#
Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D_{2}

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

**pmm**,

**pmg**,

**pgg**or

**cmm**symmetry [1]. These symmetry groups are members of the crystal class ${D}_{2}$ among the 17 two-dimensional symmetry groups [2]. We display the algorithms’ output and give enumeration tables for small values of n. This work is a continuation of our earlier works for the symmetry groups

**p3**,

**p31m**,

**p3m1**,

**p4**,

**p4g**,

**p4m**,

**p6**, and

**p6m**[3,4,5].

## 1. Introduction

**pmm**,

**pmg**,

**pgg**, and

**cmm**, which are all in the crystal class ${D}_{2}$ among the 17 two-dimensional symmetry groups [1,2]. We describe algorithms that construct the polyomino and polyiamond tiles and produce the isohedral tilings for which the tiles are fundamental domains.

**Theorem**

**1.**

**Remark.**

**Observation**

**1**

## 2. pmm

**Theorem**

**2.**

**pmm**isohedral tilings having polyominoes as fundamental domains. However, if the symmetry of tiles is ignored or is destroyed by asymmetric markings, then there are

**pmm**isohedral tilings having rectangular $kl$-ominoes as fundamental domains, where k and l are positive integers. There are no

**pmm**isohedral tilings by polyiamonds.

## 3. Displacement Points, Fault Lines and the Slide Operation

#### 3.1. Definitions and Observations

**Theorem**

**3.**

**Corollary**

**1.**

#### 3.2. Types of Tilings and the Slide Operation for **pmg** and **pgg** Groups

**pmg**or

**pgg**has as its rotation symmetry elements a rectangular lattice of 2-fold centers (these are centers for ${180}^{\circ}$ rotations in G); an example of such a lattice is shown in Figure 4. In these groups, adjacent 2-fold centers in a horizontal row are not equivalent.

**pmg**or

**pgg**group G. In our construction of these tiles T in the following two sections, the boundary of T will contain two adjacent inequivalent 2-fold centers for $\mathcal{T}$. Rotations of ${180}^{\circ}$ about these two centers and a reflection (in the case of

**pmg**) or a glide reflection (in the case of

**pgg**) generates G, and acting on T, generate $\mathcal{T}$. We classify these tilings $\mathcal{T}$ into two distinct types.

**Theorem**

**4.**

**Theorem**

**5.**

- (a)
- If $\mathcal{T}$ has displacement points, then ${\mathcal{T}}^{\prime}$ has no displacement points if $\epsilon ={\epsilon}_{0}+ks$, k any integer, and ${\mathcal{T}}^{\prime}$ has displacement points for all $\epsilon \ne {\epsilon}_{0}+ks$;
- (b)
- If $\mathcal{T}$ has no displacement points, then ${\mathcal{T}}^{\prime}$ has no displacement points if $\epsilon =ks$, k any integer, and ${\mathcal{T}}^{\prime}$ has displacement points for all $\epsilon \ne ks$;
- (c)
- There is an uncountably infinite family of tilings in general position that are obtained from $\mathcal{T}$ by ε-slide operations on $\mathcal{T}$.

## 4. pmg

#### 4.1. **pmg** Tilings and Slide Lines

**pmg**group G consists of equispaced parallel reflection axes (horizontal in Figure 8) and midway between them, rows of equispaced 2-fold centers. Glide-reflection axes are perpendicular to the reflection axes, and contain columns of 2-fold centers.

**pmg**group G, two inequivalent 2-fold centers for G must be located on the boundary of such T, and in the tiling $\mathcal{T}$ generated by G, edges of T must lie on reflection axes for $\mathcal{T}$ (observation 1 in Section 1). We will distinguish between tilings of type 1 and type 2 as defined in Section 3. Figure 9 and Figure 10 show isohedral tilings of types 1 and 2, respectively, by polyominoes and polyiamonds in which the tiles are fundamental domains for the group G.

**pmg**tiling are compositions of its 2-fold rotation and reflection symmetries, so we have the following theorem.

**Theorem**

**6.**

**pmg**group G acting on a polyomino or polyiamond fundamental domain, an ε-slide operation on $\mathcal{T}$ will produce a new isohedral tiling ${\mathcal{T}}^{\prime}$ whose symmetry group ${G}^{\prime}$ contains a copy of G, that is, the lattice of symmetries of ${G}^{\prime}$ contains a lattice of symmetries congruent to the lattice of symmetries of G.

**pmg**tilings we consider.

**Theorem**

**7.**

**pmg**group G. If $\mathcal{T}$ has a fault line l, then l must be a slide line parallel to a reflection axis, and every slide line in $\mathcal{T}$ is a fault line. The fault lines in $\mathcal{T}$ are equispaced, with the distance between two adjacent fault lines equal to the distance between two adjacent reflection axes.

**Corollary**

**2.**

**pmg**group G. If $\mathcal{T}$ has a fault line, then $\mathcal{T}$ must be a type 2 tiling.

**Theorem**

**8.**

**pmg**group G and let ${G}^{\prime}$ be the full symmetry group of $\mathcal{T}$. Let c and ${c}^{\prime}$ be the 2-fold centers for G on the boundary of T and denote by ${t}_{c{c}^{\prime}}$ the translation that is the composition of ${180}^{\circ}$ rotations about c and ${c}^{\prime}$.

- (1)
- If $\mathcal{T}$ is a type 1 tiling, let ${T}^{\prime}$ be the image of T by a ${180}^{\circ}$ rotation about c. If M and ${M}^{\prime}$ are reflection axes for G closest to c, then ${t}_{c{c}^{\prime}}$ translates T and ${T}^{\prime}$ to tile the region R bounded by M and ${M}^{\prime}$. If b is a 2-fold center for ${G}^{\prime}$ and b is not a 2-fold center for G, then T has ${C}_{2}$ or ${D}_{2}$ symmetry or b lies on a reflection axis for G.
- (2)
- If $\mathcal{T}$ is a type 2 tiling, let M be a reflection axis for G closest to the slide line L through c and ${c}^{\prime}$. Then ${t}_{c{c}^{\prime}}$ translates T to tile the region R bounded by M and L. If b is a 2-fold center for ${G}^{\prime}$ and b is not a 2-fold center for G, then b cannot lie on a slide line for the group G.

