# A New Class of Tilings with Two Prototiles

## Abstract

This paper enumerates a class of tilings with two prototiles: one regular and the other irregular. This class has been used in decorative art on at least 25 occasions. For convenience of Internet access, each sample tiling in the complete gallery is selectable individually from the classification table.

## 1  Introduction

In 1876, the Victoria and Albert Museum was fortunate enough to acquire the works of Mirza Akber who was the architect to the Court of Persia. His works contains drawings of some of the classical patterns in Islamic art - indeed it seems that only one other source of such material exists - the Topkapi Scroll [3].

Figure 1: Tiling by Mirza Akber

Figure 1 is taken from the leather scroll of Mirza Akber (and may not have been published before). The interest here is not the actual source, but the mathematics behind the tiling. It contains just two tiles, one regular (a square) and the other irregular. In addition:

1. There are symmetry operations on the tiling which are transitive on the sides of the regular prototile.
2. The symmetry operations of the tiling are transitivite with respect to both the irregular and regular prototiles.

3.
By a regular prototile, we allow regular star polygons, including those with only two points (ie, a diamond). We exclude cases in which the `irregular' tile is in fact regular. We require the regular tile to be edge-to-edge, except that the n-star can have n or 2n edges. We also require the prototiles to be topological discs and uniformed bounded, but not necessarily that the intersection of 2 tiles be connected (see B04 for an example which is not connected). Note that the regular n-gon are at cn centres and that n-stars are at dn centres (with regular polygons being a special case).

The problem that this paper addresses is the enumeration of the members of this class of tilings. In an encyclopedia of tiling patterns, about 25 instances of this class of tilings has been found in decorative art [2].

## 2  The classification

The first stage in the classification is to consider the symmetry groups involved. Firstly, we have the symmetry group of the entire tiling, which for Figure 1 is p4, then the restriction of these symmetries on the regular prototile, c4 in this case, and finally the restriction to the irregular prototile which is c1 in this case. Hence for Figure 1 we write the symmetries as [p4,c4,c1].

The overall symmetry group of the tiling must either have a cycle of length 3, 4 or 6, or have two reflections at right angles (for the diamond). Hence it cannot be cm, p1, p2, pg, pgg, pm or pmg. The gallery has examples of the remaining 10 symmetry groups.

In order to continue the classification further we must consider the topology of the tiling. To show the means of analysing this, consider Figure 2.

Figure 2: A mathematical tiling

In Figure 2 we have a tiling produced as a result of the mathematical analysis here which has not, so far as we are aware, been used in decorative art. The symmetries are [p31m,d3,c3]. Reflections are needed with any star polygon to ensure transitivity between the edges either side of the points of the star.

Given a tiling of the class we are considering, we can construct an isohedral tiling as follows: Draw lines from the vertices of the regular tile (having valency 3 or more) to its centre. Then remove the edges of the regular tile while retaining those edges as marks on the remaining tile. By virtue of the symmetry of the original tiling, we now have a marked isohedral tiling. Starting from Figure 2, we illustrate this process in Figure 3

Figure 3: Deriving a marked isohedral tiling

Note that the marking is crucial, since without it, the isohedral tiling would have the symmetry p6m rather than the p31m of Figure 2

By this construction, we can produce a marked isohedral tiling from any member of this class of tilings. Moreover, given a marked isohedral tiling, we can attempt construct a member of our class by placing a regular polygon or star polygon on the edge of the tile at a point which has the necessary symmetry.

The reason for establishing this connection with marked isohedral tilings is simply that Grünbaum and Shephard have enumerated such tilings in table 6.2.1 of [1].

From their work we can enumerate the possible IH Classes as follows (stars for marked tilings):

• cmm: 17, 26, 54, 60*, 67, 74, 78 and 91.
• pmm: 48*, 65* and 72.
• p3: 7, 10 and 33.
• p31m: 16, 18, 30, 36, 38 and 89*.
• p3m1: 19*, 35* and 87*.
• p4: 28, 55, 61, 62 and 79.
• p4g: 29, 56, 63*, 71, 73 and 81.
• p4m: 70*, 75*, 76, 80* and 82.
• p6: 11, 21, 31, 34, 39, 88 and 90.
• p6m: 20, 32, 37, 40, 77, 92* and 93.

•
We now have a means of enumerating the tilings of our class.

