# Regular and Chiral Polyhedra in Euclidean Nets

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**pcu**,

**fcu**, or

**bcu**are $P(1,0)$ and ${P}_{1}(1,0)$ in

**pcu**, and $Q(1,1)$ and ${P}_{2}(1,-1)$ in

**bcu**.

**pcu**, face-centred cubic lattice

**fcu**, and body-centred cubic lattice

**bcu**in the context of nets of Euclidean space. Rotary, chiral, and regular polyhedra are defined and described in Section 3. Finally, in Section 4, we prove Theorem 1 by enumerating the rotary polyhedra whose vertex and edge sets can be taken from those of one of the three lattices mentioned above.

## 2. Nets

**abc**that often carry information about chemical compounds whose links can be represented by that net. They are also natural structures for mathematicians to study.

**pcu**consists of all vertices and edges of $\mathcal{T}$ (see Figure 1a). Every vertex is 6-valent and has edges with three distinct direction vectors; namely, those of the coordinate axes.

**fcu**can be constructed from $\mathcal{T}$ by taking as vertices all vertices of

**pcu**whose sum of coordinates is even (one part of the natural bipartition of

**pcu**) and as edges all diagonals of squares of $\mathcal{T}$ with endpoints in the vertex set (see Figure 1b). The vertices are 12-valent, and there are six distinct directions of the edges. There are precisely four vertices of each cube of $\mathcal{T}$ in the vertex set of

**fcu**; their convex hull is a regular tetrahedron. Given any vertex of

**pcu**that is not a vertex of

**fcu**, its six neighbours in

**pcu**all belong to

**fcu**, and are the vertices of an octahedron. These tetrahedra and octahedra are the cells of tessellation $\#1$ in [13]. Each triangle of that tessellation can be extended to a plane tessellation by equilateral triangles, where all triangles are also triangles of tessellation $\#1$ in [13]. By gluing sets of six triangles together, we can obtain the vertex and edge set of a tessellation by regular hexagons as a subset of the net

**fcu**.

**bcu**has as vertex set all vertices of

**pcu**whose coordinates are either all odd or all even; two vertices are adjacent whenever they are endpoints of a diagonal of a cube of $\mathcal{T}$ (see Figure 1c, where the thin gray lines represent only those lines containing edges of $\mathcal{T}$ where two of the coordinates are even). The vertices are 8-valent, and there are four distinct directions of the edges.

**Lemma**

**1.**

- either π or $\pi /2$ if the edges are in
**pcu**; - either $\pi /3$, $\pi /2$, $2\pi /3$, or π if the edges are in
**fcu**; - and either ${cos}^{-1}(1/3)$, ${cos}^{-1}(-1/3)$, or π if the edges are in
**bcu**.

## 3. Regular and Chiral Polyhedra

#### 3.1. Definitions

- the set of vertices is discrete,
- the graph determined by all vertices and edges is connected,
- every edge belongs to exactly two faces,
- the vertex-figure at every vertex is a finite polygon. (The vertex-figure at a vertex v is the graph whose vertices are the neighbours of v, two of them joined by an edge whenever they are the neighbours of v in some face of the polyhedron.)

#### 3.2. Regular Polyhedra

**fcu**. Partial views of the polyhedra $\{4,3|4\}$ and $\{6,4|4\}$ are shown in Figure 7.

#### 3.3. Chiral Polyhedra

## 4. Polyhedra in Euclidean Lattices

**pcu**,

**fcu**, and

**bcu**such that

- the union of the polygons yields a connected graph,
- every edge of the lattice belongs to precisely two polygons, or to none of them,
- every vertex-figure is a finite polygon,
- there are abstract rotations preserving $\mathcal{S}$ along every face,
- there are abstract rotations preserving $\mathcal{S}$ around every vertex.

**pcu**,

**fcu**, and

**bcu**, including the finite polyhedra; we still mention all regular polyhedra for the sake of completeness. The main contribution of this paper, then, is the study of the chiral polyhedra that admit an embedding into the nets

**pcu**,

**fcu**, and

**bcu**.

**pcu**. As explained in Section 2,

**fcu**contains subsets of vertices and edges isometric to those of tetrahedra, octahedra, and hence also of their Petrials.

