# Symmetry and Topology: The 11 Uninodal Planar Nets Revisited

## Abstract

**:**

## 1. Introduction

**sql**according to the vector-method, including an analysis of the ideal two-dimensional space-group (plane group) of the net. Section 4 introduces a description of

**sql**using a symmetry-labeled quotient graph, showing that the full plane group of the net, including its translational symmetry, is generated by two proper rotations. The analysis of the 10 remaining uninodal two-periodic planar nets is performed along the same lines in the next sections; the different nets are analyzed in order of growing complexity. The paper ends with some general observations concerning the description of uninodal planar nets.

## 2. Methodology

**cem**,

**fsz**,

**htb**and

**tts**are missing. This paper focuses on planar nets, asking whether they can all be derived from the bouquet ${\mathcal{B}}_{n}$ exclusively using point-symmetry operations as voltages. The proposed description also follows a principle of economy, looking for the smallest possible number of loops in the bouquet. Clearly, a minimum of two loops is necessary to describe a two-periodic net, whatever the nature of the voltage. The results reported in Table 1 show that most planar nets can be obtained from ${\mathcal{B}}_{2}$ or ${\mathcal{B}}_{3}$.

## 3. The Square Lattice Net

#### 3.1. The Vector Method

**sql**net from its labeled quotient graph, the bouquet ${\mathcal{B}}_{2}$, as given in Figure 1. In this case, according to the vector method [4], voltages are vectors in ${\mathbb{Z}}^{2}$, and the generated group is a translation group of rank 2. The vertex set is defined as $\{{V}_{t}:t\in {\mathbb{Z}}^{2}\}$. If two orthogonal vectors a and b are used as a basis of the lattice in the plane, with correspondence $a=10$ and $b=01$, one obtains the derived net with the orientation as given in Figure 1. We observe that, locally, the derived net has the same structure as the voltage graph: for each color, there is one outgoing and one incoming edge at every vertex. The green edge ${a}_{t}$ links vertex ${V}_{t}$ to vertex ${V}_{t+a}$, and the red edge ${b}_{t}$ links vertex ${V}_{t}$ to vertex ${V}_{t+b}$.

**sql**net. Because

**sql**is a lattice net, it is a crystallographic net [10], which means, by definition, that its automorphism group is isomorphic to a space-group [11]. Because it is also a minimal net [12], the factor group of its space-group is isomorphic to the automorphism group of the voltage graph [2]. In order to determine the point group of

**sql**, we thus look for generators of the automorphism group of the bouquet ${\mathcal{B}}_{2}$ and then for an interpretation as symmetry operations in Euclidian space.

**sql**is the symmorphic group $p4mm$.

#### 3.2. An Example of a Symmetry-Labeled Quotient Graph

**sql**is based on translation operations in the Euclidian plane. In this section, we consider extensively a derivation of

**sql**from the bouquet ${\mathcal{B}}_{2}$ with two 4-fold rotations as voltages on the loops, as shown in Figure 2. The given representation of the net was obtained after placing the initial vertex close to the origin and considering two anticlockwise 4-fold rotations $\alpha $ and $\beta $ with centers at $(1/2,-1/2)$ and $(-1/2,1/2)$, respectively. These initial elements are shown in brown in the figure. Because voltages have order 4, each loop unwraps to a 4-cycle: starting from vertex ${V}_{1}$ at the origin, the green loop unwraps to the green 4-cycle around the center of rotation $\beta $, and similarly the red loop unwraps to the red 4-cycle around the center of rotation $\alpha $. Because we know that $\beta $ acts freely on the derived net, we also obtain by this rotation the three other red 4-cycles at the corners of the unit cell. We note that this unit cell, as drawn in Figure 2, corresponds to the space-group generated by the two rotations $\alpha $ and $\beta $, which happens to be a $2\times 2$ supercell of

