# Some New Symmetric Equilateral Embeddings of Platonic and Archimedean Polyhedra

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## Abstract

**:**

## 1. Introduction

**Note**. We provide a website with interactive three-dimensional models of the results of Section 6 and Section 7 at URL http://caagt.ugent.be/ep-embeddings/.

## 2. Definitions and Preliminaries

**Definition**

**1**.

- The Euclidian distance $d(\mu ({v}_{1}),\mu ({v}_{2}))$ is the same for every edge ${v}_{1}{v}_{2}$ of Γ, i.e., all edges in the embedding have the same length.
- If vertices ${v}_{1},{v}_{2},\dots ,{v}_{k}$ belong to the same face of V, then $\mu ({v}_{1}),\mu ({v}_{2}),\dots ,\mu ({v}_{k})$ lie in the same plane, i.e., all faces in the embedding are planar.

**Definition**

**2**.

- For each edge ${v}_{i}{v}_{j}$ of $\mathsf{\Gamma}$, the following edge equation needs to be satisfied:$${({x}_{j}-{x}_{i})}^{2}+{({y}_{j}-{y}_{i})}^{2}+{({z}_{j}-{z}_{i})}^{2}={L}^{2}.$$
- If the vertices ${v}_{i},{v}_{j},{v}_{k},\cdots $ form a face of $\mathsf{\Gamma}$, then the planarity condition holds for that face if and only if the following matrix$$\left(\begin{array}{cccc}{x}_{i}& {x}_{j}& {x}_{k}& \cdots \\ {y}_{i}& {y}_{j}& {y}_{k}& \cdots \\ {z}_{i}& {z}_{j}& {z}_{k}& \cdots \\ 1& 1& 1& \cdots \end{array}\right)$$$$\left|\begin{array}{cccc}{x}_{i}& {x}_{j}& {x}_{k}& {x}_{l}\\ {y}_{i}& {y}_{j}& {y}_{k}& {y}_{l}\\ {z}_{i}& {z}_{j}& {z}_{k}& {z}_{l}\\ 1& 1& 1& 1\end{array}\right|=0,$$

- Fix the coordinates of ${v}_{1}$ to be $(0,0,0)$. Indeed, if there is a solution, then we can always move it so that ${v}_{1}$ has exactly these coordinates.
- Similarly, we may fix ${v}_{2}$ to be of the form $({x}_{2},0,0)$, for we can always rotate a solution in such a way that ${v}_{1}{v}_{2}$ becomes the X-axis.
- Finally, we can choose ${v}_{3}$ to be of the form $({x}_{3},{y}_{3},0)$ by rotating the plane ${v}_{1}{v}_{2}{v}_{3}$ around the X-as so that it coincides with the $XY$-plane.

## 3. Analytic Solutions

## 4. Algebraic Conjugacy

## 5. Symmetric Solutions

- Extending T with the inversion $(x,y,z)\mapsto (-x,-y,-z)$ yields the pyritohedral group ${T}_{h}$. This group contains (among other elements) the three mirror symmetries with respect to the coordinate planes, mapping $(x,y,z)$ to $(-x,y,z)$, $(x,-y,z)$ and $(x,y,-z)$, respectively.
- Extending T with the coordinate transpositions $(x,y,z)\mapsto (y,x,z)$, $(x,y,z)\mapsto (z,y,x)$ and $(x,y,z)\mapsto (x,z,y)$ yields the full tetrahedral group ${T}_{d}$. The coordinate transpositions are mirror symmetries with respect to the planes $X=Y$, $Y=Z$ and $Z=X$.
- Doing both extensions above at the same time yields the full octahedral group ${O}_{h}$. This group contains (among other elements) six rotations of order 4 (i.e., with a rotation angle of $\pi /2$), with the X-, Y- and Z-axes as rotation axes:$$\begin{array}{cc}{\rho}_{X}:(x,y,z)\mapsto (x,-z,y),\text{\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & {\rho}_{X}^{-1}:(x,y,z)\mapsto (x,z,-y),\hfill \\ {\rho}_{Y}:(x,y,z)\mapsto (z,y,-x),\text{\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & {\rho}_{Y}^{-1}:(x,y,z)\mapsto (-z,y,x),\hfill \\ {\rho}_{Z}:(x,y,z)\mapsto (-y,x,z),\text{\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & {\rho}_{Z}^{-1}:(x,y,z)\mapsto (y,-x,z).\hfill \end{array}$$
- Extending T with the rotations above, but not with any of the mirror symmetries, yields the chiral octahedral group O.

