# The Roundest Polyhedra with Symmetry Constraints

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multi-Symmetric Point Arrangements on the Sphere

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Method

## 4. Results

#### 4.1. Small Values of n

#### 4.1.1. $n=8$

**Proposition**

**2.**

**Proof.**

#### 4.1.2. $n=10$

**Proposition**

**3.**

**Proof.**

#### 4.1.3. $n=14$

**Proposition**

**4.**

**Proof.**

#### 4.1.4. $n=18$

#### 4.1.5. $n=20$

#### 4.1.6. $n=22$

#### 4.2. Octahedral Goldberg Polyhedra

#### 4.2.1. $n=30$

#### 4.2.2. $n=38$

#### 4.3. Icosahedral Goldberg Polyhedra

#### 4.3.1. $n=92$

#### 4.3.2. $n=122$

#### 4.3.3. $n=132$

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The roundest polyhedron with 14 faces and with octahedral symmetry constraint. (

**a**) Minimum covering of a sphere by 14 equal circles (card model; photo: A. Lengyel); (

**b**) core of a broken turned ivory sphere, 17th century. (Grünes Gewölbe, Dresden, Germany; photo: T. Tarnai); (

**c**) Wooden die, 7th–9th centuries (Gyeongju National Museum, Korea; photo: K. Hincz).

**Figure 2.**The roundest polyhedron with 32 faces and with icosahedral symmetry constraint. (

**a**) Minimum covering of a sphere by 32 equal circles (card model; photo: A. Lengyel); (

**b**) part of a turned ivory object, around 1600 (Grünes Gewölbe, Dresden, Germany, inv. no. 255; photo: T. Tarnai); (

**c**) the Hyperball designed by P. Huybers, the underlying polyhedron is somewhat different from that in (

**a**) (photo: T. Tarnai).

**Figure 3.**The meaning of the Goldberg–Coxeter parameters b and c. The large equilateral triangle drawn with dashed lines is a face of the regular tetrahedron, octahedron or icosahedron.

**Figure 4.**Schematic view of the points where the faces of the proven and conjectured roundest polyhedra are tangent to a sphere. Triangular surface lattice on a face of the spherical tetrahedron, octahedron or icosahedron with the degrees of freedom of the points of tangency. In the legends of the subfigures, the symmetry and the Goldberg–Coxeter parameters are given. Superscripts (

**b**–

**e**) are explained in Table 2. Symbols ● ◉ denote points of tangency with zero, one, and two degrees of freedom, respectively.

**Figure 5.**Polyhedra with n faces that maximize the isoperimetric quotient under tetrahedral, octahedral, or icosahedral symmetry constraints. See main text for description.

**Table 1.**Number of points of tangency n if they lie only on q-fold and/or 2-fold and/or 3-fold rotation axes.

v | e | f | n | ||
---|---|---|---|---|---|

q = 3 | q = 4 | q = 5 | |||

1 | 0 | 0 | 4 | 6 | 12 |

0 | 1 | 0 | 6 | 12 | 30 |

0 | 0 | 1 | 4 | 8 | 20 |

0 | 1 | 1 | 10 | 20 | 50 |

1 | 0 | 1 | 8 | 14 | 32 |

1 | 1 | 0 | 10 | 18 | 42 |

1 | 1 | 1 | 14 | 26 | 62 |

**Table 2.**Polyhedra with n faces that maximize the isoperimetric quotient $IQ$ under tetrahedral, octahedral, or icosahedral symmetry constraints. Polyhedra are characterized by n, point group symmetry G, Goldberg–Coxeter parameters $(b,c)$, isoperimetric quotient $IQ$. Upper bound of $IQ$ is given via Goldberg’s formula. Particular properties are discussed in footnotes.

n | G | (b,c) | IQ | Upper Bound | Remarks |
---|---|---|---|---|---|

4 | ${T}_{d}$ | $(1,0)$ | $\mathit{0.302299894}{\phantom{\rule{3.33333pt}{0ex}}}^{\mathrm{a}}$ | $0.302299894$ | Proven, Fejes Tóth [6] |

