# Aesthetic Patterns with Symmetries of the Regular Polyhedron

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## Abstract

**:**

## 1. Introduction

## 2. Symmetry Groups of Regular Polyhedra

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- Tetrahedral group $[3,3]$:$${\Pi}_{[3,3]}^{\alpha}:\phantom{\rule{4pt}{0ex}}x+y=0,\phantom{\rule{4pt}{0ex}}{\Pi}_{[3,3]}^{\beta}:\phantom{\rule{4pt}{0ex}}y-z=0,\phantom{\rule{4pt}{0ex}}{\Pi}_{[3,3]}^{\gamma}:\phantom{\rule{4pt}{0ex}}x-y=0$$$${\alpha}_{[3,3]}=\left[\begin{array}{ccc}0& -1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right],\phantom{\rule{4pt}{0ex}}{\beta}_{[3,3]}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],\phantom{\rule{4pt}{0ex}}{\gamma}_{[3,3]}=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$$${\u25b5}_{[3,3]}=\{{(x,y,z)}^{T}\in {S}^{2}|x+y\ge 0,\phantom{\rule{4pt}{0ex}}y-z\le 0,\phantom{\rule{4pt}{0ex}}x-y\le 0\}$$
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- Octhedral group $[3,4]$:$${\Pi}_{[3,4]}^{\alpha}:\phantom{\rule{4pt}{0ex}}x-z=0,\phantom{\rule{4pt}{0ex}}{\Pi}_{[3,4]}^{\beta}:\phantom{\rule{4pt}{0ex}}x-y=0,\phantom{\rule{4pt}{0ex}}{\Pi}_{[3,4]}^{\gamma}:\phantom{\rule{4pt}{0ex}}y=0$$$${\alpha}_{[3,4]}=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],\phantom{\rule{4pt}{0ex}}{\beta}_{[3,4]}=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],\phantom{\rule{4pt}{0ex}}{\gamma}_{[3,4]}=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$$$${\u25b5}_{[3,4]}=\{{(x,y,z)}^{T}\in {S}^{2}|x-z\le 0,\phantom{\rule{4pt}{0ex}}x-y\ge 0,\phantom{\rule{4pt}{0ex}}y\ge 0\}$$
- ■
- Icosahedral group $[3,5]$:$${\Pi}_{[3,5]}^{\alpha}:\phantom{\rule{4pt}{0ex}}y=0,\phantom{\rule{4pt}{0ex}}{\Pi}_{[3,5]}^{\gamma}:\phantom{\rule{4pt}{0ex}}x=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\Pi}_{[3,5]}^{\beta}:\phantom{\rule{4pt}{0ex}}-\zeta x+y-z/\zeta =0,\phantom{\rule{4pt}{0ex}}\mathrm{where}\phantom{\rule{4pt}{0ex}}\zeta =\frac{1+\sqrt{5}}{2}$$$${\alpha}_{[3,5]}=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right],\phantom{\rule{4pt}{0ex}}{\beta}_{[3,5]}=\frac{1}{2}\left[\begin{array}{ccc}-1/\zeta & \zeta & -1\\ \zeta & 1& 1/\zeta \\ -1& 1/\zeta & \zeta \end{array}\right],\phantom{\rule{4pt}{0ex}}{\gamma}_{[3,5]}=\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$$${\u25b5}_{[3,5]}=\{{(x,y,z)}^{T}\in {S}^{2}|y\ge 0,\phantom{\rule{4pt}{0ex}}x\le 0,\phantom{\rule{4pt}{0ex}}\zeta x-y+z/\zeta \ge 0\}$$

## 3. Transform Points of ${\mathbf{S}}^{\mathbf{2}}$ into Fundamental Region Symmetrically

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 4. Colorful Spherical Patterns with $[\mathit{p},\mathit{q}]$ Symmetry

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**(

**a**) The blue spherical right triangle ${\u25b5}_{[3,4]}$ surrounded by planes ${\Pi}_{[3,4]}^{\alpha},\phantom{\rule{4pt}{0ex}}{\Pi}_{[3,4]}^{\beta}$, and ${\Pi}_{[3,4]}^{\gamma}$ forms a fundamental region associated with group $[3,4]$; (

**b**) Let $Q\in {\u25b5}_{[3,4]}$ and ${P}_{0}\notin {\u25b5}_{[3,4]}$ be two points on the different sides of ${\Pi}_{[3,4]}^{\beta}$. Then, ${\beta}_{[3,4]}\left({P}_{0}\right)$ and Q lie on the same side of ${\Pi}_{[3,4]}^{\beta}$, and the distance between them is smaller than ${P}_{0}$ and Q; and (

**c**) A schematic illustration that shows how Theorem 1 transforms ${u}^{1}\notin {\u25b5}_{[3,5]}$ into ${\u25b5}_{[3,5]}$ symmetrically. In this case, ${u}^{1}$ is first transformed by ${\gamma}_{[3,5]}$ so that ${u}^{2}={\gamma}_{[p,q]}\left({u}^{1}\right)$ goes into red tile. Then, ${u}^{2}$ is transformed by ${\beta}_{[3,5]}$ so that ${u}^{3}={\beta}_{[p,q]}\left({u}^{2}\right)$ goes into green tile. At last, ${u}^{3}$ is transformed by ${\alpha}_{[3,5]}$ so that ${u}_{0}={\alpha}_{[3,5]}\left({u}^{3}\right)\in {\u25b5}_{[3,5]}$.

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**MDPI and ACS Style**

Ouyang, P.; Wang, L.; Yu, T.; Huang, X.
Aesthetic Patterns with Symmetries of the Regular Polyhedron. *Symmetry* **2017**, *9*, 21.
https://doi.org/10.3390/sym9020021

**AMA Style**

Ouyang P, Wang L, Yu T, Huang X.
Aesthetic Patterns with Symmetries of the Regular Polyhedron. *Symmetry*. 2017; 9(2):21.
https://doi.org/10.3390/sym9020021

**Chicago/Turabian Style**

Ouyang, Peichang, Liying Wang, Tao Yu, and Xuan Huang.
2017. "Aesthetic Patterns with Symmetries of the Regular Polyhedron" *Symmetry* 9, no. 2: 21.
https://doi.org/10.3390/sym9020021