#
Icosahedral Polyhedra from D_{6} Lattice and Danzer’s ABCK Tiling

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

_{6}admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H

_{3}, its roots, and weights are determined in terms of those of D

_{6}. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D

_{6}determined by a pair of integers (m

_{1}, m

_{2}) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H

_{3}, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m

_{1}, m

_{2}) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H

_{3}, and those related transformations in 6D space with D

_{6}symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D

_{6}. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.

## 1. Introduction

## 2. Projection of D_{6} Lattice under the Icosahedral Group H_{3} and the Archimedean Polyhedra

## 3. Danzer’s ABCK Tiles and D_{6} Lattice

_{1}and ${R}_{3}$ or their conjugate groups. The octahedron $\langle K\rangle $ consists of 8K generated by a group of order eight consisting of three commuting generators ${R}_{1},{R}_{3}$ as mentioned earlier, and the third generator ${R}_{0}$ is an affine reflection [20] with respect to the golden rhombic face induced by the affine Coxeter group ${\tilde{D}}_{6}$.

_{6}image of any general vector in the 3D space in the form of c(τ

^{p}, τ

^{q}, τ

^{r}) where p, q, r ∈ $\mathbb{Z}$. Now we discuss the inflation of each tile one by one.

#### 3.1. Construction of $\tau K=B+K$

#### 3.2. Construction of $\tau B=C+4K+{B}_{1}+{B}_{2}$

#### 3.3. Construction of $\tau C={K}_{1}+{K}_{2}+{C}_{1}+{C}_{2}+A$

#### 3.4. Construction of $\tau A=3B+2C+6K$

_{1}and K

_{2}with these vertices, first rotate K by ${g}_{{K}_{1}}$ and ${g}_{{K}_{2}}$ given by the matrices

## 4. Discussions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Platonic and Archimedean icosahedral polyhedra projected from D

_{6}(vertices are orbits under the icosahedral group).

$\mathbf{Vector}\text{}\mathbf{of}\text{}\mathbf{the}\text{}\mathbf{Polyhedron}\text{}\mathbf{in}\text{}{\mathit{H}}_{3}$ $\mathit{c}\equiv \sqrt{\frac{2}{2+\mathit{\tau}}}({\mathit{m}}_{1}-{\mathit{m}}_{2}+2{\mathit{m}}_{2}\mathit{\tau})$ | $\mathbf{Corresponding}\text{}\mathbf{Vector}\text{}\mathbf{in}\text{}{\mathit{D}}_{6}$ ${\mathit{m}}_{1},{\mathit{m}}_{2}\in \mathbb{Z}$ | |
---|---|---|

Icosahedron $c\frac{{v}_{1}}{\sqrt{2}}=\frac{c}{2}\left(1,\tau ,0\right)$ | ${m}_{1}{l}_{1}+{m}_{2}({l}_{2}+{l}_{3}+{l}_{4}+{l}_{5}-{l}_{6})=\left({m}_{1}-{m}_{2}\right){\omega}_{1}+2{m}_{2}{\omega}_{5}$ ${m}_{1},{m}_{2}\in 2\mathbb{Z}\text{}\mathrm{or}\text{}2\mathbb{Z}+1$ | |

Dodecahedron $c\frac{{v}_{3}}{\sqrt{2}}=\frac{c}{2}\left(0,{\tau}^{2},1\right)$ | $\frac{1}{2}\left[\left({m}_{1}+3{m}_{2}\right)\left({l}_{1}+{l}_{2}+{l}_{3}\right)+\left({m}_{1}-{m}_{2}\right)\left({l}_{4}+{l}_{5}+{l}_{6}\right)\right]=\left({m}_{1}-{m}_{2}\right){\omega}_{6}+2{m}_{2}{\omega}_{3}$ ${m}_{1},{m}_{2}\in 2\mathbb{Z}\text{}\mathrm{or}\text{}2\mathbb{Z}+1$ | |