## 3  Tables of tilings

Below are tables of the characteristics of the tilings in this class. The entries are as follows:
[Number.] This is the number of this specific member of the class of tilings.
[Symmetries.] This consists of the three symmetry groups as noted above. The notation `...' implies that the tiling is homeomeric to the one above.
[Incidence.] This entry gives the incidence symbol for the irregular tile (see [1] chapter 6). For the regular tile, the incidence is uninformative since for the regular polygon it is [(A+)n; a+] and for the star polygon with 2n edges it is [(A+A-)n; a+]. The incidence symbol is put into a canonical form by taking the lexicographical lowest form.
[T Class.] The topological class. Starting from an edge of the regular tile, the irregular tile is traversed in the positive sense and the valence of every vertex is noted. (Hence the first two vertices are on the regular tile.)
[IH Class.] This is the number of the marked isohedral tiling as listed by Grünbaum and Shephard (table 6.2.1).
[Polygon.] This gives either the regular polygon, or the number of points of the star-polygon follow by the vertex angle. Note that an octagon is given as a 4-star polygon when it could be replaced by such a 4-star with a different vertex angle than the 3p/4. (In the tables, `triangle' means equilateral triangle.)
[Name.] This is the name used to identify the tiling (actually, a file name).

 No Symmetries Incidence T IH Polygon Name Class Class 1 [cmm,d2,c1] [a+b+c+d+e+;A+b-c+d-e-] [32.43] 54 2-star,p/3 NEW16 2 [cmm,d2,c1] [a+b+c+d+;A+b+c-d-] [3.52.4] 54 2-star,p/3 NEW17 3 [cmm,d2,c1] [a+b+c+d+;A+b-c+d-] [3.6.42] 54 2-star,p/3 NEW52 4 [cmm,d2,c1] [a+b+c+;A+b+c-] [6.52] 54 2-star,p/3 NEW53 5 [cmm,d2,c1] [a+b+c+d+;A+b-c+d-] [32.82] 78 2-star,p/3 NEW1 6 [cmm,d2,c1] [a+b+c+;A+b+c-] [3.102] 78 2-star,p/3 NEW20 7 [cmm,d2,c2] [(a+b+c+)2;A+b-c-] [(32.4)2] 60 2-star,p/3 NEWA 8 [cmm,d2,c2] [(a+b+)2;A+b-] [(3.6)2] 60 2-star,p/3 NEW54 9 [cmm,d2,d1] [a+ba-c+d+d-c-;A+bc-d+] [37] 26 2-star,p/3 NEW12 10 [cmm,d2,d1] [a+ba-c+c-;A+bc+] [4.32.42] 26 2-star,p/5 S52T 11 [cmm,d2,d1] [a+a-b+c+c-b-;A+b-c+] [3.4.34] 26 2-star,p/3 NEW14 12 [cmm,d2,d1] [a+a-b+b-;A+b+] [44] 26 2-star,p/3 NEW15 13 [cmm,d2,d1] [ab+c+d+d-c-b-;Ab-c-d+] [32.4.33.4] 26 2-star,p/3 NEW11 14 [cmm,d2,d1] [ab+c+c-b-;Ab-c+] [62.33] 26 2-star,p/3 NEW13 15 [cmm,d2,d1] [ab+c+dc-b-;Ab-c+d] [32.44] 67 2-star,p/3 NEW18 16 [cmm,d2,d1] [ab+cb-;Ab+c] [54] 67 2-star,p/3 NEW19 17 [cmm,d2,d1] [ab+c+c-b-;Ab-c+] [32.63] 91 2-star,p/3 NEW21 18 [cmm,d2,d1] [ab+b-;Ab+] [83] 91 2-star,p/3 NEW22 19 [cmm,d2,d2] [(ab+c+c-b-)2;Ab-c+] [310] 17 2-star,p/3 NEW10 20 [cmm,d2,d2] [(ab+b-)2;Ab+] [46] 17 2-star,p/4 NEW2 21 [pmm,d2,c1] [a+b+c+d+e+;A+b-c-d-e-] [32.43] 48 2-star,p/3 NEW27 22 [pmm,d2,c1] [a+b+c+d+;A+b-c-d-] [3.6.42] 48 2-star,p/2 NEW5 23 [pmm,d2,c1] [a+b+c+;A+b-c-] [62.4] 48 2-star,p/3 NEW29 24 [pmm,d2,d1] [a+ba-c+dc-;A+bc-d] [34.42] 65 2-star,p/3 NEW28 25 [pmm,d2,d1] [a+ba-c;A+bc] [6.32.6] 65 2-star,p/3 NEW30 26 [pmm,d2,d1] [a+a-b+cb-;A+b-c] [3.4.3.42] 65 2-star,p/3 NEW31 27 [pmm,d2,d1] [a+a-b;A+b] [6.4.6] 65 2-star,p/2 CM006A 28 [pmm,d2,d1] [ab+c+dc-b-;Ab-c-d] [32.44] 65 2-star,p/3 NEW23 29 [pmm,d2,d1] [ab+cb-;Ab-c] [62.42] 65 2-star,p/3 NEW25 30 [pmm,d2,d2] [(ab+cb-)2;Ab-c] [(32.42)2] 72 2-star,p/3 NEW24 31 [pmm,d2,d2] [(ab)2;Ab] [64] 72 2-star,p/3 NEW26 32 [pmm,d2,d2] [(a+ba-c)2;A+bc] [38] 72 2-star,p/2 B17 33 [pmm,d2,d2] [(a+a-b)2;A+b] [(3.4.3)2] 72 2-star,p/2 B14

