Some statistics: Version 8

Brian Wichmann

1  Introduction

This page gives some statistics concerning this release of the tiling system, having 1319 tilings.
The statistics are presented in the same order as the comprehensive search HTML form.

2  Symmetry group

Symmetry Description Number
(none) 142
cm The tiling has no rotation symmetry, but one glide reflection and one reflection 9
cmm The tiling has a rotation symmetry of order two, two glide reflections and two reflections 99
p1 The tiling has no rotations, reflections or glide reflections 3
p2 The tiling has rotations of order two, but no reflections or glide reflections 28
p3 The tiling has rotations of order three, but no reflections or glide reflections 8
p31mThe tiling has two rotations of order three, a glide reflection and a relections 29
p3m1The tiling has one rotation of order three, a glide reflection and a relections 12
p4 The tiling has rotations of order four, but no reflections or glide reflections 69
p4g The tiling has a rotation of order two another of order four, two glide reflections 69
p4m The tiling has a rotation of order four, and is its own mirror image 423
p6 The tiling has rotations of order six, but no reflections or glide reflections 60
p6m The tiling has a rotation of order six, and is its own mirror image 237
pg The tiling has two glide reflections 16
pgg The tiling has two rotations of order two and two glide reflections 15
pm The tiling has no rotations or glide reflections, but two reflections 13
pmg The tiling has two rotations of order two, one glide reflection and one reflection 23
pmm The tiling has four rotations of order two and four reflections 64
Note how unevenly the groups appear. Given a tiling of a `rare' group, it would then be easy to examine each tiling by eye for a match.

3  Two colour property

Property Number
Colouring could not be determined 163
Cannot be coloured with two colours 448
Can be coloured with two colours 708
The small number of tilings for which the colouring could not be determined is due to the software not being capable enough.

4  Tilings containing regular polygons

Polygon Number of TilingsTotal
equilateral triangle 112 253
square 336 732
regular pentagon 102 777
regular hexagon 132 182
regular heptagon 3 5
regular octagon 81 119
regular enneagon 3 3
regular decagon 2 2
12-gon 7 7
16-gon 1 1
18-gon 1 1
24-gon 1 1

5  Tilings containing regular star polygons

Points Vertex angle Tiling count Total
2 15.0 1 1
2 18.0 4 6
2 22.5 2 3
2 25.7 10 23
2 30.0 14 20
2 34.3 1 1
2 36.0 6 6
2 45.0 66 304
2 48.0 1 1
2 52.5 1 3
2 58.5 1 1
2 60.0 91 111
2 66.0 1 1
2 72.0 16 142
2 72.0 3 77
2 72.0 1 7
2 75.0 3 3
2 77.1 2 4
2 78.0 1 1
2 80.0 2 2
2 90.0 2 2
2 100.0 1 1
2 105.0 1 1
2 117.0 1 1
2 135.0 2 2
3 15.0 6 7
3 20.0 1 1
3 22.5 3 4
3 25.7 1 1
3 30.0 22 28
3 34.3 5 6
3 37.5 1 1
3 40.0 1 1
3 45.0 3 3
3 60.0 1 1
3 90.0 9 10
3 108.0 1 1
3 135.0 2 2
3 150.0 2 2
3 195.0 1 1
4 18.0 1 2
4 22.5 2 3
4 24.0 1 1
4 30.0 3 3
4 31.5 1 1
4 45.0 38 58
4 54.0 1 1
4 60.0 13 13
4 64.3 5 6
4 67.5 4 4
4 75.0 2 2
4 120.0 10 15
4 195.0 1 1
4 207.0 1 1
5 36.0 13 73
5 48.0 1 1
5 72.0 8 30
6 15.0 1 1
6 20.0 1 1
6 22.5 1 3
6 30.0 12 13
6 40.0 1 1
6 60.0 97 105
6 72.0 4 4
6 73.3 1 1
6 75.0 4 4
6 80.0 1 1
6 84.0 1 1
6 85.0 1 1
6 90.0 10 10
6 94.3 4 4
6 105.0 1 1
6 120.0 1 1
6 135.0 1 1
6 150.0 1 1
6 186.0 1 1
7 77.1 2 2
8 15.0 3 3
8 18.0 1 1
8 45.0 52 75
8 60.0 1 1
8 75.0 1 1
8 76.5 1 1
8 90.0 216 547
8 109.3 1 1
8 112.5 6 6
8 120.0 1 1
8 193.5 1 1
9 70.0 2 2
9 80.0 5 5
9 105.0 1 1
9 110.0 1 1
9 110.0 1 1
10 36.0 2 2
10 72.0 23 30
10 72.0 1 1
10 108.0 31 62
12 15.0 1 1
12 30.0 4 4
12 45.0 1 1
12 60.0 69 79
12 70.0 2 2
12 72.0 1 1
12 75.0 4 4
12 84.0 1 1
12 90.0 18 19
12 97.5 1 1
12 100.0 2 2
12 105.0 4 4
12 120.0 5 5
12 124.3 1 1
12 127.5 1 1
14 51.4 4 4
14 77.1 1 1
14 102.9 1 1
16 45.0 27 29
16 59.0 1 1
16 67.5 1 1
16 75.0 1 1
16 90.0 1 1
18 40.0 1 1
18 44.0 1 1
18 60.0 1 1
18 80.0 1 1
20 36.0 1 1
20 54.0 1 1
24 30.0 1 1
24 45.0 1 1