Icosidodecahedron $c\frac{{v}_{2}}{\sqrt{2}}=c\left(0,\tau ,0\right)$ | $\left({m}_{1}+{m}_{2}\right)\left({l}_{1}+{l}_{2}\right)+2{m}_{2}\left({l}_{3}+{l}_{4}\right)=\left({m}_{1}-{m}_{2}\right){\omega}_{2}+2{m}_{2}{\omega}_{4}$ ${m}_{1},{m}_{2}\in 2\mathbb{Z}\mathrm{or}\text{}2\mathbb{Z}+1$ | |

Truncated Icosahedron $c\frac{\left({v}_{1}+{v}_{2}\right)}{\sqrt{2}}=\frac{c}{2}\left(1,3\tau ,0\right)$ | $\left(2{m}_{1}+{m}_{2}\right){l}_{1}+\left({m}_{1}+2{m}_{2}\right){l}_{2}+{m}_{2}\left(3{l}_{3}+3{l}_{4}+{l}_{5}-{l}_{6}\right)=\left({m}_{1}-{m}_{2}\right)({\omega}_{1}+{\omega}_{2})+2{m}_{2}({\omega}_{4}+{\omega}_{5})$ ${m}_{1},{m}_{2}\in 2\mathbb{Z}\mathrm{or}\text{}2\mathbb{Z}+1$ | |

Small Rhombicosidodecahedron $c\frac{\left({v}_{1}+{v}_{3}\right)}{\sqrt{2}}=\frac{c}{2}\left(1,2\tau +1,1\right)$ | $\frac{1}{2}[\left(3{m}_{1}+3{m}_{2}\right){l}_{1}+\left({m}_{1}+5{m}_{2}\right)\left({l}_{2}+{l}_{3}\right)+\left({m}_{1}+{m}_{2}\right)\left({l}_{4}+{l}_{5}\right)+\left({m}_{1}-3{m}_{2}\right){l}_{6}]$ $=\left({m}_{1}-{m}_{2}\right)({\omega}_{1}+{\omega}_{6})+2{m}_{2}({\omega}_{3}+{\omega}_{5})$ | |

Truncated Dodecahedron $c\frac{\left({v}_{2}+{v}_{3}\right)}{\sqrt{2}}=\frac{c}{2}\left(0,3\tau +1,1\right)$ | $\frac{1}{2}[\left(3{m}_{1}+5{m}_{2}\right)\left({l}_{1}+{l}_{2}\right)+\left({m}_{1}+7{m}_{2}\right){l}_{3}+\left({m}_{1}+3{m}_{2}\right){l}_{4}+\left({m}_{1}-{m}_{2}\right)\left({l}_{5}+{l}_{6}\right)]$ $=\left({m}_{1}-{m}_{2}\right)({\omega}_{2}+{\omega}_{6})+2{m}_{2}({\omega}_{3}+{\omega}_{4})$ | |

Great Rhombicosidodecahedron $c\frac{\left({v}_{1}+{v}_{2}+{v}_{3}\right)}{\sqrt{2}}=\frac{c}{2}\left(1,4\tau +1,1\right)$ | $\frac{1}{2}[5\left({m}_{1}+{m}_{2}\right){l}_{1}+\left(3{m}_{1}+7{m}_{2}\right){l}_{2}+\left({m}_{1}+9{m}_{2}\right){l}_{3}+\left({m}_{1}+5{m}_{2}\right){l}_{4}+\left({m}_{1}+{m}_{2}\right){l}_{5}+\left({m}_{1}-3{m}_{2}\right){l}_{6}]$ $=\left({m}_{1}-{m}_{2}\right)({\omega}_{1}+{\omega}_{2}+{\omega}_{6})+2{m}_{2}({\omega}_{3}+{\omega}_{4}+{\omega}_{5})$ |

${\mathit{H}}_{\mathbf{3}}$ | ${\mathit{D}}_{\mathbf{6}}$ | |
---|---|---|

${g}_{K}$ | $\leftarrow $rotation$\to $ | ${g}_{K}:\left(1\overline{3}6\right)\left(24\overline{5}\right)$ |

${t}_{K}=\frac{1}{2}\left(\tau ,{\tau}^{2},0\right)$ | $\leftarrow $translation$\to $ | $\frac{1}{2}\left[\left({m}_{1}+5{m}_{2}\right){l}_{1}+({m}_{1}+{m}_{2}\right)\left({l}_{2}+{l}_{3}+{l}_{4}+{l}_{5}-{l}_{6}\right)]$ |

**Table 3.**Rotation and translation in H

_{3}and D

_{6}(see the text for the definition of rotations in E

_{‖}space).