Table 1: Order 2: cmm and pmm

 No Symmetries Incidence T IH Polygon Name Class Class 34 [p3,c3,c1] [a+b+c+d+e+f+g+;A+g+d+c+f+e+b+] [37] 7 triangle B16 35 [p3,c3,c1] [a+b+c+d+e+;A+c+b+e+d+] [42.3.4.3] 7 triangle NEW66 36 [p3,c3,c1] [a+b+c+d+e+;A+e+d+c+b+] [32.6.3.6] 33 triangle NEW45 37 [p3,c3,c1] [a+b+c+;A+c+b+] [92.3] 33 triangle NEW74 38 [p3,c3,c3] [(a+b+c+)3;A+c+b+] [39] 10 triangle B12 39 [p31m,c3,c1] [a+b+c+d+e+;A+e+c-d-b+] [32.4.6.4] 30 triangle B03 40 [p31m,c3,c1] [a+b+c+;A+b-c-] [63] 30 triangle NEW44 41 [p31m,c3,c1] [a+b+c+d+;A+d+c-b+] [32.122] 38 triangle NEW43 42 [p31m,c3,d1] [a+b+b-a-c+d+d-c-;A+c-b-d-] [38] 16 triangle NEW49 43 [p31m,c3,d1] [a+a-b+b-;A+b-] [52.3.5] 16 triangle NEW85 44 [p31m,c3,d1] [a+b+b-a-c+c-;A+c-b-] [(32.6)2] 36 triangle L2311 45 [p31m,c3,d3] [(a+ba-c)3;A+cb] [312] 18 triangle L2310 45 ... [(a+ba-c)3;A+cb] [312] 18 triangle J59A 45 ... [(a+ba-c)3;A+cb] [312] 18 triangle L255 46 [p31m,d3,c1] [a+b+c+d+e+;A+b-d+c+e-] [32.4.3.4] 30 3-star,p/6 NEW99 47 [p31m,d3,c1] [a+b+c+d+;A+c+b+d-] [(3.5)2] 30 3-star,p/6 NEWB 48 [p31m,d3,c1] [a+b+c+;A+c+b+] [62.3] 30 hexagon NEWS 49 [p31m,d3,c3] [(a+b+)3;A+b-] [36] 89 3-star,p/6 NEWC 50 [p31m,d3,d1] [ab+c+d+d-c-b-;Ab-d+c+] [37] 16 3-star,p/4 NEW48 50 ... [ab+c+d+d-c-b-;Ab-d+c+] [37] 16 triangle NEW46 51 [p31m,d3,d1] [ab+c+c-b-;Ac+b+] [42.3.4.3] 16 triangle C05B 51 ... [ab+c+c-b-;Ac+b+] [42.3.4.3] 16 3-star,p/6 NEW47 52 [p3m1,d3,c1] [a+b+c+d+;A+b-c-d-] [32.62] 87 3-star,p/6 NEW51 53 [p3m1,d3,c1] [a+b+c+;A+b-c-] [3.6.9] 87 3-star,p/6 NEWD 54 [p3m1,d3,d1] [ab+c+c-b-;Ab-c-] [32.6.3.6] 35 triangle B18 54 ... [ab+c+c-b-;Ab-c-] [32.6.3.6] 35 3-star,p/4 NEW42 55 [p3m1,d3,d1] [ab+b-;Ab-] [92.3] 35 triangle B13 55 ... [ab+b-;Ab-] [92.3] 35 3-star,p/4 NEW41 56 [p3m1,d3,d1] [a+b+b-a-c+c-;A+b-c-] [36] 35 3-star,p/6 NEWT 57 [p3m1,d3,d1] [a+a-b+b-;A+b-] [3.6.32] 35 3-star,p/6 NEWU 58 [p3m1,d3,d3] [(ab+b-)3;Ab-] [39] 19 3-star,2p/3 J44A