6  The angles of the tiling

Angle Number
- 72
0.1 2
0.3 1
0.5 1
1.0 1
1.5 3
2.5 1
3.0 3
4.5 2
5.0 7
6.0 3
7.5 34
9.0 4
10.0 6
12.0 8
15.0 136
18.0 13
20.0 4
22.5 94
30.0 173
36.0 109
40.0 1
45.0 340
60.0 99
72.0 2
90.0 192
120.0 8

7  Does the pattern satisfy the two polygon condition?

Property Number
False 1142
True 177

8  The interlace condition

Finite interlaces Infinite interlaces Total
0 0 789
0 1 90
0 2 109
0 3 31
0 4 4
1 0 78
1 1 66
1 2 28
1 3 1
2 0 47
2 1 27
2 2 5
2 5 1
3 0 13
3 1 5
3 2 4
4 0 7
4 1 1
4 2 1
5 0 4
5 1 3
5 4 1
6 0 2
8 2 1
15 3 1

9  Polygonal tiles used

This excludes the regular polygons and star polygons.
Reflective tilesReflective pairs No mirror image Number
0 0 0 115
0 0 1 92
0 0 2 38
0 0 3 5
0 0 4 2
0 0 6 1
0 0 8 34
0 1 0 81
1 0 0 278
1 0 1 15
1 0 2 1
1 1 0 18
2 0 0 128
2 0 1 3
2 0 2 2
2 0 6 1
2 1 0 17
2 1 1 1
2 2 0 1
2 6 0 1
3 0 0 149
3 0 1 1
3 0 2 1
3 1 0 11
3 3 0 1
3 4 0 1
3 5 0 1
4 0 0 70
4 1 0 6
4 2 0 3
5 0 0 49
5 1 0 10
5 2 0 2
5 3 0 1
6 0 0 36
6 1 0 10
6 2 0 1
6 4 0 1
7 0 0 27
7 1 0 4
7 2 0 2
8 0 0 20
8 1 0 11
8 2 0 2
8 3 0 1
9 0 0 10
9 1 0 5
9 2 0 3
10 0 0 4
10 1 0 1
10 4 0 1
11 0 0 2
11 1 0 3
11 2 0 2
12 0 0 3
12 1 0 1
12 2 0 2
13 0 0 8
13 1 0 1
13 2 0 5
13 4 0 1
14 0 0 1
14 1 0 1
15 1 0 4
16 0 0 1
17 2 0 2
18 1 0 1
19 1 0 1
19 2 0 1

10  Edge-to-edge property

Property Number
False 318
True 1001

11  Publications

Publication Number
abas 141
akber 18
aslanapa 9
backhouse 10
bain 5
bourgoin 147
briggs 10
burckhardt 3
cahier 64
calvert 19
calvert2 1
castera 40
clevenot 15
copple 1
creswell 12
critchlow 23
d-avennes 22
dawes 176
degeorge 36
dury 3
dye 121
elsaid 48
elsaid2 4
escher 1
fernandez 14
field1 10
field2 14
field4 23
gailiunas 9
gands 68
gands2 2
glassner 1
gluck 1
golomb1 29
golomb2 3
grafton 27
grube 1
hankin2 1
hautex 1
herzfeld1 1
hessemer 53
hill 28
hill2 8
hrbas 2
humbert 5
iran 50
jones 50
jones2 1
klarner 3
lee 16
maldonado 3
marshall 17
martin 1
murphy 5
myers 45
neal 5
necipoglu 17
ogel 1
orton 1
paccard 63
pc 79
pope 20
pope2 1
racinet 17
reid 4
rempel 1
rice 1
rigby1 55
scerrato 6
schatt1 2
seherr 5
smith1 1
stevens 22
stierlin 3
stock 6
stronge 17
useinov 2
volwah 1
wade 58
wahhab 13
wich2 115
wich3 2
wilber 2
wilson 11

12  Islamic Tilings

Those tilings which are referenced at least once in books about Islamic art can be counted as Islamic patterns. There are 519 of these.

13  Version records

Version Date TilingsComment
8 2008-09-29 1319See latest
7 2008-06-22 1190Tree search and Conway-Thurston notation
6 2008-05-05 1178Tilings from Stevens
5 2008-03-31 1153Islamic tilings from DeGeorge
4 2008-02-23 1130Spiral tilings added
3 2007-12-27 1106Further Islamic tilings added
2 2007-11-05 1085First version on Internet
1 2007-10-06 1076Islamic tiling added
0 2007-08-26 1050Initial system