## 1. Introduction

Polyominoes and polyiamonds and their tiling properties have been the subject of computational geometry research that investigated which polyominoes can tile the plane isohedrally and which can tile by translations alone [

4,

5]. In earlier papers [

1,

2], we gave algorithms to create polyominoes and polyiamonds that were fundamental domains for isohedral tilings having

**p3**,

**p4**, or

**p6** symmetry groups. In this article, we consider the expanded task of producing polyomino and polyiamond tiles that generate isohedral tilings of types

**p3**,

**p3m1**,

**p31m**,

**p4**,

**p4m**,

**p4g**,

**p6** or

**p6m** and for which the tiles are fundamental domains of the tiling. Recently an extension of this study was published as a separate paper [

3] in which we carried out these investigations for symmetry groups of types

**pmm**,

**pmg**,

**pgg**, and

**cmm**.

The following definitions are noted in [

3], and are included here for completeness. A polyomino (or

n-omino) is a tile homeomorphic to a disk, made up of

n unit squares that are connected at their edges; that is, the intersection of two unit squares in the polyomino is either empty or an edge of both squares. Similarly a polyiamond (or

n-iamond) is a tile homeomorphic to a disk, made up of

n unit equilateral triangles that are connected at their edges; the intersection of two unit triangles in the polyiamond is either empty or an edge of both triangles.

An isohedral tiling of the plane is a tiling by congruent tiles in which the symmetry group of the tiling acts transitively on the tiles. A fundamental domain (sometimes called a generating region) for an isohedral tiling is a region of least area that generates the whole tiling when acted on by the symmetry group of the tiling. Thus a fundamental domain for an isohedral tiling must not contain two points that are identical under the action of the symmetry group of the tiling. This implies the following fact that is important for our algorithm.

**Lemma** **1.** In an isohedral tiling in which each tile is a fundamental domain, no tile can contain a rotation center or axis of reflection for the whole tiling except on its boundary.

We note that while an isohedral tiling may have fundamental domains of many different shapes, all have the same area. The simplest fundamental domains have triangle or quadrilateral shapes [

6]. Lemma 1 and the definitions of polyomino and polyiamond immediately exclude several symmetry groups from our consideration.

**Theorem** **1.** There are no **p3**, **p31m**, **p3m1**, **p6**, or **p6m** isohedral tilings by polyominoes. There are no **p4**, **p4g**, or **p4m** isohedral tilings by polyiamonds.

Proof. Isohedral tilings of types **p3**, **p31m**, **p3m1**, **p6**, or **p6m** all have 3-fold centers, and by Lemma 1, if the tiles are fundamental domains, these centers must lie on the boundaries of the tiles. But if the tiles are polyominoes, a 120${}^{\circ}$ rotation about such a center cannot map a tile fully onto another tile (since unit squares will not be mapped to unit squares). Thus such a tiling is impossible. Similarly, a 4-fold center is impossible in an isohedral tiling by polyiamonds (since a 90${}^{\circ}$ rotation cannot map unit triangles to unit triangles), so there can be no such tilings of types **p4**, **p4g**, or **p4m**.

In each of the sections that follow, we begin with a fixed lattice of symmetry elements for a symmetry group G (that is, a fixed array of rotation centers, reflection axes and glide-reflection axes for elements of G) and give a backtracking procedure to produce a complete set of polyominoes (or polyiamonds) that are fundamental domains for G. A polyomino or polyiamond tile T is a fundamental domain for a symmetry group G if the action of G on T produces an isohedral tiling and T is a region of minimal area for which G can generate that tiling. G will be contained in (or equal to) the full symmetry group of the tiling.

As we consider symmetries of our isohedral tilings, the following theorem will be useful [

3].

**Theorem** **2.** Let G be one of the 17 two-dimensional symmetry groups and $\mathcal{T}$ an isohedral tiling generated by G acting on a tile T that is a fundamental domain for G. Let ${G}^{\prime}$ be the full symmetry group of $\mathcal{T}$. If G is a proper subgroup of ${G}^{\prime}$, there is an element of ${G}^{\prime}$ (other than the identity) that leaves T invariant. In this case, a fundamental domain for ${G}^{\prime}$ has area smaller than T.

## 7. p3m1

The lattice of reflection axes and 3-fold rotation centers of a

**p3m1** symmetry group is shown in

Figure 15; black, white, and grey circles denote three inequivalent 3-fold rotation centers. Here, unlike the

**p31m** case, all 3-fold centers lie on reflection axes. The shaded region bounded by reflection axes is a fundamental domain for the

**p3m1** group that generates the tiling by reflections in those axes. By Lemma 1, this is the only polyiamond tile possible having the area of a fundamental domain. But the full symmetry group of this tiling is type

**p6m**, which has a fundamental domain with area 1/6 that of the shaded tile. Thus,

**Theorem** **6.** There are no **p3m1** isohedral tilings having polyiamonds as fundamental domains.

We note that if the shaded region in

Figure 15 is decorated with an asymmetric motif then there are

n-iamonds (where

$n={k}^{2}$,

k a positive integer) having the shape of an equilateral triangle, for which the decorated triangle is a fundamental domain for a

**p3m1** isohedral tiling.

## 10. Enumeration Tables

Table 1,

Table 2 and

Table 3 give the number of tiles and isohedral tilings that we have generated in

Section 2 through

Section 9.