6 | ${O}_{h}$ | $(1,0)$ | $\mathit{0.523598775}{\phantom{\rule{3.33333pt}{0ex}}}^{\mathrm{a}}$ | $0.523598775$ | Proven, Fejes Tóth [6] |

8 | ${O}_{h}$ | $(1,1){\phantom{\rule{3.33333pt}{0ex}}}^{\mathrm{b}}$ | $0.604599788$ | $0.637349714$ | Proven, this work |

10 | ${T}_{d}$ | $(2,0)$ | $0.630745372$ | $0.707318712$ | Proven, this work |

12 | ${I}_{h}$ | $(1,0)$ | $\mathit{0.754697399}{\phantom{\rule{3.33333pt}{0ex}}}^{\mathrm{a}}$ | $0.754697399$ | Proven, Fejes Tóth [6] |

14 | ${O}_{h}$ | $(1,1)$ | $0.781638893$ | $0.788894402$ | Huybers [14]; proven, this work |

16 | ${T}_{d}$ | $(3,0){\phantom{\rule{3.33333pt}{0ex}}}^{\mathrm{c}}$ | $0.812189098$ | $0.814733609$ | Goldberg [5] |

18 | ${O}_{h}$ | $(2,0)$ | $0.823218074$ | $0.834942754$ | This work |

20 | ${T}_{d}$ | $(3,0)$ | $0.830222439$ | $0.851179828$ | This work |

22 | ${T}_{d}$ | $(2,2){\phantom{\rule{3.33333pt}{0ex}}}^{\mathrm{c}}$ | $0.862408738$ | $0.864510388$ | This work |

26 | ${O}_{h}$ | $(2,0){\phantom{\rule{3.33333pt}{0ex}}}^{\mathrm{d}}$ | $0.876811431$ | $0.885098414$ | Huybers [15] |

30 | O | $(2,1)$ | $0.896930384$ | $0.900256896$ | This work |

32 | ${I}_{h}$ | $(1,1)$ | $0.905798260$ | $0.906429544$ | Goldberg [5] |

38 | ${O}_{h}$ | $(3,0)$ | $0.917445003$ | $0.921082160$ | This work |

42 | ${I}_{h}$ | $(2,0)$ | $0.927651905$ | $0.928542518$ | Goldberg [5] |

62 | ${I}_{h}$ | $(2,0){\phantom{\rule{3.33333pt}{0ex}}}^{\mathrm{e}}$ | $0.945021022$ | $0.951478663$ | Huybers [15] |

72 | I | $(2,1)$ | $0.957881213$ | $0.958189143$ | Tarnai et al. [10] |

92 | ${I}_{h}$ | $(3,0)$ | $0.966957236$ | $0.967248411$ | This work |

122 | ${I}_{h}$ | $(2,2)$ | $0.975117622$ | $0.975282102$ | This work |

132 | I | $(3,1)$ | $0.976993221$ | $0.977150391$ | This work |

^{e}The system of the lattice points is supplemented with the face midpoints of the regular icosahedron.

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Lengyel, A.; Gáspár, Z.; Tarnai, T.
The Roundest Polyhedra with Symmetry Constraints. *Symmetry* **2017**, *9*, 41.
https://doi.org/10.3390/sym9030041

**AMA Style**

Lengyel A, Gáspár Z, Tarnai T.
The Roundest Polyhedra with Symmetry Constraints. *Symmetry*. 2017; 9(3):41.
https://doi.org/10.3390/sym9030041

**Chicago/Turabian Style**

Lengyel, András, Zsolt Gáspár, and Tibor Tarnai.
2017. "The Roundest Polyhedra with Symmetry Constraints" *Symmetry* 9, no. 3: 41.
https://doi.org/10.3390/sym9030041