${\mathit{H}}_{\mathbf{3}}$ | ${\mathit{D}}_{\mathbf{6}}$ | ||
---|---|---|---|

$\mathit{\tau}\mathit{K}\mathbf{=}\mathit{B}\mathbf{+}\mathit{K}$ | |||

${g}_{K}$ | ←rotation→ | $\left(1\overline{3}6\right)\left(24\overline{5}\right)$ | |

${t}_{K}=\frac{c}{2}\left(\tau ,{\tau}^{2},0\right)$ | ←translation→ | $\frac{1}{2}\left[\left({m}_{1}+5{m}_{2}\right){l}_{1}+({m}_{1}+{m}_{2}\right)\left({l}_{2}+{l}_{3}+{l}_{4}+{l}_{5}-{l}_{6}\right)]$ | |

$\mathit{\tau}\mathit{B}\mathbf{=}\mathit{C}\mathbf{+}\mathbf{4}\mathit{K}\mathbf{+}{\mathit{B}}_{\mathbf{1}}\mathbf{+}{\mathit{B}}_{\mathbf{2}}$ | |||

${g}_{B}$ | ←rotation→ | $\left(3\right)(126\overline{4}$5) | |

${g}_{4K}$ | ←rotation→ | (1)(23)(45)(6) | |

${t}_{4K}=\frac{c}{2}\left(\tau ,0,1\right)$ | ←translation→ | ${m}_{1}{l}_{5}+{m}_{2}\left({l}_{1}-{l}_{2}+{l}_{3}-{l}_{4}-{l}_{6}\right)$ | |

${g}_{{B}_{1}}$ | ←rotation→ | (1643$\overline{5}$)(2) | |

${g}_{{B}_{2}}$ | ←rotation→ | (15$\overline{2}\overline{1}\overline{5}2$), (3$\overline{6}\overline{4}\overline{3}64$) | |

${t}_{B}=\frac{c}{2}\left({\tau}^{3},0,{\tau}^{2}\right)$ | ←translation→ | $\frac{1}{2}\left[\left(3{m}_{1}+5{m}_{2}\right){l}_{5}+({m}_{1}+3{m}_{2}\right)\left({l}_{1}-{l}_{2}+{l}_{3}-{l}_{4}-{l}_{6}\right)]$ | |

$\mathit{\tau}\mathit{C}\mathbf{=}{\mathit{K}}_{\mathbf{1}}\mathbf{+}{\mathit{K}}_{\mathbf{2}}\mathbf{+}{\mathit{C}}_{\mathbf{1}}\mathbf{+}{\mathit{C}}_{\mathbf{2}}\mathbf{+}\mathit{A}$ | |||

${g}_{{K}_{1}}$ | ←rotation→ | (5$\overline{4}$621)(3) | |

(identity translation) | |||

${g}_{{K}_{2}}$ | ←rotation→ | (1$\overline{1}$)(63$\overline{4}5\overline{2}\overline{6}\overline{3}4\overline{5}2$) | |

${t}_{K}=\frac{c}{2}\left({\tau}^{2},\tau ,1\right)$ | ←translation→ | $\left({m}_{1}+{m}_{2}\right)\left({l}_{1}+{l}_{5}\right)+2{m}_{2})\left({l}_{3}-{l}_{6}\right)$ | |

${t}_{C}=-\frac{c}{2}\left(\tau ,\tau ,\tau \right)$ | ←translation→ | $-\frac{1}{2}\left[\left({m}_{1}+3{m}_{2}\right)({l}_{1}+{l}_{3}+{l}_{5}\right)+\left({m}_{1}-{m}_{2}\right)\left({l}_{2}-{l}_{4}-{l}_{6}\right)]$ | |

