadratic forms. Part II. Usp. Mat. Nauk.
**1938**, 4, 102–164. [Google Scholar] - Conway, J.H.; Sloane, N.J.A. Sphere Packings, Lattices and Groups; Springer: New York, NY, USA, 1988. [Google Scholar]
- Conway, J.H.; Sloane, N.J.A. Miscellanea Mathematica; Hilton, P., Hirzebruch, F., Remmert, R., Eds.; Springer: New York, NY, USA, 1991; pp. 71–108. [Google Scholar]
- Deza, M.; Grishukhin, V. Nonrigidity degrees of root lattices and their duals. Geom. Dedicate
**2004**, 104, 15–24. [Google Scholar] [CrossRef] - Engel, P. Geometric Crystallography: An Axiomatic Introduction to Crystallography; Springer: Dordrecht, The Netherlands, 1986. [Google Scholar]
- Engel, P.; Michel, L.; Senechal, M. Lattice geometry. Preprint IHES /P/04/45. 1994. [Google Scholar]
- Moody, R.V.; Patera, J. Voronoi and Delaunay cells of root lattices: Classification of their faces and facets by Coxeter-Dynkin diagrams. J. Phys. A Math. Gen.
**1992**, 25, 5089–5134. [Google Scholar] [CrossRef] - Koca, M.; Ozdes Koca, N.; Al-Siyabi, A.; Koc, R. Explicit construction of the Voronoi and Delaunay cells of W(An) and W(Dn) lattices and their facets. Acta Crystallogr. Sect. A Found. Adv.
**2018**, 74, 499–511. [Google Scholar] - Ausloos, M.; Bartolacci, F.; Castellano, N.G.; Cerqueti, R. Exploring how innovation strategies at time of crisis influence performance: A cluster analysis perspective. Technol. Anal. Strateg. Manag.
**2018**, 30, 484–497. [Google Scholar] [CrossRef] - Coxeter, H.S.M. Regular Polytopes, 3rd ed.; Dover Publications: New York, NY, USA, 1973. [Google Scholar]
- Grünbaum, B. Convex Polytopes; Wiley: New York, NY, USA, 1967. [Google Scholar]
- Humphreys, J.E. Cambridge Studies in Advanced Mathematics, Reflection Groups and Coxeter Groups; Cambridge University Press: Cambridge, UK, 1990; Volume 29. [Google Scholar]
- Michel, L. Bravais Classes, Voronoi Cells, Delone Cells, Symmetry and Structural Properties of Condensed Matter; Lulek, T., Florek, W., Walcerz, S., Eds.; World Scientific: Singapore, 1995; p. 279. [Google Scholar]
- Michel, L. Complete Description of the Voronoi Cell of the Lie Algebra A
_{n}Weight Lattice. On the Bounds for the Number of d-Faces of the n-Dimensional Voronoi Cells, Algebraic Methods in Physics; Saint-Aubin, Y., Vinet, L., Eds.; Springer: Berlin/Heidelberg, Germany, 2001; pp. 149–171. [Google Scholar] - Koca, M.; Koca, N.O.; Koc, R. Affine Coxeter group Wa (A
_{4}), quaternions, and decagonal quasicrystals. Int. J. Geom. Meth. Mod. Phys.**2014**, 11, 1450031. [Google Scholar] [CrossRef] - Carter, R.W. Simple Groups of Lie Type; John Wiley & Sons: New York, NY, USA, 1972; pp. 158–169. [Google Scholar]
- Nischke, K.P.; Danzer, L. A construction of inflation rules based on n-fold symmetry. Discret. Comput. Geom.
**1996**, 15, 221–236. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Coxeter–Dynkin diagram of ${a}_{n}$, (

**b**) extended Coxeter–Dynkin diagram of ${a}_{n}$.

**Figure 2.**A typical rhombic face of the rhombic dodecahedron (note that edges are represented by the vectors ${k}_{2}$ and ${k}_{3}$ with an angle 109.5°).

**Figure 5.**Projections of the lattices ${A}_{3}{}^{*}\mathrm{and}{A}_{3}$, leading to two overlapping square lattices, ${A}_{3}$ is the sublattice of the other.

**Figure 6.**A 3-dimensional rhombohedron, the rhombohedral facet of the Voronoi cell of the root lattice ${A}_{4}.$

**Figure 7.**Patch of the Penrose rhombic tiling obtained by the cut and project method from the lattice ${A}_{4}{}^{*}.$ The four types of vertices are distinguished by numbers as stated in the text.

**Figure 11.**Some patches (a-b-c) from three triangular prototiles from Delone cells of the root lattice ${A}_{5}$, obtained by substitution rule.

**Figure 13.**Two 7-fold symmetric patches (a-b) obtained from the projection of the Delone cells of the root lattice ${A}_{6}.$

**Figure 14.**The patch obtained from the projection of the Voronoi cell of the root lattice ${A}_{7}$.

**Figure 15.**Patches (a-b) of prototiles from Delone cells of the root lattice ${A}_{7}\mathrm{with}$ 8-fold symmetry.

**Figure 16.**Three rhombuses illustrated with different colors obtained from the Voronoi cell of ${A}_{11}$ and a patch is depicted.

**Figure 17.**A 12-fold symmetric patch obtained by substitution from triangular prototiles originating from Delone cells of the root lattice ${A}_{11}.$

**Table 1.**Rhombic prototiles projected from Voronoi cells of the root lattice ${A}_{n}$ (rhombuses with angles $\left(\frac{2\pi m}{h},\text{}\pi -\frac{2\pi m}{h}\right),m\in N$).