**sql**.

**sql**can be achieved by analyzing the labeled quotient graph $N/T$, as follows.

**sql**can be worked out through labeled quotient graphs, as indicated in Figure 3. There are indeed two freely acting automorphisms of $N/T$ that leave the voltages over cycles unchanged and that should be interpreted as images of translations in N, thus extending the group T. We first define the automorphism ${\theta}_{v}=(T,T\beta )(T{\beta}^{2},T{\beta}^{3})$ exchanging (i) T with $T\beta $ and $T{\beta}^{2}$ with $T{\beta}^{3}$; (ii) green and red horizontal edges between T and $T\beta $, as well as those between $T{\beta}^{2}$ and $T{\beta}^{3}$; and (iii) vertical edges between T and $T{\beta}^{3}$ with those of the same color between $T\beta $ and $T{\beta}^{2}$. A second automorphism ${\theta}_{h}=(T,T{\beta}^{3})(T\beta ,T{\beta}^{2})$ is defined similarly, this time exchanging colors for vertical edges and keeping colors for horizontal edges; ${\theta}_{v}$ and ${\theta}_{h}$ act on $N/T$ as reflections in the blue v and h lines, respectively. The graph $N/T$ thus has a single vertex class and two edge classes for the automorphism group $\langle {\theta}_{h},{\theta}_{v}\rangle $; the four horizontal edges form a first class and the four vertical edges form a second class. Hence its quotient is the bouquet ${\mathcal{B}}_{2}$ with all horizontal edges mapped on the upper loop and all vertical edges mapped on the lower loop. More precisely, both 2-cycles with voltage 02 (resp. 20) are wrapped on the upper (resp. lower) loop. This means that the voltages on the loops are respectively 01 and 10.

## 4. The Kagome Net

**kgm**) net. As given in Table 1, we use as voltages on the loops of the bouquet ${\mathcal{B}}_{2}$ the 3-fold rotations $\alpha $ and $\beta $ with centers at $(1/3,-1/3)$ and $(-1/3,1/3)$, respectively. These symmetry operations can be represented by the following $3\times 3$ matrices:

**sql**. For instance, according to the voltages assigned to loops in the bouquet ${\mathcal{B}}_{2}$, the green edge outgoing from $T\alpha $ in $N/T$ goes to $T\alpha \beta $. After this vertex has been rewritten in the form $xT{\alpha}^{m}$, one can see that the respective edge goes from $T\alpha $ to $T{\alpha}^{m}$ and must be assigned voltage x. In particular, we have

**kgm**can be performed by using, for instance, the program SYSTRE [13].

**kgm**, any vertex admits one incoming and one outgoing red (resp. green) edge, in accordance with the bouquet ${\mathcal{B}}_{2}$. The nature of the respective symmetry operation can be read in the open space between the two edges of the same color, here as a 3-fold rotation. This observation applies to any net in further sections. We note also the two-color 6-cycle (strong ring) associated with the relator ${\left(\alpha \beta \right)}^{3}$. It is a general fact that generators form one-color cycles while relators give rise to multicolor cycles.

## 5. The Honeycomb Net

**hcb**) net with vertices of degree 3, which is identified with an alternative Cayley color graph. To this end, we consider the bouquet ${\mathcal{B}}_{2}$ with loops assigned a 6-fold rotation $\alpha $ and a 2-fold rotation $\beta $ with respective centers at $(1,2)$ and $(-1/2,-1)$, represented by the following matrices:

**hcb**runs as in the previous cases. We first note that the two combinations $a=\alpha \beta {\alpha}^{2}$ and $b={\alpha}^{3}\beta $ generate a translation (abelian) group T of rank 2, which is normal in the abstract group $G=\langle (\alpha ,\beta )|{\alpha}^{6},{\beta}^{2},{\left(\alpha \beta \right)}^{3}\rangle $; there are six T-cosets that may be written as $T{\alpha}^{n}$ with $0\le n\le 5$. Using these cosets, the bouquet can be unwrapped to the labeled quotient graph shown in Figure 6b. The analysis of the automorphism $(T,T{\alpha}^{2},T{\alpha}^{4})(T\alpha ,T{\alpha}^{5},T{\alpha}^{3})$ shows that it can be associated to the existence of a translation $\tau $ of the derived net with $3\tau =a-b$, indicating that the derived net is indeed isomorphic to

**hcb**, with a triple unit cell.

## 6. Decorated sql and hcb Nets

**fes**(decorated

**sql**) and

**hca**(decorated

**hcb**) are described by the bouquet ${\mathcal{B}}_{2}$ with two rotations as voltages: (i) a 4-fold rotation $\alpha $ and a 2-fold rotation $\beta $ with an extra relator ${\left(\alpha \beta \right)}^{4}$ for

**fes**, and (ii) a 3-fold rotation $\alpha $ and a 2-fold rotation $\beta $ with an extra relator ${\left(\alpha \beta \right)}^{6}$ for

**hca**. These relators are interpreted as 8- and 12-cycles in the respective derived nets. A translation (normal, abelian) group T is generated by the two combinations ${\alpha}^{2}\beta $ and $\alpha \beta \alpha $ with four cosets $T{\alpha}^{i}$$(i=0,1,2,3)$ in the case of

**fes**. The translation group T of

**hca**is generated by the two combinations $\alpha \beta \alpha \beta \alpha $ and ${\alpha}^{2}\beta \alpha \beta $. There are six cosets: $T{\alpha}^{i}$ and $T\beta {\alpha}^{i}$$(i=0,1,2)$. Because the derivation is rather straightforward, only the final nets are shown in Figure 7. We note that the edges associated to 2-fold rotations have no orientation.