**Corollary**

**3.**

#### 4.2. Creating Polyominoes as Fundamental Domains of **pmg** Symmetry Groups

**pmg**group G. At first we create tiles whose isohedral tilings generated by the group G will not have any displacement points. These tilings will have a single underlying lattice of unit squares, and symmetries in G will keep this lattice invariant.

**pmg**group G can be generated by a reflection and two inequivalent 2-fold rotations whose centers lie on a line parallel to the reflection axis, and are nearest the reflection axis. By observation 1 in Section 1 and the symmetries of the lattice of unit squares, the 2-fold centers must be located at vertices or midpoints of edges of unit squares, and reflection axes must be located on edges of unit squares.

**pmg**symmetry group generated by ${180}^{\circ}$ rotations about the black and white 2-fold centers and reflections in one of the two reflections axes we have placed. Figure 11 shows the three essentially different locations for the origin relative to the lattice of unit squares and gives restrictions on x and y for each case.

- For $0<k\le n$, ${T}_{k}$ is a set of unit squares that form a k-omino, $B({T}_{k})$ the set of unit squares that are edge-adjacent to those in ${T}_{k}$, and ${\mathcal{T}}_{n}$ a set of inequivalent n-omino tiles. In the set ${T}_{k}$, all squares are taken from the region between the two reflection axes placed at the outset of the process (see Figure 11). (In [5], ${T}_{k}$ is denoted by T.)
- For the empty set ∅ we define $B(\varnothing )$ to be the set of unit squares that contain the origin.
- We define $E({T}_{k})$, the Boolean function of ${T}_{k}$, which is true if $k=n$ and the white 2-fold center is on the boundary of ${T}_{k}$. Otherwise $E({T}_{k})$ is false.

**pmg**group G. For each tile T in Figure 12 and $n\le 5$, Figure 13 shows the corresponding tiling by T generated in this manner, or in the case when T is a type 2 tile, a representative of an infinite family of tilings. When T is a type 2 tile, the black 2-fold center can be at any point between a vertex and midpoint of the edge on which it lies; Figure 12 shows only a representative tile T with 2-fold centers in the original positions placed at the beginning of the process (as in Figure 11). More details on these tilings and their symmetries are in the next section.

#### 4.3. Symmetries of the Polyomino Tilings

**pmg**group G that generates an isohedral tiling $\mathcal{T}$. However, it is possible that $\mathcal{T}$ has additional symmetries, that is, G is a proper subgroup of the full symmetry group ${G}^{\prime}$ of $\mathcal{T}$ and T is not a fundamental domain for ${G}^{\prime}$ (Theorem 1). The

**pmg**group G can be a proper subgroup of a symmetry group of any of these types:

**pmm**,

**cmm**,

**p4m**,

**p4g**, or

**pmg**.

#### 4.3.1. $\mathcal{T}$ Is a Type 1 Tiling

**pmg**symmetry group larger than G (e.g., tiling 4-3-2 in Figure 13). When T has only ${D}_{1}$ symmetry and the 2-fold centers on T are symmetric with respect to the mirror axis for T, $\mathcal{T}$ will have

**cmm**symmetry (e.g., tiling 4-5 in Figure 13). When T has ${D}_{2}$ symmetry, T is a brick. All type 1 tilings by bricks have

**pmm**symmetry (e.g., tilings 2-1-3, 2-1-4 in Figure 13) unless the brick is square, and in that case, the tiling has

**p4m**symmetry (e.g., tilings 1-1-2, 4-2-2 in Figure 13).

#### 4.3.2. $\mathcal{T}$ Is a Type 2 Tiling

**Theorem**

**9.**

**pmg**symmetry group G. If T is not a brick, then every new tiling ${\mathcal{T}}_{1}$ obtained from $\mathcal{T}$ by an ε-slide operation is also a

**pmg**isohedral tiling whose symmetry group ${G}_{1}$ is a copy of G. If T is a brick, then the special tilings in Figure 10b,c are the only tilings ${\mathcal{T}}_{1}$ obtained from $\mathcal{T}$ by an ε-slide operation that have symmetry group ${G}_{1}$ not a copy of G.

**pmg**symmetry group G. Let ${\mathcal{T}}^{\prime}$ be obtained from $\mathcal{T}$ by an $\epsilon $-slide operation and let ${G}^{\prime}$ be the full symmetry group of ${\mathcal{T}}^{\prime}$. Then by Theorem 6, ${G}^{\prime}$ contains a subgroup ${G}_{1}$ that is a copy of G, so ${G}_{1}$ is a

**pmg**symmetry group with fundamental domain T. Suppose c is a 2-fold center for ${G}^{\prime}$ and c is not a 2-fold center for ${G}_{1}$. Then c must lie in a region R bounded by a reflection axis M for ${G}_{1}$ and a slide line L for ${G}_{1}$ nearest M. By Theorem 8, c cannot lie on L, so c must lie on M or lie in the interior of R. Also by Theorem 8 and its corollary, opposite edges of T lie on L and M and the other two edges of T must be parallel.

**pmg**symmetry group ${G}_{1}$ which is identical to G, only differing in the placement of the 2-fold centers on the slide lines (and on the edges of tiles on those slide lines). In addition, T will be a fundamental domain for each of these tilings. We underline the label of each of these tiles in Figure 12 and the label for one corresponding representative tiling in general position in Figure 13.

#### 4.4. Creating Polyiamonds as Fundamental Domains for **pmg** Symmetry Groups

**pmg**group G; the procedure is similar to that in Section 4.2. At first we create tiles whose isohedral tilings generated by the group G will not have any displacement points. These tilings will have a single underlying lattice of unit triangles, and symmetries in G will keep this lattice invariant.

**pmg**group G can be generated by a reflection and two inequivalent 2-fold rotations whose centers lie on a line parallel to the reflection axis, and are nearest the reflection axis. By observation 1 in Section 1 and the symmetries of the lattice of unit triangles, the 2-fold centers must be located at vertices or midpoints of edges of unit triangles, and reflection axes must be located on edges of triangles.

**pmg**symmetry group generated by ${180}^{\circ}$ rotations about the black and white 2-fold centers and reflections in one of the two reflections axes we have placed. Figure 15 shows the three essentially different locations for the origin relative to the lattice of unit triangles, and gives restrictions on x and y for each case.