${N}_{n}$ is the number of inequivalent tiles

T in

${\mathcal{T}}_{n}$ for a given symmetry group

G of type

**p3**,

**p31m**,

**p3m1**,

**p4**,

**p4m**,

**p4g**,

**p6**, or

**p6m** that generate an isohedral tiling having the

n-omino or

n-iamond tiles as fundamental domain. Tiles are equivalent only if they generate the same tiling by the action of the group

G when each tile is marked with an asymmetric motif. For example, the tiles 5-2 and 5-2-2 in

Figure 3 are congruent and their corresponding tilings are the same, but the placement of their 4-fold centers is different, and so if the tiles are marked with an asymmetric motif, they generate different isohedral tilings.

Marking each tile with an asymmetric motif also guarantees that the group that generates the tiling is the full symmetry group of the tiling. 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.

${S}_{n}$ is the corresponding number of the ${N}_{n}$ tilings when the asymmetric motif of each tile is removed. That is, ${S}_{n}$ is the number of isohedral tilings having full symmetry group G and having (unmarked) n-ominoes or n-iamonds as fundamental domains. In our figures that depict the isohedral tilings for small values of n, these tilings do not have parentheses around their labels. These are the most important “counting” results in this article. ${N}_{n}^{\prime}$ is the number of non-congruent tiles in the ${N}_{n}$ tilings, counted by ignoring rotation centers attached to tiles; similarly, ${S}_{n}^{\prime}$ is the number of non-congruent tiles in the ${S}_{n}$ tilings.

For example, for symmetry group

**p6** and

$n=7$, there are 20 tiles in

Figure 17, with corresponding isohedral tilings in

Figure 18, so

${N}_{7}=20$. Since tiling 7-1 of

Figure 18 has parentheses around its label,

${S}_{7}=19$. From

Figure 17, we can see that four pairs of tiles are congruent: 7-1 and 7-1-2; 7-3 and 7-3-2; 7-5 and 7-5-2; 7-12 and 7-12-2. Thus

${N}_{7}^{\prime}=20-4=16$. Among the 19 tiles counted for

${S}_{7}$, there are also 16 non-congruent tiles, so

${S}_{7}^{\prime}=16$.

## 11. Summary

We have described computer algorithms that can enumerate and display isohedral tilings by n-omino or n-iamond tiles for given n in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. Their symmetry groups are of types **p3**, **p31m**, **p4**, **p4g**, and **p6**. We have shown that there are no isohedral tilings with symmetry groups of types **p3m1**, **p4m**, or **p6m** that have polyominoes or polyiamonds as fundamental domains. For symmetry groups of types **p3**, **p31m**, **p4**, **p4g**, and **p6** we used the backtracking Procedure 1 to obtain a set ${\mathcal{T}}_{n}$ of n-omino or n-iamond tiles where each tile produced one isohedral tiling, generated by a given symmetry group G of one of these five types. We can denote ${\mathcal{T}}_{n}\left(G\right)$ as the set ${\mathcal{T}}_{n}$ for that symmetry group G and ${\mathcal{T}}_{n}^{*}\left(G\right)$ the corresponding set of isohedral tilings.

We investigated the symmetries of tilings in the set

${\mathcal{T}}_{n}^{*}\left(G\right)$ and noted those tilings that satisfy the following two conditions: (1) the full symmetry group of the tiling is

G; and (2) the tiles are fundamental domains for

G. We denote the subset of

${\mathcal{T}}_{n}^{*}\left(G\right)$ that satisfies (1) and (2) as

${\mathcal{S}}_{n}^{*}\left(G\right)$ and the corresponding set of tiles as

${\mathcal{S}}_{n}\left(G\right)$. (For small values of

n, these tiles and their tilings were displayed with labels without parentheses.) The enumeration of

${\mathcal{S}}_{n}^{*}\left(G\right)$ is the main counting result of this article. Although the

n-omino or

n-iamond tiles produced by our algorithm are not always fundamental domains for the isohedral tilings they generate, if we mark these tiles with an asymmetric motif, then the set

${\mathcal{T}}_{n}^{*}\left(G\right)$ is the set of all isohedral tilings with symmetry group

G in which the corresponding tiles in

${\mathcal{T}}_{n}\left(G\right)$ are fundamental domains. The set

${\mathcal{T}}_{n}^{*}\left(G\right)$ can then also include a marked fundamental domain for a

**p3m1** symmetry group. In

Table 1,

Table 2 and

Table 3 of

Section 10, we used the notation

${N}_{n}=\#{\mathcal{T}}_{n}^{*}\left(G\right)$ and

${S}_{n}=\#{\mathcal{S}}_{n}^{*}\left(G\right)$.