Table 2: Order 3: p3, p31m and p3m1

 No Symmetries Incidence T IH Polygon Name Class Class 59 [p4,c4,c1] [a+b+c+d+e+f+;A+f+c+e+d+b+] [34.4.3] 28 square P16 59 ... [a+b+c+d+e+f+;A+f+c+e+d+b+] [34.4.3] 28 square S28A 60 [p4,c4,c1] [a+b+c+d+;A+b+d+c+] [44] 28 square B01 60 ... [a+b+c+d+;A+b+d+c+] [44] 28 square R4 61 [p4,c4,c1] [a+b+c+d+e+;A+e+d+c+b+] [32.43] 55 square NEW55 62 [p4,c4,c1] [a+b+c+;A+c+b+] [62.4] 55 square NEW56 63 [p4,c4,c1] [a+b+c+d+;A+d+c+b+] [32.82] 79 square NEW57 64 [p4,c4,c2] [(a+b+c+)2;A+c+b+] [(32.4)2] 61 square L3010 64 ... [(a+b+c+)2;A+c+b+] [(32.4)2] 61 square B02 65 [p4,c4,c4] [(a+b+)4;A+b+] [38] 62 square NEW4 66 [p4g,c4,c1] [a+b+c+d+e+;A+e+c-d-b+] [32.43] 56 square NEWE 67 [p4g,c4,c1] [a+b+c+;A+b-c-] [62.4] 56 square NEW58 68 [p4g,c4,c1] [a+b+c+d+;A+d+c-b+] [32.82] 81 square NEW59 69 [p4g,c4,d1] [a+b+b-a-c+dc-;A+c-b-d] [37] 29 square H2C12 69 ... [a+b+b-a-c+dc-;A+c-b-d] [37] 29 square K09A 70 [p4g,c4,d1] [a+a-b;A+b] [53] 29 square H2C34 71 [p4g,c4,d1] [a+b+b-a-c+c-;A+c-b-] [(32.4)2] 71 square G8 71 ... [a+b+b-a-c+c-;A+c-b-] [(32.4)2] 71 square L3311 72 [p4g,c4,d2] [(a+ba-c)2;A+cb] [38] 73 square K09B 73 [p4g,d2,c1] [a+b+c+d+e+;A+b-d+c+e-] [32.43] 56 2-star,p/3 NEWF 74 [p4g,d2,c1] [a+b+c+d+;A+b-d+c+] [3.5.4.5] 56 2-star,p/3 NEW60 75 [p4g,d2,c1] [a+b+c+;A+c+b+] [62.4] 56 2-star,p/3 NEW61 76 [p4g,d2,c4] [(a+b+)4;A+b-] [38] 63 2-star,p/3 NEWG 77 [p4g,d2,d1] [ab+c+d+d-c-b-;Ab-d+c+] [33.4.3.4.3] 29 2-star,p/3 NEW62 78 [p4g,d2,d1] [ab+c+c-b-;Ac+b+] [45] 29 2-star,p/3 NEWH