## 7. The htb Net

**htb**(hexagonal tungsten bronze) net, shown in Figure 8, is described by the bouquet ${\mathcal{B}}_{2}$ with a 6-fold rotation $\alpha $ and a 3-fold rotation $\beta $ as voltages, with relator ${\left(\alpha \beta \right)}^{2}$. The translation group T is generated by the two combinations ${\alpha}^{4}\beta $ and $\beta {\alpha}^{4}$ and admits six cosets given as $T{\alpha}^{i}$$(i=0,\dots ,5)$. We note again the presence in the net of a two-color 4-cycle (strong ring), associated to the relator.

## 8. The Hexagonal Lattice Net

**hxl**) is regular, of degree 6 and can be described using the bouquet ${\mathcal{B}}_{3}$ with voltages $\alpha $, $\beta $ and $\gamma $, corresponding to 3-fold rotations, which can be represented by the following $3\times 3$ matrices:

**hxl**net given in Figure 9c. The net

**hxl**is thus isomorphic to the Caley color graph of $p3$ with the three generators $\alpha $, $\beta $ and $\gamma $. We note again the correlation between strong rings of the net and relators of the space-group, in particular, the relator $\alpha \beta \gamma $ associated to the green–blue–red 3-cycle.

## 9. fsz

**fsz**shown in Figure 10. As for the

**htb**net, the translation group is generated by the two combinations $\beta {\alpha}^{4}$ and ${\alpha}^{4}\beta $. We note the blue–red–green cycle associated with the relator $\gamma \beta \alpha $.

## 10. tts

**tts**shown in Figure 11. We note the strong similarity of this representation of

**tts**with that of

**sql**given in Figure 2. This fact clearly reflects the similarity of their symmetry-labeled quotient graphs, as that of

**tts**is obtained from that of

**sql**by adding a third loop with voltage $\gamma =\alpha \beta $. The relationship is evident in Table 1.

## 11. fxt

**fxt**only admits three reflections, ${\sigma}_{1}$, ${\sigma}_{2}$ and ${\sigma}_{3}$, as voltages of the bouquet ${\mathcal{B}}_{3}$. These reflections are coupled as a 6-fold rotation $\alpha ={\sigma}_{1}{\sigma}_{2}$, a 3-fold rotation $\beta ={\sigma}_{1}{\sigma}_{3}$ and a 2-fold rotation $\gamma ={\sigma}_{2}{\sigma}_{3}$. As a result, the alternative Cayley color graph has no orientation and its space-group is the full group $p6mm$. The determination of the translation group T can be performed by comparison with

**htb**. Indeed, only even combinations of reflections can generate a translation, and the pairing of reflections yields then a combination of the three rotations $\alpha $, $\beta $ and $\gamma $, which is quite similar to the expression of translations for

**htb**whose generators obey similar relations. Hence, we can take the two combinations ${\alpha}^{4}\beta $ and $\beta {\alpha}^{4}$ as generators of T, which admits 12 cosets, given as $T{\alpha}^{i}$ and $T{\alpha}^{i}{\sigma}_{1}$$(i=0,\dots ,5)$. The derived net is shown in Figure 12.

## 12. cem

## 13. Final Considerations

**r**is a relator if and only if

**r**= 1 is a relation. We observe also that more than one set of generators is usually possible to describe the same net.

**cem**are generated by using rotations or reflections as voltages of some bouquet ${\mathcal{B}}_{n}$ with $n=2,3,4$. Even in the case of

**cem**, it must be said that a single voltage corresponds to a translation, and this translation is given as the combination of two 2-fold rotations, themselves given as voltages. Hence, in all cases, the whole space-group is generated by no more than three point-symmetry operations. Of course, these operations have distinct fixed points; otherwise they would generate a point-group. This is quite different from the conventional description of space-groups in International Tables, for instance, where given relations strictly concern the linear part of symmetry operations, because translations are always implicitly assumed and explicitly given as generators. In fact, even the geometrical interpretation as point-group operations, of generators given in Table 1, is needless. The space-group is abstractly generated from the relations listed in Table 1. Following general theorems from topological graph theory, the generated group acts freely (without fixed points) on the derived graph, that is, on the alternative Cayley color graph of the group, so that the generated space-group is a subgroup of the full space-group of the respective net.