**pcu**in the obvious way. They can also be found in

**fcu**, for example, by considering only the vertices and edges of the net whose third coordinates equal to 0. As mentioned in Section 2, the vertices and edges of each of the remaining four planar polyhedra—$\{3,6\}$, $\{6,3\}$, ${\{\infty ,3\}}_{6}$, and ${\{\infty ,6\}}_{3}$—can be seen as subsets of those of

**fcu**.

**pcu**,

**fcu**, and

**bcu**(see Lemma 1).

**pcu**. A sample square of one embedding of $\{4,4\}\#\left\{\right\}$ in

**fcu**has vertices

**bcu**has vertices

**pcu**,

**fcu**, and

**bcu**.

**pcu**,

**fcu**, and

**bcu**. A sample hexagon of $\{6,3\}\#\left\{\right\}$ in each of these lattices has vertex set

**fcu**and $a=1$ for the remaining two nets.

**pcu**. A sample helix of one embedding of $\{4,4\}\#\left\{\infty \right\}$ in

**fcu**has vertices

**bcu**has vertices

**pcu**,

**fcu**, and

**bcu**. A sample hexagonal helix of $\{3,6\}\#\left\{\infty \right\}$ in each of these lattices has vertex set

**pcu**,

**fcu**, or

**bcu**.

**pcu**and

**dia**have more symmetries than the blended poyhedra they carry.

**pcu**,

**fcu**, or

**bcu**, we use the standard inner product to take the cosine of the angle between them and compare with the cosine of the angles described in Lemma 1. That is, the cosine must equal 0 or $-1$ if the edges are in

**pcu**; $1/2$, 0, $-1/2$, or $-1$ if the edges are in

**fcu**; and $1/3$, $-1/3$, or $-1$ if the edges are in

**bcu**. It will then remain to determine if the parameters yield a polyhedron; in particular, if the polyhedron in question has finite faces, we still have to verify if the obtained parameters are rational multiples of each other, or if one of them is 0.

#### 4.1. Polyhedra $P(a,b)$

**pcu**; the faces are some Petrie polygons of the cubes in the cubic tiling. The six faces around the origin are illustrated in the left of Figure 8. Two of the six faces are in solid lines, two in dotted lines and two in dashed lines.

**pcu**,

**fcu**, or

**bcu**.

#### 4.2. Polyhedra $Q(c,d)$

**bcu**are the union of 8 disjoint copies of the vertices of 2

**bcu**, the net similar to

**bcu**whose edges are twice as long. The following list contains a representative in each of these copies:

**bcu**; the edges are those of

**bcu**after removing:

- all edges with direction vector $(1,1,1)$ at vertices in 2
**bcu**; - all edges with direction vector $(1,-1,1)$ at vertices in $(-1,1,1)+2$
**bcu**; - all edges with direction vector $(1,1,-1)$ at vertices in $(1,-1,1)+2$
**bcu**; - all edges with direction vector $(-1,1,1)$ at vertices in $(1,1,-1)+2$
**bcu**.

**bcu**, and hence the vertices of $Q(1,1)$ are 6-valent. The faces are skew quadrilaterals congruent to the base quadrilateral with vertices $(0,0,0)$, $(1,-1,-1)$, $(0,-2,0)$, and $(1,-1,1)$. The corresponding net then has only one kind of vertex and one kind of edge under $G\left(Q\right(1,1\left)\right)$, it is bipartite (as a subnet of

**bcu**), and its smallest rings have 4-edges. The author does not know if this net already has a name. The six faces of this polyhedron at the origin are shown in Figure 8. Two of the six faces are in solid lines, two in dotted lines and two in dashed lines.

#### 4.3. Polyhedra $Q{(c,d)}^{*}$

**pcu**,

**fcu**, or

**bcu**.

#### 4.4. Polyhedra ${P}_{1}(a,b)$

**bcu**, although the angles between consecutive edges of a face suggest that they could. To see this, we recall that the three neighbours of $(0,0,0)$ in ${P}_{1}(1,b)$ are $(b,1,0)$, $(1,0,b)$, and $(0,b,1)$, and note that the neighbours of $(b,1,0)$ are $(0,0,0)$, $(b-1,1,b)$, and $(b,1-b,1)$ (see ([11], Page 198)). This implies that the directions of the edges at $(0,0,0)$ are $(b,1,0)$, $(1,0,b)$, and $(0,b,1)$; and that at $(b,1,0)$, there are edges in the directions of $(-1,0,b)$ and $(0,-b,1)$. Therefore ${P}_{1}(1,b)$ has edges with at least five different directions. Since

**bcu**has edges in only four different directions (the main diagonals of a cube of the cubic tiling), there is no chiral polyhedron ${P}_{1}(a,b)$ with $a,b\ne 0$ whose vertex and edge sets are subsets of

**pcu**,

**fcu**, or

**bcu**.

**pcu**. The three edges at every vertex in ${P}_{1}(1,0)$ are in the directions of the canonical axes. The axes of the helices are in the directions of the diagonals of a cube of the cubic tiling. The 1-skeleton of ${P}_{1}(1,0)$ is illustrated in the left of Figure 9. The three helical faces at some point are shown in the right part of the same figure.

