A New Class of Tilings with Two Prototiles

Brian Wichmann
Surrey, UK

John Dawes ¹
Berkshire, UK

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:

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

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.

Footnotes:

¹ John Dawes died on the 19th January 2002 and hence this article is dedicated to his memory.

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