Table 3: Order 4: p4, and p4g

 No Symmetries Incidence T IH Polygon Name Class Class 79 [p4m,d2,c1] [a+b+c+d+;A+b-c-d-] [32.82] 80 2-star,p/3 NEWI 80 [p4m,d2,c1] [a+b+c+;A+b-c-] [3.12.8] 80 2-star,p/5 NEWV 81 [p4m,d2,d1] [a+b+b-a-c+c-;A+b-c-] [(32.4)2] 70 2-star,p/3 NEWJ1 82 [p4m,d2,d1] [a+a-b+b-;A+b-] [3.8.3.4] 70 2-star,p/3 NEW65 83 [p4m,d2,d1] [ab+c+c-b-;Ab-c-] [32.8.4.8] 82 2-star,p/3 NEWJ2 84 [p4m,d2,d1] [ab+b-;Ab-] [122.4] 82 2-star,p/3 NEW34 85 [p4m,d2,d4] [(ab+b-)4;Ab-] [(32.4)4] 76 2-star,p/2 J51A 86 [p4m,d4,c1] [a+b+c+d+;A+b-c-d-] [32.4.8] 80 4-star,p/4 NEW8 87 [p4m,d4,c1] [a+b+c+;A+b-c-] [3.4.12] 80 4-star,p/4 NEW36 88 [p4m,d4,c1] [a+b+c+;A+b-c-] [3.6.8] 80 4-star,p/4 NEW37 89 [p4m,d4,d1] [ab+c+c-b-;Ab-c-] [32.43] 70 square B04B 89 ... [ab+c+c-b-;Ab-c-] [32.43] 70 4-star,p/4 NEW32 90 [p4m,d4,d1] [ab+b-;Ab-] [62.4] 70 square J25A 90 ... [ab+b-;Ab-] [62.4] 70 4-star,3p/4 B07 91 [p4m,d4,d1] [a+ba-c+c-;A+bc-] [34.4] 82 4-star,p/4 NEWK 92 [p4m,d4,d1] [a+a-b+b-;A+b-] [(3.4)2] 82 4-star,p/4 NEW35 93 [p4m,d4,d1] [a+a-b;A+b] [3.8.3] 82 4-star,p/4 NEW38 94 [p4m,d4,d1] [ab+cb-;Ab-c] [32.82] 82 square B19 94 ... [ab+cb-;Ab-c] [32.82] 82 4-star,p/4 NEW39 95 [p4m,d4,d1] [ab;Ab] [122] 82 4-star,p/4 NEW40 96 [p4m,d4,d2] [(ab+b-)2;Ab-] [(32.4)2] 75 square A221 96 ... [(ab+b-)2;Ab-] [(32.4)2] 75 4-star,p/4 NEW33 97 [p4m,d4,d4] [(ab)4;Ab] [38] 76 4-star,p/4 NEWL

Table 4: Order 4: p4m

 No Symmetries Incidence T IH Polygon Name Class Class 98 [p6,c3,c1] [a+b+c+d+e+f+;A+f+c+e+d+b+] [34.6.3] 21 triangle B09 99 [p6,c3,c1] [a+b+c+d+;A+b+d+c+] [43.6] 21 triangle NEW67 100 [p6,c3,c1] [a+b+c+d+e+;A+e+d+c+b+] [32.4.6.4] 31 triangle NEW68 101 [p6,c3,c1] [a+b+c+;A+c+b+] [63] 31 triangle NEW69 102 [p6,c3,c1] [a+b+c+d+;A+d+c+b+] [32.122] 39 triangle NEW70 103 [p6,c3,c2] [(a+b+c+)2;A+c+b+] [(32.6)2] 34 triangle L4212 103 ... [(a+b+c+)2;A+c+b+] [(32.6)2] 34 triangle B10 104 [p6,c3,c6] [(a+b+)6;A+b+] [312] 11 triangle F242F 104 ... [(a+b+)6;A+b+] [312] 11 triangle B11 105 [p6,c6,c1] [a+b+c+d+e+f+;A+f+c+e+d+b+] [36] 21 hexagon P010 106 [p6,c6,c1] [a+b+c+d+;A+b+d+c+] [43.3] 21 hexagon B08 106 ... [a+b+c+d+;A+b+d+c+] [43.3] 21 hexagon NEW71 107 [p6,c6,c1] [a+b+c+d+e+;A+e+d+c+b+] [32.4.3.4] 31 hexagon NEW72 108 [p6,c6,c1] [a+b+c+;A+c+b+] [62.3] 31 hexagon NEW73 109 [p6,c6,c1] [a+b+c+d+;A+d+c+b+] [32.62] 88 hexagon NEW75 110 [p6,c6,c2] [(a+b+c+)2;A+c+b+] [36] 34 hexagon F135 110 ... [(a+b+c+)2;A+c+b+] [36] 34 hexagon F49 110 ... [(a+b+c+)2;A+c+b+] [36] 34 hexagon W39 110 ... [(a+b+c+)2;A+c+b+] [36] 34 hexagon RS1 111 [p6,c6,c3] [(a+b+)3;A+b+] [36] 90 hexagon F242E 111 ... [(a+b+)3;A+b+] [36] 90 hexagon NEW76 111 ... [(a+b+)3;A+b+] [36] 90 hexagon B06