Root Lattice | h | # of Prototiles | Rhombuses with Pairs of Angles |
---|---|---|---|

${A}_{3}$ | 4 | 1 | $\left(\frac{\pi}{2},\frac{\pi}{2}\right)$, (square) |

${A}_{4}$ | 5 | 2 | $\left(\frac{2\pi}{5},\frac{3\pi}{5}\right),\left(\frac{4\pi}{5},\frac{\pi}{5}\right)$, (Penrose’s thick and thin rhombuses) |

${A}_{5}$ | 6 | 1 | $\left(\frac{\pi}{3},\frac{2\pi}{3}\right)$ |

${A}_{6}$ | 7 | 3 | $\left(\frac{2\pi}{7},\frac{5\pi}{7}\right),\left(\frac{4\pi}{7},\frac{3\pi}{7}\right),\left(\frac{6\pi}{7},\frac{\pi}{7}\right)$ |

${A}_{7}$ | 8 | 2 | $\left(\frac{\pi}{4},\frac{3\pi}{4}\right),\left(\frac{\pi}{2},\frac{\pi}{2}\right)$, (Amman-Beenker tiles) |

${A}_{8}$ | 9 | 4 | $\left(\frac{2\pi}{9},\frac{7\pi}{9}\right),\left(\frac{4\pi}{9},\frac{5\pi}{9}\right),\left(\frac{6\pi}{9},\frac{3\pi}{9}\right),\left(\frac{8\pi}{9},\frac{\pi}{9}\right)$ |

${A}_{9}$ | 10 | 2 | $\left(\frac{2\pi}{5},\frac{3\pi}{5}\right),\left(\frac{\pi}{5},\frac{4\pi}{5}\right),$ (Penrose’ thick and thin rhombuses) |

${A}_{10}$ | 11 | 3 | $\left(\frac{2\pi}{11},\frac{9\pi}{11}\right),\left(\frac{4\pi}{11},\frac{7\pi}{11}\right),\left(\frac{6\pi}{11},\frac{5\pi}{11}\right)$ |

${A}_{11}$ | 12 | 3 | $\left(\frac{\pi}{6},\frac{5\pi}{6}\right),\left(\frac{\pi}{3},\frac{2\pi}{3}\right),\left(\frac{\pi}{2},\frac{\pi}{2}\right)$ |

**Table 2.**Triangular prototiles with angles $\left(\frac{{n}_{1}\pi}{h},\frac{{n}_{2}\pi}{h},\text{}\frac{{n}_{3}\pi}{h}\right)$ projected from Delone cells of the root lattice ${A}_{n}$.

Root Lattice | h | # of Prototiles | Triangles Denoted by Triple Natural Numbers $\left({\mathit{n}}_{1},{\mathit{n}}_{2},{\mathit{n}}_{3}\right);\text{}{\mathit{n}}_{1}+{\mathit{n}}_{2}+{\mathit{n}}_{3}=\mathit{h}$ |
---|---|---|---|

${A}_{2}$ | 3 | 1 | $\left(1,1,1\right)$ |

${A}_{3}$ | 4 | 1 | $\left(1,1,2\right),$ (right triangle) |

${A}_{4}$ | 5 | 2 | $\left(1,1,3\right),\left(1,2,2\right),$ (Robinson triangles) |

${A}_{5}$ | 6 | 3 | $\left(1,1,4\right),\left(1,2,3\right),\left(2,2,2\right)$ |

${A}_{6}$ | 7 | 4 | $\left(1,1,5\right),\left(1,2,4\right),\left(1,3,3\right),\left(2,2,3\right),$ (Danzer triangles) |

${A}_{7}$ | 8 | 5 | $\left(1,1,6\right),\left(1,2,5\right),\left(1,3,4\right),\left(2,2,4\right),\left(2,3,3\right)$ |

${A}_{8}$ | 9 | 7 | $\left(1,1,7\right),\left(1,2,6\right),\left(1,3,5\right),\left(1,4,4\right),\left(2,2,5\right),$ $\left(2,3,4\right),\left(3,3,3\right)$ |

${A}_{9}$ | 10 | 8 | $\left(1,1,8\right),\left(1,2,7\right),\left(1,3,6\right),\left(1,4,5\right),\left(2,2,6\right),$ $\left(2,3,5\right),\left(2,4,4\right),\left(3,3,4\right)$ |

${A}_{10}$ | 11 | 10 | $\left(1,1,9\right),\left(1,2,8\right),\left(1,3,7\right),\left(1,4,6\right),\left(1,5,5\right),$ $\left(2,2,7\right),\left(2,3,6\right),\left(2,4,5\right),\left(3,3,5\right),\left(3,4,4\right)$ |

${A}_{11}$ | 12 | 12 | $\left(1,1,10\right),\left(1,2,9\right),\left(1,3,8\right),\left(1,4,7\right),\left(1,5,6\right),$ $\left(2,2,8\right),\left(2,3,7\right),\left(2,4,6\right),\left(2,5,5\right),\left(3,3,6\right),$ $\left(3,4,5\right),\left(4,4,4\right)$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ozdes Koca, N.; Al-Siyabi, A.; Koca, M.; Koc, R.
Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice *A _{n}*.

*Symmetry*

**2019**,

*11*, 1082. https://doi.org/10.3390/sym11091082

**AMA Style**

Ozdes Koca N, Al-Siyabi A, Koca M, Koc R.
Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice *A _{n}*.

*Symmetry*. 2019; 11(9):1082. https://doi.org/10.3390/sym11091082

**Chicago/Turabian Style**

Ozdes Koca, Nazife, Abeer Al-Siyabi, Mehmet Koca, and Ramazan Koc.
2019. "Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice *A _{n}*"

*Symmetry*11, no. 9: 1082. https://doi.org/10.3390/sym11091082