**fes**admits 4- and 8-rings. We emphasize again that translations appear in this analysis as a simple consequence of point-symmetry operations; of course, this does not generalize to an arbitrary net.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**Left**) A geometric realization (embedding) of the square lattice net (

**sql**) in the Euclidian plane with oriented edges, and (

**Right**) its labeled quotient graph with voltages in ${\mathbb{Z}}^{2}$. The two classes of edges in the net are given the same color as the representative loop in ${\mathcal{B}}_{2}$.

**Figure 2.**(

**Left**) The square lattice net and a symbolic representation of the space-group generated by a single vertex and two 4-fold rotation centers (in brown) using (

**Right**) the bouquet ${\mathcal{B}}_{2}$ with the respective rotations $\alpha $ and $\beta $ as voltages. The two classes of edges in the net are given the same color as the representative loop in ${\mathcal{B}}_{2}$. Note that the initial vertex has been slightly shifted in relation to the origin, thus providing a truly $p4$ embedding with a $2\times 2$ unit cell.

**Figure 3.**(

**Left**) The labeled quotient graph $N/T$ of the net N derived from the symmetry-labeled bouquet ${\mathcal{B}}_{2}$ in Figure 2; vertices correspond to T-cosets in $G=\langle \alpha ,\beta \rangle $; edge colors match the classes according to the two generators $\alpha $ and $\beta $, as given in Figure 2. The quotient by the reflections ${\theta}_{v}$ and ${\theta}_{h}$ through blue lines h and v is (

**Right**) the bouquet ${\mathcal{B}}_{2}$ with half voltages 10 and 01 (see text).

**Figure 4.**(

**Left**) The bouquet ${\mathcal{B}}_{2}$ with loops assigned 3-fold rotations, and (

**Right**) the labeled quotient graph $N/T$ of the derived net $N=\mathbf{kgm}$ with translation group $T=\langle 20,02\rangle $ (see text).

**Figure 5.**The

**kgm**net derived from the bouquet ${\mathcal{B}}_{2}$ with loops assigned 3-fold rotations (in the center of the unit cell).

**Figure 6.**(

**a**) The bouquet ${\mathcal{B}}_{2}$ with loops assigned 6-fold and 2-fold rotations, (

**b**) the labeled quotient graph unwrapped from the bouquet ${\mathcal{B}}_{2}$, and (

**c**) the derived

**hcb**net showing all symmetry elements in the extended unit cell. We note the absence of orientation on red edges obtained after the identification of ingoing and outgoing edges associated to the 2-fold rotation $\beta $ (see text).

**Figure 7.**(

**a**) The

**fes**net and (

**b**) the

**hca**net generated from the bouquet ${\mathcal{B}}_{2}$. We note the absence of orientation on red edges obtained after identification of ingoing and outgoing edges associated to the respective 2-fold rotations (see text).

**Figure 8.**The

**htb**net generated from the bouquet ${\mathcal{B}}_{2}$ with 6- and 3-fold rotations as voltages.

**Figure 9.**(

**a**) The bouquet ${\mathcal{B}}_{3}$ with loops assigned 3-fold rotations. (

**b**) The labeled quotient graph unwrapped from (

**a**). (

**c**) The derived

**hxl**net showing all symmetry elements in the extended unit cell (see text).

**Figure 10.**The net

**fsz**as derived from the bouquet ${\mathcal{B}}_{3}$ with loops assigned 6-, 3- and 2-fold rotations. We ntoe the absence of orientation on the blue edge associated to the 2-fold rotation.

**Figure 11.**The net

**tts**as derived from the bouquet ${\mathcal{B}}_{3}$ with loops assigned two 4-fold rotations and one 2-fold rotation. We note the absence of orientation on the blue edge associated to the 2-fold rotation and the three-color 3-cycle associated to the relator.

**Figure 12.**The net

**fxt**as derived from the bouquet ${\mathcal{B}}_{3}$, where each loop has been assigned a reflection; we note the absence of orientation.

**Figure 13.**Symmetric representation in $p2mm$ of the alternative Cayley color graph isomorphic to the net

**cem**, as derived from the bouquet ${\mathcal{B}}_{4}$, where three out of the four loops have been assigned a reflection; only symmetry elements from $p2$ are shown. We note the absence of orientation along the respective edges.