**pmg**group G.

#### 4.5. Symmetries of the Polyiamond Tilings

**pmg**group G that generates an isohedral tiling $\mathcal{T}$. As noted in Section 4.3, the

**pmg**group G can be a proper subgroup of the full symmetry group ${G}^{\prime}$ of $\mathcal{T}$. For polyiamond tiles T, ${G}^{\prime}$ might be a

**cmm**,

**p6m**, or

**pmg**symmetry group. Also, as pointed out in Section 4.3, in order to determine whether $\mathcal{T}$ has symmetries that are not in G, we only need to look for 2-fold centers that are not symmetry elements in G. We consider two cases separately.

#### 4.5.1. $\mathcal{T}$ Is a Type 1 Tiling

**pmg**symmetry group ${G}^{\prime}$ larger than G (e.g., tilings 4-1-3 and 6-3-2 in Figure 17); in particular, every type 1 tiling $\mathcal{T}$ by a parallelogram tile T will have symmetry group ${G}^{\prime}$ of type

**pmg**. When T has mirror symmetry and the 2-fold centers on T are symmetric to a mirror axis for T, $\mathcal{T}$ will have

**p6m**symmetry (e.g., tiling 4-3 in Figure 17) or

**cmm**symmetry (see Figure 18; tiles 8–23 and 8–25 in Figure 16 will also produce

**cmm**tilings).

#### 4.5.2. $\mathcal{T}$ Is a Type 2 Tiling

**pmg**symmetry group with fundamental domain T. If ${G}_{1}$ is not the full symmetry group of $\mathcal{T}$, then as with type 1 tilings, we look for 2-fold rotation symmetries of $\mathcal{T}$ that are not in G. The argument in Theorem 9 is valid for a type 2 tiling with polyiamond fundamental domain T, and polyiamonds cannot be bricks.

**Theorem**

**10.**

**pmg**symmetry group G. Then every new tiling ${\mathcal{T}}_{1}$ obtained from $\mathcal{T}$ by an ε-slide operation is also a

**pmg**isohedral tiling whose symmetry group ${G}_{1}$ is a copy of G.

## 5. pgg

#### 5.1. **pgg** Tilings, Fault Lines, and Slide Lines

**pgg**group G consists of two sets of equispaced parallel glide-reflection axes, one set perpendicular to the other (vertical and horizontal in Figure 19) and midway between them, rows of equispaced 2-fold centers. The generating glide vectors are defined by the (horizontal and vertical) distances between two adjacent 2-fold centers; these are also equal to the distances between two adjacent parallel glide-reflection axes.

**pgg**isohedral tiling $\mathcal{T}$ having a polyomino or polyiamond tile T as fundamental domain, fault lines can occur in two distinct ways. The first is along slide lines. For type 2 tilings, $\epsilon $-slides along slide lines can produce an uncountably infinite family of tilings having slide lines as fault lines (Theorem 5). Also, by Theorem 4, an $\epsilon $-slide operation merely shifts the 2-fold centers a distance of $\epsilon /2$ in the direction of the slide, and the lattice of 2-fold centers of the new tiling ${\mathcal{T}}^{\prime}$ contains this shifted copy of the lattice of 2-fold centers of $\mathcal{T}$. Glide-reflection axes parallel to the slide lines are midway between slide lines and so are preserved by the slide, and so are their glide vectors, since these are determined by the lattice of 2-fold centers. A glide reflection g whose axis is perpendicular to slide lines is the composition of a ${180}^{\circ}$ rotation about a 2-fold center and a glide reflection whose glide-reflection axis is parallel to a slide line, so the glide vector for g is preserved, but its glide reflection axis is shifted by $\epsilon /2$. We state these observations in the following theorem.

**Theorem**

**11.**

**pgg**group G. Then an ε-slide operation on $\mathcal{T}$ will produce a new isohedral tiling ${\mathcal{T}}^{\prime}$ whose symmetry group ${G}^{\prime}$ contains a copy of G, that is, the lattice of symmetries of ${G}^{\prime}$ contains a lattice of symmetries congruent to the lattice of symmetries of G.

**pgg**symmetries. Thus midpoints of edges of unit squares (or triangles) are the only positions possible for displacement points along ${L}_{g}$ fault lines. Theorem 12 states that slide lines and ${L}_{g}$ glide-reflection axes are the only possibilities for fault lines in $\mathcal{T}$.

**Theorem**

**12.**

**pgg**group G. If $\mathcal{T}$ has a fault line l, then l is (1) a glide-reflection axis or (2) a slide line parallel to a glide-reflection axis. In case (1), every glide reflection axis in $\mathcal{T}$ parallel to l is a fault line, and in case (2) every slide line in $\mathcal{T}$ is a fault line. The fault lines in $\mathcal{T}$ are equispaced.

**pgg**isohedral tiling $\mathcal{T}$ having a polyomino or polyiamond tile T as fundamental domain.

**pgg**tilings in which the edges of tiles make up lines that partition the tiling into horizontal strips, and in each of these cases, the boundaries of the strips are fault lines. Slide lines for type 2 tilings will always partition the tiling into parallel strips. However, in

**pgg**tilings, glide-reflection axes can be boundaries of parallel strips that partition the tiling even when these axes are not fault lines. In many of the

**pgg**tilings we consider, the tiling is not composed of strips of tiles bounded by lines that are unions of edges of tiles. This feature is most prevalent in tilings in which the glide-reflection axes are at a ${45}^{\circ}$ angle to the horizontal. Most of these tilings have a pronounced diagonal pattern.

#### 5.2. Creating Polyominoes as Fundamental Domains for **pgg** Symmetry Groups

**pgg**group G. At first we create tiles whose isohedral tilings generated by the group G will not have any displacement points on slide lines. Displacement points for these tilings can only occur on glide-reflection axes, at midpoints of edges of unit squares.

**pgg**group G can be generated by a glide reflection and two adjacent inequivalent 2-fold rotations whose centers lie on a line parallel to the glide-reflection axis and are nearest to that axis. By observation 1 in Section 1 and the symmetries of the lattice of unit squares, the 2-fold centers must be located at vertices or midpoints of edges of unit squares. Glide-reflection axes must be located on (1) lines in the lattice of unit squares, or (2) lines through midpoints of two edges of a unit square, or (3) lines through opposite vertices of unit squares.