Table 5: Order 6: p6

 No Symmetries Incidence T IH Polygon Name Class Class 112 [p6m,d2,c1] [a+b+c+d+;A+b-c-d-] [32.6.12] 77 2-star,p/3 NEWM 113 [p6m,d2,c1] [a+b+c+;A+b-c-] [3.9.12] 77 2-star,p/3 NEW77 114 [p6m,d2,c1] [a+b+c+;A+b-c-] [3.18.6] 77 2-star,p/6 NEW78 115 [p6m,d2,d1] [a+b+b-a-c+c-;A+b-c-] [35.6] 32 2-star,p/2 G7 116 [p6m,d2,d1] [a+a-b+b-;A+b-] [(3.6)2] 32 2-star,p/6 NEW79 117 [p6m,d2,d1] [a+a-b+b-;A+b-] [3.12.32] 32 2-star,p/6 NEW80 118 [p6m,d2,d1] [a+b+c+c-b-;A+b-c-] [32.12.3.12] 40 2-star,p/3 NEW81 119 [p6m,d2,d1] [ab+b-;Ab-] [182.3] 40 2-star,p/6 NEW82 120 [p6m,d2,d1] [ab+c+c-b-;Ab-c-] [32.63] 92 2-star,p/3 NEW83 121 [p6m,d2,d1] [ab+b-;Ab-] [92.6] 92 2-star,p/3 NEW84 122 [p6m,d2,d3] [(ab+b-)3;Ab-] [(32.6)3] 93 2-star,p/2 NEW3 123 [p6m,d2,d6] [(ab+b-)6;Ab-] [318] 20 2-star,p/2 B04 124 [p6m,d3,c1] [a+b+c+d+;A+b-c-d-] [32.4.12] 77 3-star,2p/3 NEW7 125 [p6m,d3,c1] [a+b+c+;A+b-c-] [3.6.12] 77 3-star,p/6 NEWN 126 [p6m,d3,c1] [a+b+c+;A+b-c-] [3.18.4] 77 3-star,p/6 NEW86 127 [p6m,d3,d1] [ab+c+c-b-;Ab-c-] [32.4.6.4] 32 3-star,2p/3 NEW6 127 ... [ab+c+c-b-;Ab-c-] [32.4.6.4] 32 3-star,p/6 NEWP2 128 [p6m,d3,d1] [ab+b-;Ab-] [63] 32 3-star,p/6 NEW87 129 [p6m,d3,d1] [ab+cb-;Ab-c] [32.122] 40 3-star,p/6 NEW88 130 [p6m,d3,d1] [ab;Ab] [182] 40 3-star,p/6 NEW89 131 [p6m,d3,d1] [a+ba-c+c-;A+bc-] [34.6] 92 3-star,2p/3 NEW9 131 ... [a+ba-c+c-;A+bc-] [34.6] 92 3-star,p/6 NEWP1 132 [p6m,d3,d1] [a+a-b+b-;A+b-] [3.4.3.6] 92 3-star,p/6 NEW90 133 [p6m,d3,d1] [a+a-b;A+b] [3.12.3] 92 3-star,p/6 NEW91 134 [p6m,d3,d2] [(ab+b-)2;Ab-] [(32.6)2] 37 3-star,p/6 NEWQ 135 [p6m,d3,d6] [(ab)6;Ab] [312] 20 3-star,2p/3 J59B 136 [p6m,d6,c1] [a+b+c+d+;A+b-c-d-] [32.4.6] 77 6-star,p/3 NEWR 137 [p6m,d6,c1] [a+b+c+;A+b-c-] [3.62] 77 6-star,p/6 NEW92 138 [p6m,d6,c1] [a+b+c+;A+b-c-] [3.9.4] 77 6-star,p/3 NEW93 139 [p6m,d6,d1] [ab+c+c-b-;Ab-c-] [32.4.3.4] 32 hexagon C09B 139 ... [ab+c+c-b-;Ab-c-] [32.4.3.4] 32 6-star,p/3 S14A 140 [p6m,d6,d1] [ab+b-;Ab-] [62.3] 32 6-star,p/3 NEW94 141 [p6m,d6,d1] [a+ba-c+c-;A+bc-] [35] 40 6-star,p/3 NEW96 142 [p6m,d6,d1] [a+a-b;A+b] [3.6.3] 40 6-star,p/3 NEW97 143 [p6m,d6,d1] [a+a-b+b-;A+b-] [3.4.32] 40 6-star,p/3 NEW98 144 [p6m,d6,d1] [ab+cb-;Ab-c] [32.62] 92 hexagon VA1 145 [p6m,d6,d1] [ab;Ab] [92] 92 6-star,p/3 NEW95 146 [p6m,d6,d2] [(ab+b-)2;Ab-] [36] 37 hexagon B20 147 [p6m,d6,d3] [(ab)3;Ab] [36] 93 6-star,p/3 B15 147 ... [(ab)3;Ab] [36] 93 hexagon B05