**Table 1.**Generators used as voltages for bouquets in the description of uninodal two-periodic nets, respective relators, space-groups and unit cell size of an embedding of the alternative Cayley color graph in comparison with those of the net.

Net | Bouquet | Generators | Relators | Space-Groups | Unit Cell |
---|---|---|---|---|---|

cem | ${\mathcal{B}}_{4}$ | $\alpha $, $\beta $, $\gamma $, $\delta $ | ${\alpha}^{2}$, ${\beta}^{2}$, ${\gamma}^{2}$, $\alpha \beta \delta $, [$\delta ,\gamma $] | $p2$/$c2mm$ | $1\times 1$ |

fes | ${\mathcal{B}}_{2}$ | $\alpha $, $\beta $ | ${\alpha}^{4}$, ${\beta}^{2}$, ${\left(\alpha \beta \right)}^{4}$ | $p4$/$p4mm$ | $1\times 1$ |

fsz | ${\mathcal{B}}_{3}$ | $\alpha $, $\beta $, $\gamma $ | ${\alpha}^{6}$, ${\beta}^{3}$, ${\gamma}^{2}$, $\gamma \beta \alpha $ | $p6$/$p6$ | $1\times 1$ |

fxt | ${\mathcal{B}}_{3}$ | ${\sigma}_{1}$, ${\sigma}_{2}$, ${\sigma}_{3}$ | ${\sigma}_{1}^{2}$, ${\sigma}_{2}^{2}$, ${\sigma}_{3}^{2}$, ${\left({\sigma}_{1}{\sigma}_{2}\right)}^{6}$, ${\left({\sigma}_{1}{\sigma}_{3}\right)}^{3}$, ${\left({\sigma}_{2}{\sigma}_{3}\right)}^{2}$ | $p6mm$/$p6mm$ | $1\times 1$ |

hca | ${\mathcal{B}}_{2}$ | $\alpha $, $\beta $ | ${\alpha}^{3}$, ${\beta}^{2}$, ${\left(\alpha \beta \right)}^{6}$ | $p6$/$p6mm$ | $1\times 1$ |

hcb | ${\mathcal{B}}_{2}$ | $\alpha $, $\beta $ | ${\alpha}^{6}$, ${\beta}^{2}$, ${\left(\alpha \beta \right)}^{3}$ | $p6$/$p6mm$ | $3\times 1$ |

htb | ${\mathcal{B}}_{2}$ | $\alpha $, $\beta $ | ${\alpha}^{6}$, ${\beta}^{3}$, ${\left(\alpha \beta \right)}^{2}$ | $p6$/$p6mm$ | $1\times 1$ |

hxl | ${\mathcal{B}}_{3}$ | $\alpha $, $\beta $, $\gamma $ | ${\alpha}^{3}$, ${\beta}^{3}$, ${\gamma}^{3}$, $\alpha \beta \gamma $ | $p3$/$p6mm$ | $3\times 1$ |

kgm | ${\mathcal{B}}_{2}$ | $\alpha $, $\beta $ | ${\alpha}^{3}$, ${\beta}^{3}$, ${\left(\alpha \beta \right)}^{3}$ | $p3$/$p6mm$ | $1\times 1$ |

sql | ${\mathcal{B}}_{2}$ | $\alpha $, $\beta $ | ${\alpha}^{4}$, ${\beta}^{4}$, ${\left(\alpha \beta \right)}^{2}$ | $p4$/$p4mm$ | 2 × 2 |

tts | ${\mathcal{B}}_{3}$ | $\alpha $, $\beta $, $\gamma $ | ${\alpha}^{4}$, ${\beta}^{4}$, ${\gamma}^{2}$, $\alpha \beta \gamma $ | $p4$/$p4gm$ | $1\times 1$ |

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Eon, J.-G. Symmetry and Topology: The 11 Uninodal Planar Nets Revisited. *Symmetry* **2018**, *10*, 35.
https://doi.org/10.3390/sym10020035

**AMA Style**

Eon J-G. Symmetry and Topology: The 11 Uninodal Planar Nets Revisited. *Symmetry*. 2018; 10(2):35.
https://doi.org/10.3390/sym10020035

**Chicago/Turabian Style**

Eon, Jean-Guillaume. 2018. "Symmetry and Topology: The 11 Uninodal Planar Nets Revisited" *Symmetry* 10, no. 2: 35.
https://doi.org/10.3390/sym10020035