**pgg**symmetry group generated by ${180}^{\circ}$ rotations about the black and white 2-fold centers and glide reflections in one of the two glide-reflection axes we have placed. The area S of a fundamental domain for G is given by

**pgg**group G.

#### 5.3. Symmetries of the Polyomino Tilings

**pgg**group G that generates an isohedral tiling $\mathcal{T}$. However, it is possible that $\mathcal{T}$ has additional symmetries, that is, G is a proper subgroup of the full symmetry group ${G}^{\prime}$ of $\mathcal{T}$ and T is not a fundamental domain for ${G}^{\prime}$ (Theorem 1). The

**pgg**group G can be a proper subgroup of a symmetry group of any of these types:

**pgg**,

**pmm**,

**cmm**,

**p4m**,

**p4g**, or

**pmg**. As argued in Section 4.3, to determine if $\mathcal{T}$ has symmetries that are not in G, we only need to look for 2-fold centers for ${G}^{\prime}$ that are not 2-fold centers for G. In order to seek 2-fold centers not in G, we look at all vertices, midpoints of edges and centers of unit squares in a polyomino T in the tiling $\mathcal{T}$ except for those equivalent to the original 2-fold centers we have chosen (black and white centers in Figure 21), and investigate whether or not they can be new 2-fold centers of $\mathcal{T}$. If a new 2-fold center is found, we indicate that G is not the full symmetry group of $\mathcal{T}$ by putting parentheses around the labels of the tile T in Figure 22 and its corresponding tiling $\mathcal{T}$ in Figure 23. In the case of a type 2 tiling, although there may be a finite number of $\epsilon $-slides of the tiling that produce a tiling with larger symmetry group than G, we put parentheses only around the labels of those type 2 tilings for which every tiling in the infinite family has a symmetry group larger than G.

**pmg**,

**pmm**, and

**cmm**tilings in Figure 7 can be generated by G, as well as

**p4m**tilings. Brick tilings unique to this

**pgg**group G are those of “herringbone” type in which the 2-fold centers on a brick tile are on adjacent edges (e.g., tiles and tilings 2-1-1, 3-1-1 in Figure 22 and Figure 23); a square brick of this type generates a

**p4m**tiling (e.g., tiles and tilings 1-1-2 and 4-4-1 in Figure 22 and Figure 23). Other brick tilings have the inequivalent 2-fold centers on a brick tile on opposite edges, where a line through them makes a ${45}^{\circ}$ angle with those edges (e.g., tiling 2-1-2 which has

**p4g**symmetry, and 4-1-2 in Figure 23).

#### 5.3.1. $\mathcal{T}$ Is a Type 1 Tiling

**pmg**(e.g., tiling 4-3-6 in Figure 23). Figure 24a shows the situation for an arbitrary ${C}_{2}$ tile T: the 2-fold center for T is a 2-fold center for ${G}^{\prime}$ and produces new glide-reflection axes in ${G}^{\prime}$; horizontal glide-reflection axes in G are reflection axes in ${G}^{\prime}$.

**pmg**(e.g., tilings 3-2-2 and 4-5-3 in Figure 23). Figure 24b illustrates the situation for a 6-iamond chevron: the mirror symmetry of T produces new 2-fold centers on the horizontal glide reflection axes of $\mathcal{T}$ and new glide-reflection axes through the original 2-fold centers in the full symmetry group ${G}^{\prime}$, and vertical glide-reflection axes in G are reflection axes in ${G}^{\prime}$.

**cmm**. Figure 24c illustrates the situation for a 6-iamond bow tie: the additional symmetries of T produce new 2-fold centers and new glide-reflection axes through the original 2-fold centers, and reflection axes for T are reflection axes in ${G}^{\prime}$.

#### 5.3.2. $\mathcal{T}$ Is a Type 2 Tiling

**pmg**group, there can be 2-fold centers on slide lines that are not in the group G. This happens only when the original 2-fold centers for a tile T (placed at the outset) are at slide points of the tiling $\mathcal{T}$, and the new 2-fold center is midway between these. Moreover, the existence of the new 2-fold centers requires that T have a mirror line on the glide-reflection axis for $\mathcal{T}$ parallel to the slide line and that the edges of T not on the slide line are parallel. (The composition of the new 2-fold rotation with a glide reflection in an axis perpendicular to the slide line produces a reflection in the glide-reflection axis parallel to the slide line; the composition of an original 2-fold rotation with the new 2-fold rotation produces a translation that must carry T to an adjacent tile.) For such tilings $\mathcal{T}$ in which T is not a brick, $\mathcal{T}$ will have

**pmg**symmetry.

**pmg**,

**cmm**, or in the case T is a polyiamond,

**p6m**symmetry (e.g., tiling 4-5-2 in Figure 23; also see Figure 25). Except for these special positions, the tilings will have

**pgg**symmetry group G.

#### 5.4. Creating Polyiamonds as Fundamental Domains for **pgg** Symmetry Groups

**pgg**group G, we follow a pattern similar to that in Section 5.2 and create tiles whose isohedral tilings generated by the group G will have either no displacement points or displacement points only on glide-reflection axes ${L}_{g}$. As before, the 2-fold centers must be located at vertices or midpoints of edges of unit triangles. Glide-reflection axes must be located on (1) lines in the lattice of unit triangles, or (2) lines through midpoints of two edges of a unit triangle, or (3) lines that join a vertex and midpoint of an opposite edge of a unit triangle, or (4) lines through adjacent midpoints of two joined triangles. Since glide reflections in axes of types (1) or (2) combine with 2-fold rotations to produce glide reflections in axes of types (3) or (4), we only need to consider the first two types.

**pgg**symmetry group generated by ${180}^{\circ}$ rotations about the black and white 2-fold centers and glide reflections in one of the two glide-reflection axes we have placed. The area S of a fundamental domain for G is given by

**pgg**group G. Figure 28 shows, for each tile T in Figure 27 and $n\le 5$, the corresponding tiling by T generated in this manner, or in the case when T is a type 2 tile, a representative of an infinite family of tilings. When T is a type 2 tile, the black 2-fold center can be at any point between a vertex and midpoint of the edge on which it lies; Figure 27 shows only a representative tile T with 2-fold centers in the original positions placed at the outset. More details on these tilings and their symmetries are in the next section.