Table 6: Order 6: p6m

changes done are in John's letter, 25th June 2001

Summaries of the tilings from these tables are as follows:

The enumeration is completed by showing how to obtain the tilings of this class from a marked isohedral tiling. We do those by example, taking the marked isohedral class 92 which is shown in Figure 4

Figure 4: Marked isohedral tiling, class 92

The regular tile must have its centre on the edge of the isohedral tiling. There are just three positions for this, marked A (centre for d6), B (centre for d3) and C (centre for d2) in Figure 4

We now consider in turn those three positions:

[A:] The regular tile must be either a regular hexagon or a six-pointed star. If this regular polygon is small, it does not extend to meet any other vertex (other than A which it replaces), then we have tiling VA1. On the other hand, if the regular polygon is large, is must extend to C and therefore be a 6-pointed star, that is tiling NEW95
[B:] Here we must have a 3-pointed star. For the small case we have NEW9 (the 3-pointed star being a regular hexagon), or for a proper star, the tiling NEWP1. (NEW9 and NEWP1 are homeomeric.)
The situation with the large case is more complex. The star can extend to C to meet another star, or the star can extend to A. The extension to C gives tiling NEW90, while the extension to A gives NEW91
[C:] Here we must have a diamond. If the diamond is small we have tiling NEW83, and if it is large we have NEW84

Hence we conclude that there are seven classes of our tiling derived from class 92 of the marked isohedral classification.

Note that it is not always possible to place a regular tile on the edge of the isohedral tiling, even when the point in question has the appropriate symmetry. Consider the tiling NEW45, but with the triangles removed, ie, the original marked isohedral tiling. A triangle cannot be placed at the vertex of valency 6 although that point has the necessary symmetry (because it would not be edge-to-edge).

In the case of the irregular tile, it may be necessary to avoid a specific construction which would introduce an unwanted symmetry. For an example of this, see NEW97

In some cases, a potential tiling of this class is not possible, since the irregular tile must, in fact, be regular.

## 4  Conclusions

We conclude that there are 147 homeomeric types of tiling of this class. The enumeration is made possible from that of the marked isohedral tilings provided by Grünbaum and Shephard. We would hope that this enumeration would enable the variety of tilings used in decorative art to be increased.

It would seem that a similar reasoning to that applied here would allow for the enumeration of, say, two (distinct) regular tiles and an irregular one. However, the number of cases could make such an enumeration very tedious.

## References

[1]
B. Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman & Co., New York, NY, 1987.
[2]
B. A. Wichmann, The World of Patterns, CD-ROM and booklet. World Scientific, 2001. See tile/maths/twopoly on the CD. ISBN 981-02-4619-6
[3]
G. Necipoglu, The Topkapi Roll - Geometry and ornament in Islamic architecture. The Getty Center for the History of Art and the Humanities. 1995. ISBN 0-89236-335-5

Each of the large tilings are generated by a computer program which outputs some statistical information. For the details of this, see [2], but the information can be ignored in the context of this paper. The tilings used in decorative art reference the source on the CD.