#### 4.5. Polyhedra ${P}_{2}(c,d)$

**bcu**. This can be seen by noting that the direction of the three edges at $(0,0,0)$ of this polyhedron are $(1,-1,-1)$, $(-1,1,-1)$, and $(-1,-1,1)$; and that the isometries ${S}_{1}$ and ${S}_{2}$ (the latter mapping $(x,y,z)$ to $(y,z,x)$) that generate the symmetry group of the polyhedron preserve the set of directions $\left\{\right(1,-1,-1),(-1,1,-1),(-1,-1,1),(1,1,1\left)\right\}$, all directions of edges of

**bcu**. In fact, this polyhedron has its vertices and edges in the diamond net

**dia**, which is contained in

**bcu**. In the left of Figure 10, we show a portion of the 1-skeleton of ${P}_{2}(1,1)$; the three helices at a point are illustrated in the right of the same figure.

#### 4.6. Polyhedra ${P}_{3}(c,d)$

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Blended polyhedra $\{4,4\}\#\left\{\right\}$ (

**a**); ${\{\infty ,4\}}_{4}\#\left\{\right\}$ (

**b**); $\{4,4\}\#\left\{\infty \right\}$ (

**c**) and ${\{\infty ,4\}}_{4}\#\left\{\infty \right\}$ (

**d**).

**Figure 7.**The polyhedra $\{4,3|4\}$ and $\{6,4|4\}$. Squares and hexagons in the same shade of gray represent polygons in parallel planes

Polyhedra | Net | Remarks | Polyhedra | Net | Remarks | |
---|---|---|---|---|---|---|

$\{3,3\}$, ${\{4,3\}}_{3}$ | fcu | finite | $\{3,6\}$, ${\{\infty ,6\}}_{3}$ | fcu | planar | |

$\{3,4\}$, ${\{6,4\}}_{3}$ | fcu | finite | $\{6,3\}$, ${\{\infty ,3\}}_{6}$ | fcu | planar | |

$\{4,3\}$, ${\{6,3\}}_{4}$ | pcu | finite | $\{4,4\}$, ${\{\infty ,4\}}_{4}$ | pcu, fcu | planar |

Polyhedra | Ambient Net | Net |
---|---|---|

$\{3,6\}\#\left\{\right\}$, ${\{\infty ,6\}}_{3}\#\left\{\right\}$ | none | hxl |

$\{6,3\}\#\left\{\right\}$, ${\{\infty ,3\}}_{6}\#\left\{\right\}$ | pcu, fcu, bcu | hcb |

$\{4,4\}\#\left\{\right\}$, ${\{\infty ,4\}}_{4}\#\left\{\right\}$ | fcu, bcu | sql |

$\{3,6\}\#\left\{\infty \right\}$, ${\{\infty ,6\}}_{3}\#\left\{\infty \right\}$ | pcu, fcu, bcu | pcu |

$\{6,3\}\#\left\{\infty \right\}$, ${\{\infty ,3\}}_{6}\#\left\{\infty \right\}$ | none | acs |

$\{4,4\}\#\left\{\infty \right\}$, ${\{\infty ,4\}}_{4}\#\left\{\infty \right\}$ | fcu, bcu | dia |

Polyhedra | Ambient Net | Net |
---|---|---|

$\{4,6|4\}$, ${\{\infty ,6\}}_{4,4}$ | pcu | pcu |

$\{6,4|4\}$, ${\{\infty ,4\}}_{6,4}$ | fcu | sod |

$\{6,6|3\}$, ${\{\infty ,6\}}_{6,3}$ | fcu | crs |

${\{4,6\}}_{6}$, ${\{6,6\}}_{4}$ | fcu | hxg |

${\{6,4\}}_{6}$, ${\{\infty ,4\}}_{\xb7,*3}$ | pcu | nbo |

${\{\infty ,3\}}^{\left(a\right)}$, ${\{\infty ,3\}}^{\left(b\right)}$ | fcu | srs |

Polyhedra | Ambient Net | Net |
---|---|---|

$P(1,0)$ | pcu | pcu |

$Q(1,1)$ | bcu | unknown |

${P}_{1}(1,0)$ | pcu | srs |

${P}_{2}(1,-1)$ | bcu | srs |

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Pellicer, D.
Regular and Chiral Polyhedra in Euclidean Nets. *Symmetry* **2016**, *8*, 115.
https://doi.org/10.3390/sym8110115

**AMA Style**

Pellicer D.
Regular and Chiral Polyhedra in Euclidean Nets. *Symmetry*. 2016; 8(11):115.
https://doi.org/10.3390/sym8110115

**Chicago/Turabian Style**

Pellicer, Daniel.
2016. "Regular and Chiral Polyhedra in Euclidean Nets" *Symmetry* 8, no. 11: 115.
https://doi.org/10.3390/sym8110115