#### 5.5. Symmetries of the Polyiamond Tilings

**pgg**group G that generates an isohedral tiling $\mathcal{T}$. However, it is possible that G is a proper subgroup of the full symmetry group ${G}^{\prime}$ of $\mathcal{T}$ and T is not a fundamental domain for ${G}^{\prime}$ (Theorem 1). The

**pgg**group G can be a proper subgroup of a symmetry group of any of these types:

**pgg**,

**cmm**,

**pmg**, or

**p6m**. In Section 5.3, we described additional symmetries possible for the full symmetry group ${G}^{\prime}$ for both polyomino and polyiamond tiles having symmetry. Tilings 2-1-2, 4-1-4, and 4-1-7 in Figure 28 have tiles with ${C}_{2}$ symmetry and have

**pmg**symmetry group. In particular, all tilings by polyiamond parallelogram tiles that fill out parallel strips bounded by adjacent glide-reflection axes will have

**pmg**symmetry group. Tilings 3-1-5, 4-2-1, and 4-3-5 in Figure 28 have tiles with ${D}_{1}$ symmetry and have

**pmg**symmetry group. Tiling 2-1-1 in Figure 28 has a tile with ${D}_{2}$ symmetry and has

**cmm**symmetry group.

## 6. cmm

#### 6.1. **cmm** Tilings

**cmm**group G consists of two sets of equispaced parallel reflection axes, one set perpendicular to the other (vertical and horizontal in Figure 29) and midway between them, glide-reflection axes. Perpendicular reflection axes are inequivalent, and perpendicular glide-reflection axes are inequivalent. Every 2-fold center is at the intersection of two reflection axes or the intersection of two glide-reflection axes. The generating glide vectors are defined by the (horizontal and vertical) distances between two adjacent parallel reflection axes or parallel glide-reflection axes.

**cmm**group G, two inequivalent 2-fold centers for G must be located on the boundary of such T, and in the tiling, edges of T must lie on reflection axes (observation 1 in Section 1). Thus the grid of perpendicular reflection axes in the

**cmm**lattice in Figure 29 must lie on lattice lines in the underlying lattice of unit squares (for polyomino tilings) or unit triangles (for polyiamond tilings). This is impossible for the triangle lattice.

**Theorem**

**13.**

**cmm**isohedral tiling having a polyiamond as fundamental domain.

**cmm**group G, a ${180}^{\circ}$ rotation about the 2-fold center on T that is not on a reflection axis will fill out a rectangle bounded by four reflection axes that lie on lattice lines in the underlying lattice of unit squares. This rotation preserves the underlying lattice of unit squares within this rectangle, and reflections in the axes that bound it will also preserve the full underlying lattice of unit squares.

**Theorem**

**14.**

**cmm**isohedral tiling having a polyomino as fundamental domain.

#### 6.2. Creating Polyominoes as Fundamental Domains for **cmm** Symmetry Groups

**cmm**group G. G can be generated by a 2-fold rotation about a center not on a reflection axis and reflections in two perpendicular axes nearest that 2-fold center. As before, the 2-fold centers for G must be located at vertices or midpoints of edges of unit squares, and the reflection axes must be along edges of unit squares.

**cmm**symmetry group generated by ${180}^{\circ}$ rotations about the black 2-fold center and reflections in two perpendicular reflection axes we have placed. The area S of a fundamental domain for G is given by

**cmm**group G. Figure 32 shows the set of tilings generated in this manner, corresponding to the tiles in Figure 31 for $n\le 8$. More details on these tilings and their symmetries are in the next section.

#### 6.3. Symmetries of the Polyomino Tilings

**cmm**group G that generates an isohedral tiling $\mathcal{T}$. When G is a proper subgroup of the full symmetry group ${G}^{\prime}$ of $\mathcal{T}$, the tile T must have symmetry, and this symmetry is an element of ${G}^{\prime}$ (Theorem 1). Since our tiles are polyominoes, the

**cmm**group G can be a proper subgroup of a group of any of these types:

**pmm**,

**cmm**,

**p4g**,

**p4m**.

**cmm**group G produces a tile T bounded by reflection axes on three sides, with its fourth side a rectilinear edge having 2-fold symmetry (see Figure 33).

**cmm**lattice are those in which the fourth side is a straight edge, that is, T is a brick. When T is a brick, a ${180}^{\circ}$ rotation about the 2-fold center on its edge places a copy of T edge-to-edge with T, and reflections in the bounding reflection axes produce an isohedral edge-to-edge tiling.

**Theorem**

**15.**

**cmm**group G. If T is not a brick, then G is the full symmetry group of $\mathcal{T}$. If T is a brick, then T has

**pmm**symmetry, unless T is a square, in which case $\mathcal{T}$ has

**p4m**symmetry.

## 7. Enumeration Tables

**pmg**,

**pgg**and

**cmm**having n-ominoes or n-iamonds as fundamental domains for small values of n.

**pmg**,

**pgg**, or

**cmm**symmetry group G that generates an isohedral tiling having ${T}_{n}$ as a fundamental domain. Recall that we defined tiles ${T}_{n}$ to be equivalent only if they generate the same tiling by the action of the group G when each tile is marked with an asymmetric motif. So ${N}_{n}$ is also the number of isohedral tilings having G as full symmetry group and having n-omino or n-iamond tiles as fundamental domain, when each tile is marked with an asymmetric motif. Here the infinite family of type 2 tilings for a single ${T}_{n}$ is counted as one tiling.

**pgg**symmetry group having 4-iamonds as fundamental domains, there are 14 tiles in Figure 27 whose label begins with 4, so ${N}_{4}=14$. But 6 of the corresponding tilings in Figure 28 have additional symmetry (namely 4-1-4, 4-1-7, 4-2-1, 4-3-1, 4-3-4, and 4-3-5) indicated by parentheses around their labels, so ${S}_{4}=14-6=8$. Also in Figure 27 we can see that among the 4-iamonds tiles, there are only 3 distinct shapes, indicated by the tile numbers: a parallelogram (tiles 4-1), an equilateral triangle (tile 4-2), and a chevron (tiles 4-3). Thus ${N}_{4}^{\prime}=3$. Similarly, among the eight 4-iamonds counted by ${S}_{4}$ (tiles 4-1-1, 4-1-2, 4-1-3, 4-1-5, 4-1-6, 4-2-2, 4-3-2, and 4-3-3), there are also these three distinct shapes, so ${S}_{4}^{\prime}=3$.

