# Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{1}$\oplus $ A

_{1}$\oplus $ A

_{1}and ${A}_{3}$, although they describe the achiral polyhedra such as the families of regular and irregular tetrahedron and icosahedron, respectively. We explicitly show that achiral polyhedra possess larger proper rotational symmetries transforming them to their mirror images. We organize the paper as follows. In Section 2 we introduce quaternions and construct the Coxeter groups in terms of quaternions [17]. We extend the group W (A

_{1}$\oplus $ A

_{1}$\oplus $ A

_{1}) to the octahedral group by the symmetry group$\text{}Sym\left(3\right)\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{Coxeter}\text{}\mathrm{diagram}$ A

_{1}$\oplus $ A

_{1}$\oplus $ A

_{1}. In Section 3 we obtain the proper rotation subgroup of the Coxeter group W (A

_{1}$\oplus $ A

_{1}$\oplus $ A

_{1}) and determine the vertices of an irregular tetrahedron. In Section 4 we discuss a similar problem for the Coxeter-Dynkin diagram ${A}_{3}$ leading to an icosahedron and again prove that it can be transformed by the group W(B

_{3})

^{+}to its mirror image, which implies that neither the tetrahedron nor icosahedron are chiral solids. We focus on the irregular icosahedra constructed either by the proper tetrahedral group or its extension pyritohedral group and construct the related dual solids tetartoid and pyritohedron. We also construct the irregular polyhedra taking the mid-points of edges of the irregular icosahedron as vertices. Section 5 deals with the construction of irregular and regular snub cube and their dual solids from the proper rotational octahedral symmetry W(B

_{3})

^{+}using the same technique employed in the preceding sections. The chiral polyhedron taking the mid-points as vertices of the irregular snub cube is also discussed. In Section 6 we repeat a similar technique for the constructions of irregular snub dodecahedra and their dual solids using the proper icosahedral group W(H

_{3})

^{+}, which is isomorphic to the group of even permutations of five letters$\text{}Alt\left(5\right)$. The chiral polyhedra whose vertices are the mid-points of the edges of the irregular snub dodecahedron are constructed. Irregular polyhedra transform to regular polyhedra when the parameter describing irregularity turns out to be the solution of certain cubic equations. Section 7 involves the discussion of the technique for the construction of irregular chiral polyhedra.

## 2. Quaternionic Constructions of the Coxeter Groups

_{1}$\oplus $ A

_{1}$\oplus $ A

_{1}) $\approx $ ${C}_{2}\text{}\times \text{}{C}_{2}\times \text{}{C}_{2}:=\text{}{2}^{3}$ of order 8. The scaled root system $(\pm {e}_{1},\pm {e}_{2},\pm {e}_{3})$ represents the vertices of an octahedron and has more symmetries than the Coxeter group W (A

_{1}$\oplus $ A

_{1}$\oplus $ A

_{1}). The automorphism group of the root system can be obtained by extending the Coxeter group of order 8 by the Dynkin diagram symmetry $Sym\left(3\right)$ of the Coxeter diagram in Figure 1. This is obvious from the diagram where the generators of the symmetric group $Sym\left(3\right)$ of order 6 can be chosen as:

## 3. The Orbit ${\mathit{C}}_{\mathbf{2}}\times {\mathit{C}}_{\mathbf{2}}\left({\mathit{a}}_{\mathbf{1}}{\mathit{a}}_{\mathbf{2}}{\mathit{a}}_{\mathbf{3}}\right)$ as an Irregular Tetrahedron

## 4. The Regular and Irregular Icosahedron Derived from the Orbit $\mathit{W}{\left({\mathit{A}}_{\mathbf{3}}\right)}^{+}\left({\mathit{a}}_{\mathbf{1}}{\mathit{a}}_{\mathbf{2}}{\mathit{a}}_{\mathbf{3}}\right)$

#### 4.1. Dual of an Irregular Icosahedron

#### 4.2. Regular and Irregular Icosidodecahedron

## 5. The Regular and Irregular Snub Cubes Derived from $\mathit{W}{\left({\mathit{B}}_{\mathbf{3}}\right)}^{+}\left({\mathit{a}}_{\mathbf{1}}{\mathit{a}}_{\mathbf{2}}{\mathit{a}}_{\mathbf{3}}\right)$

- 6 equilateral triangles with edge length: $A=\sqrt{2\left(1+x+{x}^{2}\right)}$,
- 8 squares with edge length:$\text{}B=\sqrt{2\left(1+y+{y}^{2}\right)},\text{}$
- 24 scalene triangles with edge lengths: $A,B,\text{}C=\sqrt{2\left({x}^{2}+\frac{1}{2}{y}^{2}\right)}$.