- The orbit size for the point $(0,y,y)$ remains 12 in all cases.
- The orbit size for the point $(x,y,y)$, with $x\ne 0,y$ remains 12 for ${T}_{h}$ and becomes 24 for ${T}_{d}$, O and ${O}_{h}$.
- The orbit size for the point $(0,y,z)$, with $0\ne y\ne z\ne 0,y$ remains 12 for ${T}_{d}$ and becomes 24 for ${T}_{h}$, O and ${O}_{h}$.

## 6. Archimedean and Platonic Solids

**Tetrahedron (3.3.3)**The four vertices of the (graph of the) tetrahedron form a single orbit of T. If the EP-embedding is to be invariant for T, then the coordinates of these vertices must be of the form ${(\pm a,\pm a,\pm a)}_{\mathrm{even}}$, as explained in Section 5. Every vertex is connected to every other vertex by an edge. The edges form a single orbit of T, and hence there is only a single edge equation. Because the faces are triangles, there are no face equations.

**Octahedron (3.3.3.3)**. The six vertices of the (graph of the) octahedron form a single orbit and therefore map to coordinates $(\pm b,0,0)$, $(0,\pm b,0)$ and $(0,0,\pm b)$, as explained in Section 5. The edges form a single orbit, where $(b,0,0)\xb7(0,b,0)$ is a typical edge. This representation leads to a single edge equation $2{b}^{2}=2$ (this time we choose edge length $L=\sqrt{2}$). There are no face equations. Both solutions $b=\pm 1$ lead to the same result, i.e., the regular octahedron with vertices $(\pm 1,0,0)$, $(0,\pm 1,0)$ and $(0,0,\pm 1)$.

**Cube (4.4.4)**. The eight vertices of the (graph of the) cube form two orbits of size 4, as denoted by filled and unfilled vertices in Figure 4a, with coordinates ${(a,a,a)}_{\mathrm{even}}$ and ${(c,c,c)}_{\mathrm{even}}$. (They form a single orbit under the full automorphism group of the graph, but here we only consider the subgroup T.) The 12 edges form a single orbit, where $(a,-a,-a)\xb7(c,c,c)$ is a typical edge. This leads to the edge equation

**Three-fold rotational symmetry**. Before proceeding with the more complex polyhedra, we would like to illustrate how the three-fold rotational symmetry that we impose helps to simplify the face equations for certain hexagonal faces. Consider a hexagon through points with coordinates of the following form:

**Truncated tetrahedron (3.6.6)**. The 12 vertices of the truncated tetrahedron form a single orbit with representative $(x,y,z)$ (Figure 4b). There are two orbits of edges, as denoted in the picture by two thicknesses of edges. One orbit, generated by $(x,y,z)\xb7(z,x,y)$ has size 12. The other orbit, generated by $(x,y,z)\xb7(-x,y,-z)$, has size 6. There is one orbit of four triangles, and one orbit of four hexagons. (One hexagon H lies “behind” the others in the picture.) Each hexagon is rotation invariant for one of the rotation axes of T through a vertex of the reference tetrahedron, as is evident for H in the picture.

**Cuboctahedron (3.4.3.4)**. The 12 vertices of a cuboctahedron form a single orbit. The edges form two orbits of 12 edges generated by $(x,y,z)\xb7(z,x,y)$ and $(x,y,z)\xb7(-y,z,-x)$. There are two orbits of triangles and one of quadrangles (Figure 4c).

**Icosahedron (3.3.3.3.3)**. The 12 vertices of an icosahedron still form a single orbit under the subgroup T of its full symmetry group ${I}_{h}$ (Figure 4d). The edges, on the other hand, now form three orbits (as denoted in the picture by three different colors) generated by

**Dodecahedron (5.5.5)**. The 20 vertices of the dodecahedron come in two orbits of size 4, generated by $(a,a,a)$ and $(c,c,c)$, and one orbit of size 12, generated by $(x,y,z)$ (Figure 4e).