## 8. Summary

**pmm**,

**pmg**,

**pgg**and

**cmm**symmetry. These symmetry groups belong to crystal class ${D}_{2}$ among the 17 two-dimensional symmetry groups.

**pmm**that have polyominoes or polyiamonds as fundamental domains. For symmetry groups of types

**pmg**,

**pgg**and

**cmm**we used the backtracking procedure described in Section 4.2 to obtain the set ${\mathcal{T}}_{n}$ of n-omino or n-iamond tiles where each tile produced one isohedral tiling, generated by a given symmetry group of one of these three types. If we let G denote the symmetry group we can denote ${\mathcal{T}}_{n}(G)$ as the set ${\mathcal{T}}_{n}$ for that symmetry group, and ${\mathcal{T}}_{n}^{*}(G)$ the corresponding set of isohedral tilings. Note that if some tiling in ${\mathcal{T}}_{n}^{*}(G)$ is type 2, it belongs to an infinite family in ${\mathcal{T}}_{n}^{*}(G)$.

## Acknowledgments

## References and Notes

- Schattschneider, D. The plane symmetry groups: their recognition and notation. Am. Math. Mon.
**1978**, 85, 439–450. [Google Scholar] [CrossRef] - Coxeter, H.S.M.; Moser, W.O.J. Generators and Relations for Discrete Groups; Springer-Verlag: New York, NY, USA, 1965. [Google Scholar]
- Fukuda, H.; Mutoh, N.; Nakamura, G.; Schattschneider, D. A Method to Generate Polyominoes and Polyiamonds for Tilings with Rotational Symmetry. Graphs Comb.
**2007**, 23, 259–267. [Google Scholar] [CrossRef] - Fukuda, H.; Mutoh, N.; Nakamura, G.; Schattschneider, D. Enumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry. Lect. Notes Comput. Sci.
**2008**, 4535, 68–78. [Google Scholar] - Fukuda, H.; Mutoh, N.; Nakamura, G.; Schattschneider, D. Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry. Available online: http://arnetminer.org/viewpub.do?pid=2855920 (accessed on 2 June 2011).
- Golomb, S.W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed.; Princeton University Press: Princeton, NJ, USA, 1994. [Google Scholar]
- Grünbaum, B.; Shephard, G. Tilings and Patterns; W.H. Freeman: New York, NY, USA, 1987. [Google Scholar]
- Martin, G.E. Polyominoes: A guide to Puzzles and Problems in Tiling; Mathematical Association of America: Washington, DC, USA, 1991. [Google Scholar]
- Myers, J. Polyomino, polyhex and polyiamond tiling. Available online: www.srcf.ucam.org/jsm28/tiling/ (accessed on 2 June 2011).
- Rhoads, G.C. Tilings by Polyomoinoes, Polyhexes, and Polyiamonds. J. Comput. Appl. Math.
**2005**, 174, 329–353. [Google Scholar] [CrossRef] - In [10], G. Rhoads has used the term “fundamental domain” in a non-traditional way; what he calls a fundamental domain is more commonly called a unit cell, or translation unit, which is a smallest patch that can tile the plane using only translations. He has investigated the question of which polyominoes, polyhexes, and polyiamonds can serve as translation units for isohedral tilings. In many cases, these isohedral tilings will not have a polyomino or polyiamond fundamental domain in the traditional sense.

**Figure 1.**The symmetry elements of a

**pmm**symmetry group G. Black, white, gray, and dark gray circles mark the four inequivalent 2-fold rotation centers; black and gray horizontal lines and black and gray vertical lines are the four inequivalent mirror reflection axes. The shaded area is a fundamental domain for G.

**Figure 2.**The point p is a displacement point for an isohedral tiling by (

**a**) a 4-omino T and (

**b**) a 2-iamond T, shown shaded. Heavy lines are edges of the tiles; thin lines show the units that make up the tiles to which p belongs. In (a), note that the tile T has ${C}_{2}$ symmetry, but the rotation that leaves T invariant does not leave the whole tiling invariant (see Remark, page 326).

**Figure 3.**Enlargement of the region near the displacement point p for the tilings in Figure 2. Here the shaded polyomino (polyiamond) tiles T, ${T}_{1}$, ${T}_{2}$ are only partially shown, edges of unit squares or triangles that are not edges of tiles are thin lines, vertices of tiles are black dots, and the edge e of the unit square (triangle) that contains p in its interior is a thick black segment.

**Figure 5.**Slide regions for the tiling $\mathcal{T}$. Bounding lines for these regions are slide lines. Regions with the same label are congruent by a vertical translation.

**Figure 6.**(

**a**) A tiling by $2\times 6$ polyomino rectangles; (

**b**) A tiling by 20-iamond parallelograms. White circles mark 2-fold centers along a slide line for the tilings. The point q is a slide point, v and w are neighboring vertices of unit squares or unit triangles on the slide line. Here ${\epsilon}_{0}$ is the distance from q to v and is less than $s/2$, so these tilings are in general position.

**Figure 7.**Type 2 isohedral tilings by bricks that are m unit squares high and n unit squares wide. One brick (shaded) is a fundamental domain for the group G generated by 2-fold centers marked by white circles. The tilings are related by an $\epsilon $-slide operation. (

**a**) One tiling in an uncountably infinite family of tilings in general position; all are the same topological type and have

**pmg**symmetry group; (

**b**) 2-fold centers coincide with vertices and midpoints of edges of the bricks; ${\epsilon}_{0}=0$. The tiling has

**pmm**symmetry group unless $m=n$, when it is type

**p4m**; (

**c**) 2-fold centers and slide points are equispaced along the slide lines. When n is even, ${\epsilon}_{0}=0$; when n is odd ${\epsilon}_{0}=s/2$. The symmetry group is type

**cmm**.

**Figure 8.**The lattice of symmetries of a

**pmg**group G. Reflection axes are solid lines, glide-reflection axes are dotted, and the black and white circles mark inequivalent 2-fold centers. The shaded region is a fundamental domain for G. G is generated by a reflection in one axis and ${180}^{\circ}$ rotations about two adjacent black and white centers nearest that axis and on the same side of the axis.