${g}_{{C}_{1}}$ | ←rotation→ | (1)(4)(2$\overline{6}$)(35) | |

${g}_{{C}_{2}}$ | ←rotation→ | (1)(35$\overline{6}42)$ | |

${t}_{C}=\frac{c}{2}\left({\tau}^{2},{\tau}^{2},{\tau}^{2}\right)$ | ←translation→ | $\left({m}_{1}+2{m}_{2}\right)\left({l}_{1}+{l}_{3}+{l}_{5}\right)+{m}_{2}\left({l}_{2}-{l}_{4}-{l}_{6}\right)$ | |

${g}_{A}$ | ←rotation→ | (1)(6)(23)(45) | |

${t}_{A}=\frac{c}{2}\left({\tau}^{2},0,\tau \right)$ | ←translation→ | $\frac{1}{2}\left[\left({m}_{1}+5{m}_{2}\right){l}_{5}+\left({m}_{1}+{m}_{2}\right)\left({l}_{1}-{l}_{2}+{l}_{3}-{l}_{4}-{l}_{6}\right)\right]$ | |

$\mathit{\tau}\mathit{A}\mathbf{=}\mathit{C}\mathbf{+}{\mathit{K}}_{\mathbf{1}}\mathbf{+}{\mathit{K}}_{\mathbf{2}}\mathbf{+}\mathit{B}\mathbf{+}\mathit{\tau}\mathit{B}$ | |||

${t}_{C}=-\frac{c}{2}\left(\tau ,\tau ,\tau \right)$ | ←translation→ | $-\frac{1}{2}\left[\left({m}_{1}+3{m}_{2}\right)({l}_{1}+{l}_{3}+{l}_{5}\right)+\left({m}_{1}-{m}_{2}\right)\left({l}_{2}-{l}_{4}-{l}_{6}\right)]$ | |

${g}_{C}$ | ←rotation→ | $(1\overline{1})$ (2$\overline{4}$)(36)(5$\overline{5}$) | |

${g}_{{K}_{1}}$ | ←rotation→ | $\left(1\right)$ (2543$\overline{6}$) | |

${g}_{{K}_{2}}$ | ←rotation→ | $\left(1\right)\left(4\right)$ (2$\overline{6}$)(35) | |

${t}_{K}=\frac{c}{2}\left(\tau ,0,1\right)$ | ←translation→ | ${m}_{1}{l}_{5}+{m}_{2}\left({l}_{1}-{l}_{2}+{l}_{3}-{l}_{4}-{l}_{6}\right)$ | |

${g}_{B}$ | ←rotation→ | $\left(65\overline{1}\overline{6}\overline{5}1\right),$ ($\overline{4}\overline{2}\overline{3}423)$ | |

${t}_{B}=\frac{c}{2}\left({\tau}^{3},0,{\tau}^{2}\right)$ | ←translation→ | $\frac{1}{2}\left[\left(3{m}_{1}+5{m}_{2}\right){l}_{5}+({m}_{1}+3{m}_{2}\right)\left({l}_{1}-{l}_{2}+{l}_{3}-{l}_{4}-{l}_{6}\right)]$ |

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**MDPI and ACS Style**

Al-Siyabi, A.; Ozdes Koca, N.; Koca, M.
Icosahedral Polyhedra from *D*_{6} Lattice and Danzer’s *ABCK* Tiling. *Symmetry* **2020**, *12*, 1983.
https://doi.org/10.3390/sym12121983

**AMA Style**

Al-Siyabi A, Ozdes Koca N, Koca M.
Icosahedral Polyhedra from *D*_{6} Lattice and Danzer’s *ABCK* Tiling. *Symmetry*. 2020; 12(12):1983.
https://doi.org/10.3390/sym12121983

**Chicago/Turabian Style**

Al-Siyabi, Abeer, Nazife Ozdes Koca, and Mehmet Koca.
2020. "Icosahedral Polyhedra from *D*_{6} Lattice and Danzer’s *ABCK* Tiling" *Symmetry* 12, no. 12: 1983.
https://doi.org/10.3390/sym12121983