**Truncated cube (3.8.8)**. The 24 vertices of the truncated cube come in two orbits of size 12, generated by $(d,e,f)$ and $(x,y,z)$ (Figure 6a). There are three edge orbits of size 12, with the following representatives:

- The classical embedding of the truncated cube, with coordinates$$(x,y,z)=\pm (1,1+\sqrt{2},1+\sqrt{2}),\phantom{\rule{1.em}{0ex}}(d,e,f)=\pm (-1,1+\sqrt{2},1+\sqrt{2}).$$
- The algebraic conjugate of the above, with coordinates$$(x,y,z)=\pm (1,1-\sqrt{2},1-\sqrt{2}),\phantom{\rule{1.em}{0ex}}(d,e,f)=\pm (-1,1-\sqrt{2},1-\sqrt{2}).$$
- An embedding with coordinates$$(x,y,z)=\pm (1+\sqrt{2},1,1),\phantom{\rule{1.em}{0ex}}(d,e,f)=\pm (-1+\sqrt{2},1,1),$$

**Truncated octahedron (4.6.6)**. The EP-embeddings of the truncated octahedron with 24 vertices turn out to be much easier to compute than those of the truncated cube. There are two orbits of 12 vertices, generated by $(d,e,f)$ and $(x,y,z)$ (Figure 4f), and three orbits of 12 edges, with the following representatives:

**Rhombicuboctahedron (3.4.4.4)**. The 24 vertices of the rhombicuboctahedron come in two orbits of 12 vertices, generated by $(d,e,f)$ and $(x,y,z)$ (Figure 6b). There are four orbits of 12 edges, with the following representatives:

**Snub cube (3.3.3.3.4)**. The last Archimedean solid with 24 vertices again has two orbits of 12 vertices, generated by $(d,e,f)$ and $(x,y,z)$ (Figure 6c), and this time five orbits of 12 edges:

**Icosidodecahedron (3.5.3.5)**. The 24 vertices come in two orbits of size 12, generated by $(d,e,f)$ and $(x,y,z)$, and one orbit of size 6, generated by $(b,0,0)$ (Figure 6d). There are five orbits of 12 edges, with the following representatives:

**Truncated cuboctahedron (4.6.8)**. (For this case and the cases that follow, we have omitted the corresponding pictures as they become too crowded.) The 48 vertices of the truncated cuboctahedron come in 4 orbits of size 12 with generators $(x,y,z)$, $(d,e,f)$, $(X,Y,Z)$, $(D,E,F)$. The edges have the following representatives (6 orbits of size 12):

**Rhombicosidodecahedron (3.4.5.4)**. The 60 vertices of the rhombicosidodecahedron come in five orbits of 12 vertices with representatives $(p,q,r)$, $(d,e,f)$, $(D,E,F)$, $(x,y,z)$ and $(X,Y,Z)$. There are 10 orbits of 12 edges each, with the following representatives

## 7. The Remaining Cases

**Truncated icosahedron (5.6.6)**The 60 vertices of this polyhedron are split into one orbit of 12 vertices, with coordinates of the form

**Truncated dodecahedron (3.10.10)**The 60 vertices of this polyhedron come in three orbits, and we may choose the same representatives $(0,p,q)$, $(d,e,f)$ and $(x,y,z)$ as in the case of the truncated icosahedron. There are five orbits of edges:

**Truncated icosidodecahedron (4.6.10)**There are 120 vertices in five orbits of 24. We choose $(p,q,r)$, $(x,y,z)$, $(X,Y,Z)$, $(d,e,f)$ and $(D,E,F)$ as representatives for these orbits. There is one orbit of 12 decagons. The following matrix displays the coordinates of the vertices of one representative:

**Snub dodecahedron (3.3.3.3.5)**Because the graph of the snub dodecahedron has an automorphism group (i.e., I) that does not contain ${T}_{h}$ as a subgroup, we cannot find solutions with this symmetry. Enforcing the even larger symmetry group I on our solution yields only the classical embedding and its three (real) conjugates. Indeed, because of the enforced symmetry, the pentagonal faces embed as regular pentagons and hence the embedding is a uniform polyhedron, all of which have been classified before [4].