**Figure 9.**Isohedral tilings of type 1 by (

**a**) a 6-omino T and (

**b**) a 6-iamond T; in each, T is a fundamental domain for the

**pmg**group G. Black and white circles mark inequivalent 2-fold centers for the tilings.

**Figure 10.**Type 2 isohedral tilings $\mathcal{T}$ by a brick polyomino tile T (shaded) that is k unit squares high and n unit squares wide; the tilings are related by $\epsilon $-slide operations. T is a fundamental domain for the

**pmg**group G whose lattice of 2-fold centers (white circles) and reflection axes (thin lines) are shown in (

**a**), where $\mathcal{T}$ is in general position; In (

**b**), new black 2-fold centers for $\mathcal{T}$ are at intersections of reflection axes of G with new vertical reflection axes (one is shown as a thin line). When n is even, ${\epsilon}_{0}=0$, and when n is odd, ${\epsilon}_{0}=s/2$. $\mathcal{T}$ has a

**cmm**symmetry group; In (

**c**), $\mathcal{T}$ is edge-to-edge; the ${D}_{2}$ symmetry of T induces additional new gray 2-fold centers and new horizontal reflection axes for $\mathcal{T}$ (one is shown as a thin line). Here ${\epsilon}_{0}=0$ and $\mathcal{T}$ has a

**pmm**symmetry group unless $k=n$, when it has a

**p4m**symmetry group.

**Figure 11.**The square lattice for a

**pmg**group G. The black and white circles are inequivalent 2-fold centers, and black lines are reflection axes; these symmetry elements generate G. (

**a**)–(

**c**) illustrate the three essentially different locations for the origin relative to the lattice of unit squares. The positive integers x and y that determine the placement of the white center and the reflection axes are restricted as follows. In (

**a**) and (

**b**), y must be even, and in (

**c**) x must be even and y must be odd. For these examples, (a) $x=4$, $y=4$; (b) $x=4$, $y=4$; (c) $x=4$, $y=3$.

**Figure 12.**List of n-ominoes for $n\le 6$ produced by the procedure in Section 4.2; these are fundamental domains for the

**pmg**groups G used to generate their corresponding isohedral tilings in Figure 13. Labels indicate n followed by the tile number for that n. Parentheses indicate that the corresponding tiling has symmetry group larger than G. Underline indicates that the corresponding tiling is type 2 and so has slide lines; there is an infinite family of tilings by that tile having the same slide lines.

**Figure 13.**List of isohedral tilings by the n-ominoes in Figure 12 for $n\le 5$; the n-ominoes are fundamental domains for the

**pmg**groups G that generate the tilings. Labels correspond to those in Figure 12. Tilings with underlined labels are in general position and represent an infinite family of tilings. Red lines indicate slide lines.

**Figure 14.**The 2-fold center c is on M, and E represents an edge of tile T. E reflects in M onto edge ${E}_{1}$ and E rotates ${180}^{\circ}$ about c onto edge ${E}_{2}$. In $\mathcal{T}$, ${E}_{1}$ and ${E}_{2}$ must be parallel or coincide. (

**a**) c at a vertex on M; (

**b**) c at the midpoint of an edge of T on M.

**Figure 15.**The triangular lattice for a

**pmg**group G. The black and white circles are inequivalent 2-fold centers, and black lines are reflection axes; these symmetry elements generate G. (

**a**)–(

**c**) illustrate the three essentially different locations of the origin relative to the lattice of unit triangles. The positive integers x and y that determine the placement of the white center and the reflection axes are restricted as follows. In (

**a**) and (

**b**), y must be even, and in (c) y must be odd. For these examples, (a) $x=4$, $y=4$; (b) $x=4$, $y=4$; (c) $x=3$, $y=3$.

**Figure 16.**List of n-iamonds for $n\le 8$ produced by the procedure in Section 4.4; these are fundamental domains for the

**pmg**groups G used to generate their corresponding isohedral tilings in Figure 17. Labels indicate n followed by the tile number for that n. Parentheses indicate that the corresponding tiling has symmetry group larger than G. Underlines indicate that the corresponding tiling has slide lines, so there is an infinite family of tilings by that tile having the same slide lines and same

**pmg**symmetry group.

**Figure 17.**List of isohedral tilings by the n-iamonds in Figure 16 for $n\le 6$; the n-iamonds are fundamental domains for the

**pmg**groups G that generate the tilings. Labels correspond to those in Figure 16. Tilings with underlined labels are in general position and represent an infinite family of tilings. Red lines indicate slide lines.

**Figure 18.**(

**a**) A type 1 isohedral tiling $\mathcal{T}$ by polyiamond tile 8-7-2 in Figure 16; this tile is a fundamental domain for the

**pmg**group G generated by the 2-fold centers marked by black and white circles and the horizontal black reflection axes; glide-reflection axes for G are vertical dotted lines; (

**b**) The full symmetry group ${G}^{\prime}$ of $\mathcal{T}$ is type

**cmm**, with additional vertical reflection axes (thin lines), 2-fold centers (gray circles), and horizontal glide-reflection axes.

**Figure 19.**The lattice of symmetries of a

**pgg**group G. Glide-reflection axes are dotted lines and their minimal glide vectors are shown as dotted vectors. Black and white circles mark inequivalent 2-fold centers. Each shaded region is a fundamental domain for G. G is generated by a ${180}^{\circ}$ rotation about a 2-fold center and a glide reflection whose glide-reflection axis is closest to that 2-fold center.

**Figure 20.**Three different possibilities for fault lines in a

**pgg**isohedral tiling $\mathcal{T}$ having a polyomino or polyimond tile T as fundamental domain. Dotted glide-reflection axes in relation to one tile are shown. (a) $\mathcal{T}$ is type 2, T is a 3-omino; slide lines are fault lines. (b) $\mathcal{T}$ is type 1, T is a 3-omino; glide-reflection axes are fault lines. (c) $\mathcal{T}$ is type 2, T is a 4-iamond; both slide lines and glide-reflection axes are fault lines.