^{3}− x

^{2}− x − 1 ≠ 0

^{3}− x

^{2}− x − 1 ≠ 0

#### 5.1. Dual of the Irregular Snub Cube

#### 5.2. Chiral Polyhedra with Vertices at the Edge Mid-Points of the Irregular Snub Cube

## 6. The Regular and Irregular Snub Dodecahedron Derived from $\mathit{W}{\left({\mathit{H}}_{\mathbf{3}}\right)}^{+}\left({\mathit{a}}_{\mathbf{1}}{\mathit{a}}_{\mathbf{2}}{\mathit{a}}_{\mathbf{3}}\right)$

- (i)
- $x=1$ and $y=0:\mathrm{Truncated}\text{}\mathrm{icosahedron},$
- (ii)
- $x=0$ and $y=1:$ Truncated dodecahedron,
- (iii)
- $x=y$ $=0:$ Icosidodecahedron.

#### 6.1. Dual of the Irregular Snub Dodecahedron

#### 6.2. Chiral Polyhedra with Vertices at the Edge Mid-Points of the Irregular Snub Dodecahedron

## 7. Concluding Remarks

_{3}and H

_{3}to obtain the quaternionic description of the relevant chiral groups. Employing the same technique for the diagrams A

_{1}$\oplus $ A

_{1}$\oplus $ A

_{1}and A

_{3}, the irregular tetrahedra and icosahedra have been included although they are not chiral as they can be transformed to their mirror images by the proper rotational subgroup of the octahedral group.

_{4}Coxeter diagram [29], and its irregular form can be obtained using the same technique used for the irregular icosahedron.

## Author Contributions

## Conflicts of Interest

## References

- Coxeter, H.S.M.; Moser, W.O.J. Generators and Relations for Discrete Groups; Springer: Berlin, Germany, 1965. [Google Scholar]
- Cotton, F.A.; Wilkinson, G.; Murillo, C.A.; Bochmann, M. Advanced Inorganic Chemistry, 6th ed.; Wiley-Interscience: New York, NY, USA, 1999. [Google Scholar]
- Caspar, D.L.D.; Klug, A. Cold spring harbor symp. Quant. Biol.
**1962**, 27, 1. [Google Scholar] [CrossRef] - Twarock, R. Mathematical virology: A novel approach to the structure and assembly of viruses. Philos. Trans. R. Soc.
**2006**, 364, 3357–3373. [Google Scholar] [CrossRef] [PubMed] - Jaric, M.V. (Ed.) Introduction to the Mathematics of Quasicrystals; Academic Press: New York, NY, USA, 1989. [Google Scholar]
- Senechal, M. Quasicrystals and Geometry; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Suck, J.B.; Schreiber, M.; Haussler, P. (Eds.) Quasicrystals (An Introduction to Structure, Physical Properties, and Applications); Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Koca, M.; Koc, R.; Al-Ajmi, M. Polyhedra obtained from Coxeter groups and quaternions. J. Math. Phys.
**2007**, 48, 113514. [Google Scholar] [CrossRef] - Koca, M.; Koca, N.O.; Koc, R. Catalan solids derived from 3D-root systems. J. Math. Phys.
**2010**, 51, 043501. [Google Scholar] [CrossRef] - Koca, M.; Al-Ajmi, M.; Al-Shidhani, S. Quasi regular polyhedra and their duals with Coxeter symmetries represented by quaternions II. Afr. Rev. Phys.
**2011**, 1006, 53. [Google Scholar] - McMullen, P.; Schulte, E. Abstract Regular Polytopes; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Schulte, E.; Weiss, A.I. Chiral polytopes. In Applied Geometry and Discrete Mathematics (The Victor Klee Festchrift); DIMACS Series in Discrete Mathematics and Theoretical Computer Science; Gritzmann, P., Sturmfels, B., Eds.; American Mathematical Society: Providence, RI, USA; The Association for Computing Machinery: New York, NY, USA, 1991; Volume 4, pp. 493–516. [Google Scholar]
- Schulte, E.; Weiss, A.I. Chirality and projective linear groups. Discret. Math.
**1994**, 131, 221–261. [Google Scholar] [CrossRef] - Schulte, E.; Weiss, A.I. Free extensions of chiral polytopes. Can. J. Math.
**1995**, 47, 641–651. [Google Scholar] [CrossRef] - Huybers, P.; Coxeter, H.S.M. A new approach to the chiral Archimedean solids. Math. Rep. Acad. Sci. Can.
**1979**, 1, 259–274. [Google Scholar] - Weissbach, B.; Martini, H. Beitrage zur Algebra und Geometrie (Contributions to Algebra and Geometry); Springer: Berlin/Heidelberg, Germany, 2002; Volume 43, p. 121. [Google Scholar]
- Conway, J.H.; Smith, D.A. On Quaternion’s and Octonions: Their Geometry, Arithmetics, and Symmetry; Peters, A.K., Ltd.: Natick, MA, USA, 2003. [Google Scholar]
- Carter, R.W. Simple Groups of Lie Type; John Wiley & Sons Ltd.: Hoboken NJ, USA, 1972. [Google Scholar]
- Humphreys, J.E. Reflection Groups and Coxeter Groups; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Koca, M.; Koc, R.; Barwani, M.A. Non-crystallographic Coxeter Group H
_{4}in E_{8}. J. Phys. A Math. Gen.**2001**, 11201, A34. [Google Scholar] - Koca, N.O.; Al-Mukhaini, A.; Koca, M.; Al-Qanobi, A. Symmetry of the pyritohedron and lattices. SQU J. Sci.
**2016**, 21, 140–150. [Google Scholar] [CrossRef] - Koca, M.; Koc, R.; Al-Barwani, M. Quaternionic roots of SO(8), SO(9), F
_{4}and the related Weyl groups. J. Math. Phys.**2003**, 44, 3123. [Google Scholar] [CrossRef] - Koca, M.; Koc, R.; Al-Barwani, M. Quaternionic root systems and subgroups of the Aut (F
_{4}). J. Math. Phys.**2006**, 47, 043507. [Google Scholar] [CrossRef] - Koca, M.; Koc, R.; Al-Barwani, M.; Al-Farsi, S. Maximal subgroups of the Coxeter group W(H
_{4}) and quaternions. Linear Algebra Appl.**2006**, 412, 441. [Google Scholar] [CrossRef] - Du Val, P. Homographies, Quaternions, and Rotations; Oxford University Press: Oxford, UK, 1964. [Google Scholar]
- Slansky, R. Group theory for unified model building. Phys. Rep.
**1981**, 79, 1–128. [Google Scholar] [CrossRef] - Coxeter, H.S.M. Regular Polytopes, 3rd ed.; Dover Publications: New York, NY, USA, 1973. [Google Scholar]
- Koca, M.; Koc, R.; Al-Ajmi, M. Group theoretical analysis of 4D polytopes 600-cell and 120-cell with quaternions. J. Phys. A Math. Theor.
**2007**, 40, 7633. [Google Scholar] [CrossRef] - Koca, M.; Koca, N.O.; Al-Barwani, M. Snub 24-Cell derived from the Coxeter-Weyl group W(D
_{4}). Int. J. Geom. Methods Mod. Phys.**2012**, 9, 15. [Google Scholar] [CrossRef]