## 8. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Algebraic conjugation of the pentagonal pyramid (

**a**) yields a pyramid with a pentagram base (

**b**). Algebraic conjugation of the icosahedron (

**c**) yields a great dodecahedron (

**d**). Algebraic conjugation of the dodecahedron (

**e**) yields a great stellated dodecahedron (

**f**).

**Figure 3.**Schematic representation of the group T: group elements (

**a**); vertex orbits of size 4 (

**b**,

**c**); vertex orbits of size 6 (

**d**); and vertex orbits of size 12 (

**e**).

**Figure 4.**Orbit structure of the cube (

**a**); the truncated tetrahedron (

**b**); the cuboctahedron (

**c**); the icosahedron (

**d**); the dodecahedron (

**e**); and the truncated octahedron (

**f**).

**Figure 5.**EP-embeddings of the dodecahedron: the equilateral dodecastemma (

**a**), showing a typical face (

**b**); the conjugate of the dodecastemma (

**c**,

**d**); a “decorated” tetrahedron (

**e**) with collinear edges and coplanar faces; and a fifth embedding (

**f**) with intersecting faces.

**Figure 6.**Orbit structure of: the truncated cube (

**a**); the rhombicuboctahedron (

**b**); the snub cube (

**c**); and the icosidodecahedron (

**d**).

Platonic | Archimedean (cntd.) | ||
---|---|---|---|

Tetrahedron | 3.3.3 | Rhombicuboctahedron | 3.4.4.4 |

Cube | 4.4.4 | Truncated cuboctahedron | 4.6.8 |

Octahedron | 3.3.3.3 | Snub cube | 3.3.3.3.4 |

Dodecahedron | 5.5.5 | Icosidodecahedron | 3.5.3.5 |

Icosahedron | 3.3.3.3.3 | Truncated dodecahedron | 3.10.10 |

Archimedean | Truncated icosahedron | 5.6.6 | |

Truncated tetrahedron | 3.6.6 | Rhombicosidodecahedron | 3.4.5.4 |

Cuboctahedron | 3.4.3.4 | Truncated icosidodecahedron | 4.6.10 |

Truncated cube | 3.8.8 | Snub dodecahedron | 3.3.3.3.5 |

Truncated octahedron | 4.6.6 |

**Table 2.**Number of elements, number of orbits of group T and measure of difficulty for Platonic solids.

3.3.3 | 4.4.4 | 3.3.3.3 | 5.5.5 | 3.3.3.3.3 | ||
---|---|---|---|---|---|---|

Vertices | 4 | 8 | 6 | 20 | 12 | |

Edges | 6 | 12 | 12 | 30 | 30 | |

Faces | 4 | 6 | 8 | 12 | 20 | |

Vertex orbits | size 4 | 1 | 2 | 2 | ||

size 6 | 1 | |||||

size 12 | 1 | 1 | ||||

Edge orbits | size 6 | 1 | 1 | 1 | ||

size 12 | 1 | 1 | 2 | 2 | ||

Face orbits | 4 triangles | 1 | 2 | 2 | ||

12 triangles | 1 | |||||

6 quadrangles | 1 | |||||

12 pentagons | 1 | |||||

Variables | 1 | 1 | 1 | 5 | 3 | |

Bézout | 2 | 2 | 2 | ${2}^{3}{3}^{2}$ | ${2}^{3}$ |

**Table 3.**Number of elements, number of orbits of group T and measure of difficulty for Archimedean solids with full symmetry that is tetrahedral or octahedral.

3.6.6 | 3.4.3.4 | 3.8.8 | 4.6.6 | 3.4.4.4 | 4.6.8 | 3.3.3.3.4 | ||
---|---|---|---|---|---|---|---|---|