**Figure 21.**The square lattice for a

**pgg**group G. The black and white circles are inequivalent 2-fold centers, and dotted black lines are glide-reflection axes; these symmetry elements generate G. (

**a**)–(

**d**) illustrate the four different locations of the 2-fold centers relative to the lattice of unit squares and placement of glide-reflection axes. Restrictions on positive integers x and y used to place the white centers and glide-reflection axes are as follows. In (

**a**), when x is odd, y must be even; In (

**b**), x must be even; In (

**c**), both x and y must be even; In (

**d**), x and y must have opposite parities. In the examples shown, (

**a**) x = 4, y = 4; (

**b**) x = 4, y = 2; (

**c**) x = 4, y = 4; (

**d**) x = 3, y = 4.

**Figure 22.**List of n-ominoes for $n\le 5$ produced by the procedure in Section 5.2; these are fundamental domains for the

**pgg**groups G used to generate their corresponding isohedral tilings in Figure 23. Labels indicate n followed by the tile number for that n. Parentheses indicate that the corresponding tiling has symmetry group larger than G. Underlines indicate that the corresponding tiling is type 2 and so has slide lines; there is an infinite family of tilings by that tile having the same slide lines.

**Figure 23.**List of isohedral tilings by the n-ominoes in Figure 22 for $n\le 4$; the n-ominoes are fundamental domains for the

**pgg**groups G that generate the tilings. Labels correspond to those in Figure 22. Tilings with underlined labels are in general position and represent an infinite family of tilings. Red lines indicate slide lines.

**Figure 24.**Three type 1 tilings $\mathcal{T}$ generated by a

**pgg**group G whose lattice of symmetry elements is shown. The full symmetry group ${G}^{\prime}$ of each tiling is larger than G. (

**a**) A general tile with ${C}_{2}$ symmetry; ${G}^{\prime}$ is type

**pmg**; (

**b**) A tile with ${D}_{1}$ symmetry; ${G}^{\prime}$ is type

**pmg**; (

**c**) A tile with ${D}_{2}$ symmetry; ${G}^{\prime}$ is type

**cmm**.

**Figure 25.**Four special positions of type 2 tilings generated by a

**pgg**group whose lattice of 2-fold centers is shown. (

**a**) and (

**b**) are tilings in the family 4-5-2 of Figure 23; (

**a**) is type

**pmg**; (

**b**) is type

**cmm**; (

**c**) and (

**d**) are tilings by a trapezoidal 3-iamond; (

**c**) is type

**pmg**; (

**d**) is type

**cmm**.

**Figure 26.**The triangular lattice for a

**pgg**group G. The black and white circles are inequivalent 2-fold centers, and dotted black lines are glide-reflection axes; these symmetry elements generate G. (

**a**)–(

**c**) illustrate the three different locations of the black 2-fold centers relative to the lattice of unit triangles and placement of glide-reflection axes. Restrictions on positive integers x and y that determine the placement of the white center and glide-reflection axes are as follows. In (

**a**) and (

**b**), x even and y odd is excluded, and in (

**c**), x and y both even is excluded. In these examples, (

**a**) $x=4$, $y=4$; (

**b**) $x=5$, $y=3$; (

**c**) $x=4$, $y=3$.

**Figure 27.**List of n-iamonds for $n\le 6$ produced by the procedure in Section 5.4; these are fundamental domains for the

**pgg**groups G used to generate their corresponding isohedral tilings in Figure 28. Labels indicate n followed by the tile number for that n. Parentheses indicate that the corresponding tiling has symmetry group larger than G. Underline indicates that the corresponding tiling is type 2 and so has slide lines; there is an infinite family of tilings by that tile having the same slide lines.

**Figure 28.**List of isohedral tilings by the n-iamonds in Figure 27 for $n\le 5$; the n-iamonds are fundamental domains for the

**pgg**groups G that generate the tilings. Labels correspond to those in Figure 27. Tilings with underlined labels are in general position and represent an infinite family of tilings. Red lines indicate slide lines.

**Figure 29.**The lattice of symmetries of a

**cmm**group G. Reflection axes are solid lines, glide-reflection axes are dotted lines and their minimal glide vectors are shown as dotted vectors. Black and white circles mark inequivalent 2-fold centers. The shaded region is a fundamental domain for G. G is generated by a ${180}^{\circ}$ rotation about a black 2-fold center and reflections in perpendicular reflection axes nearest that center.

**Figure 30.**The square lattice for a

**cmm**group G. The black circle is a 2-fold center, and black lines are reflection axes; these symmetry elements generate G. (

**a**) and (

**b**) illustrate the two different locations of the 2-fold centers relative to the lattice of unit squares. In (

**a**) and (

**b**) x is any positive integer, but in (

**a**) y must be an even positive integer, and in (

**b**) y must be an odd positive integer. In these two examples (

**a**) $x=2$, $y=2$; (

**b**) $x=2$ and $y=3$.

**Figure 31.**List of n-ominoes for $n\le 10$ produced by the procedure in Section 6.2; these are fundamental domains for the

**cmm**groups G used to generate their corresponding isohedral tilings in Figure 32. Labels indicate n followed by the tile number for that n. Parentheses indicate that the corresponding tiling has symmetry group larger than G.

**Figure 33.**Two polyomino fundamental domains (shaded) for a

**cmm**group G. The black circle is a 2-fold center, and the edges of the bounding rectangle are reflection axes. Although the shaded fundamental domain on the left has mirror symmetry, the mirror is not compatible with the

**cmm**reflection axes.

**Table 1.**The number of isohedral tilings of types

**pmg**,

**pgg**and

**cmm**having n-ominoes or n-iamonds as fundamental domains.

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Fukuda, H.; Kanomata, C.; Mutoh, N.; Nakamura, G.; Schattschneider, D.
Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D_{2}. *Symmetry* **2011**, *3*, 325-364.
https://doi.org/10.3390/sym3020325

**AMA Style**

Fukuda H, Kanomata C, Mutoh N, Nakamura G, Schattschneider D.
Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D_{2}. *Symmetry*. 2011; 3(2):325-364.
https://doi.org/10.3390/sym3020325

**Chicago/Turabian Style**

Fukuda, Hiroshi, Chiaki Kanomata, Nobuaki Mutoh, Gisaku Nakamura, and Doris Schattschneider.
2011. "Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D_{2}" *Symmetry* 3, no. 2: 325-364.
https://doi.org/10.3390/sym3020325