**Figure 6.**The vertices connected to the general vertex$\text{}\mathsf{\Lambda}$ (note that all vertices are not in the same plane).

**Figure 7.**(

**a**) A face transitive irregular dodecahedron (tetartoid) under rotational tetrahedral symmetry; (

**b**) the dual irregular icosahedron.

**Figure 9.**(

**a**) Regular$\text{}\mathrm{icosadodecahedron}\left(x=y=\tau \right);\text{}$ (

**b**) irregular icosidodecahedron ($x=1,y=2)$.

**Figure 12.**The irregular snub cube with squares, equilateral triangles, isosceles triangles and squares (for $x=1,\text{}y=-2\tau ).$

**Figure 13.**The irregular snub cube with equilateral triangles, scalene triangles and squares (for$\text{}x=2$ and $y=4).$

**Figure 14.**Dual of the snub cube (pentagonal icositetrahedron) with its pentagonal face. If we substitute$\text{}x=1,\text{}y=0$ in (46) we obtain the Catalan solid tetrakis hexahedron (dual of the truncated octahedron).

**Figure 15.**Dual of the irregular snub cube of Figure 13.

**Figure 22.**Chiral polyhedron whose vertices are the mid-points of the edges of the snub dodecahedron.

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Koca, N.O.; Koca, M.
Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions. *Symmetry* **2017**, *9*, 148.
https://doi.org/10.3390/sym9080148

**AMA Style**

Koca NO, Koca M.
Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions. *Symmetry*. 2017; 9(8):148.
https://doi.org/10.3390/sym9080148

**Chicago/Turabian Style**

Koca, Nazife Ozdes, and Mehmet Koca.
2017. "Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions" *Symmetry* 9, no. 8: 148.
https://doi.org/10.3390/sym9080148