Vertices | 12 | 12 | 24 | 24 | 24 | 48 | 24 | |

Edges | 18 | 24 | 36 | 36 | 48 | 72 | 60 | |

Faces | 8 | 14 | 14 | 14 | 26 | 26 | 38 | |

Vertex orbits | size 12 | 1 | 1 | 2 | 2 | 2 | 4 | 2 |

Edge orbits | size 6 | 1 | ||||||

size 12 | 1 | 2 | 3 | 3 | 4 | 6 | 5 | |

Face orbits | 4 triangles | 1 | 2 | 2 | 2 | 2 | ||

12 triangles | 2 | |||||||

6 quadrangles | 1 | 1 | 1 | 1 | ||||

12 quadrangles | 1 | 1 | ||||||

4 hexagons | 1 | 2 | 2 | |||||

6 octagons | 1 | 1 | ||||||

Variables | 2 | 2 | 6 | 3 | 2 | 7 | 5 | |

Bézout | ${2}^{2}$ | ${2}^{2}$ | ${2}^{6}$ | ${2}^{3}$ | ${2}^{2}$ | ${2}^{7}$ | ${2}^{5}$ |

**Table 4.**Number of elements, number of orbits of group T and measure of difficulty for Archimedean solids with full symmetry that is icosahedral.

3.5.3.5 | 3.10.10 | 5.6.6 | 3.4.5.4 | 4.6.10 | 3.3.3.3.5 | ||
---|---|---|---|---|---|---|---|

Vertices | 30 | 60 | 60 | 60 | 120 | 60 | |

Edges | 60 | 90 | 90 | 120 | 180 | 150 | |

Faces | 32 | 32 | 32 | 62 | 62 | 92 | |

Vertex orbits | size 6 | 1 | |||||

size 12 | 2 | 5 | 5 | 5 | 10 | 5 | |

Edge orbits | size 6 | 1 | 1 | 1 | |||

size 12 | 5 | 7 | 7 | 10 | 15 | 12 | |

Face orbits | 4 triangles | 2 | 2 | 2 | 2 | ||

12 triangles | 1 | 1 | 1 | 6 | |||

6 quadrangles | 1 | 1 | |||||

12 quadrangles | 2 | 2 | |||||

12 pentagons | 1 | 1 | 1 | 1 | |||

4 hexagons | 2 | 2 | |||||

12 hexagons | 1 | 1 | |||||

12 decagons | 1 | 1 | |||||

Variables | 7 | 15 | 13 | 8 | 22 | 15 | |

Bézout | ${2}^{5}{3}^{2}$ | ${2}^{15}$ | ${2}^{8}{3}^{5}$ | ${2}^{6}{3}^{2}$ | ${2}^{22}$ | ${2}^{13}{3}^{2}$ |

**Table 5.**Number of new EP-embeddings found, subdivided according to full geometric symmetry group, not including the classical embeddings and their algebraic conjugates. (Polyhedra for which we found no new embeddings are not listed, except for the truncated icosidodecahedron where the EP-equations could only partially be solved.)

Enforced T Symmetry | # | ${\mathit{T}}_{\mathit{h}}$ | ${\mathit{T}}_{\mathit{d}}$ | T |
---|---|---|---|---|

Dodecahedron (5.5.5) | 4 | 2 | 1 | 1 |

Truncated cube (3.8.8) | 2 | 1 | 1 | |

Snub cube (3.3.3.3.4) | 1 | 1 | ||

Icosidodecahedron (3.5.3.5) | 9 | 2 | 1 | 6 |

Truncated cuboctahedron (4.6.8) | 2 | 2 | ||

Rhombicosidodecahedron (3.4.5.4) | 24 | 8 | 16 | |

Enforced ${\mathbf{T}}_{\mathbf{h}}$ Symmetry | ||||

Truncated icosahedron (5.6.6) | 6 | 6 | ||

Truncated dodecahedron (3.10.10) | 16 | 16 | ||

Truncated icosidodecahedron (4.6.10) | 0 |

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## Share and Cite

**MDPI and ACS Style**

Coolsaet, K.; Schein, S.
Some New Symmetric Equilateral Embeddings of Platonic and Archimedean Polyhedra. *Symmetry* **2018**, *10*, 382.
https://doi.org/10.3390/sym10090382

**AMA Style**

Coolsaet K, Schein S.
Some New Symmetric Equilateral Embeddings of Platonic and Archimedean Polyhedra. *Symmetry*. 2018; 10(9):382.
https://doi.org/10.3390/sym10090382

**Chicago/Turabian Style**

Coolsaet, Kris, and Stan Schein.
2018. "Some New Symmetric Equilateral Embeddings of Platonic and Archimedean Polyhedra" *Symmetry* 10, no. 9: 382.
https://doi.org/10.3390/sym10090382