Some statistics: Version 16

Brian Wichmann

1  Introduction

This page gives some statistics concerning this release of the tiling system, having 1695 tilings.
Note that the numerical counts in the tables are actually hypertext links which give a single instance of a pattern having that characteristic.
The statistics are presented in the same order as the comprehensive search HTML form.

2  Symmetry group

Symmetry Description Number
(none) 165
*X (cm) The tiling has no rotation symmetry, but one glide reflection and one reflection 9
2*22 (cmm) The tiling has a rotation symmetry of order two, two glide reflections and two reflections 133
O (p1) The tiling has no rotations, reflections or glide reflections 3
2222 (p2) The tiling has rotations of order two, but no reflections or glide reflections 29
333 (p3) The tiling has rotations of order three, but no reflections or glide reflections 8
3*3 (p31m)The tiling has two rotations of order three, a glide reflection and a relections 35
*333 (p3m1)The tiling has one rotation of order three, a glide reflection and a relections 14
442 (p4) The tiling has rotations of order four, but no reflections or glide reflections 80
4*2 (p4g) The tiling has a rotation of order two another of order four, two glide reflections 92
*442 (p4m) The tiling has a rotation of order four, and is its own mirror image 569
632 (p6) The tiling has rotations of order six, but no reflections or glide reflections 75
*632 (p6m) The tiling has a rotation of order six, and is its own mirror image 331
XX (pg) The tiling has two glide reflections 16
22X (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 15
22* (pmg) The tiling has two rotations of order two, one glide reflection and one reflection 24
*2222 (pmm) The tiling has four rotations of order two and four reflections 82
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 223
Cannot be coloured with two colours 548
Can be coloured with two colours 924
Most of the cases in 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 176 485
square 441 1021
regular pentagon 122 860
regular hexagon 172 292
regular heptagon 5 8
regular octagon 108 156
regular enneagon 4 4
regular decagon 3 7
12-gon 10 10
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 (undef) 1 1
2 15.0 1 1
2 18.0 4 6
2 22.5 2 3
2 25.7 10 23
2 30.0 15 21
2 34.3 1 1
2 36.0 7 11
2 45.0 89 459
2 48.0 1 1
2 50.0 1 1
2 51.4 1 1
2 52.5 1 3
2 58.5 1 1
2 60.0 102 129
2 66.0 1 1
2 72.0 25 302
2 72.0 3 77
2 72.0 1 7
2 75.0 4 10
2 77.1 4 10
2 78.0 1 1
2 80.0 3 3
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 (undef) 1 1
3 15.0 7 8
3 18.0 4 4
3 20.0 2 2
3 22.5 3 4
3 25.7 1 1
3 30.0 26 33
3 34.3 5 6
3 37.5 1 1
3 40.0 3 3
3 45.0 3 3
3 60.0 2 2
3 90.0 13 18
3 102.9 1 1
3 105.0 1 1
3 108.0 1 1
3 112.5 1 1
3 135.0 2 2
3 150.0 2 2
3 195.0 1 1
4 (undef) 1 1
4 18.0 2 3
4 22.5 2 3
4 24.0 1 1
4 30.0 7 7
4 31.5 1 1
4 45.0 53 88
4 48.0 1 1
4 51.4 1 3
4 54.0 1 1
4 60.0 21 22
4 64.3 5 6
4 67.5 4 4
4 75.0 2 2
4 90.0 1 1
4 120.0 11 16
4 126.0 4 5
4 195.0 1 1
4 207.0 1 1
5 36.0 17 123
5 48.0 1 1
5 72.0 11 39
6 (undef) 5 6
6 15.0 1 1
6 18.0 1 1
6 20.0 1 1
6 22.5 1 3
6 30.0 17 18
6 40.0 1 1
6 48.0 3 3
6 60.0 122 131
6 72.0 5 5
6 73.3 1 1
6 75.0 5 5
6 78.0 1 1
6 80.0 3 3
6 84.0 1 1
6 85.0 2 2
6 90.0 15 15
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 (undef) 1 1
7 77.1 4 4
8 (undef) 3 3
8 15.0 9 10
8 18.0 1 1
8 45.0 76 146
8 50.0 1 1
8 52.5 1 1
8 60.0 2 2
8 70.0 1 1
8 71.3 2 2
8 72.0 3 3
8 73.1 1 1
8 75.0 2 2
8 76.5 1 1
8 90.0 299 836
8 105.0 1 1
8 108.0 1 1
8 109.3 1 1
8 112.5 6 6
8 120.0 1 1
8 193.5 1 1
9 20.0 4 4
9 32.0 1 1
9 70.0 2 2
9 72.0 1 1
9 72.5 1 1
9 80.0 6 6
9 100.0 1 1
9 105.0 1 1
9 110.0 1 1
9 110.0 1 1
9 120.0 1 1
10 (undef) 2 2
10 36.0 2 2
10 72.0 35 52
10 72.0 1 1
10 108.0 40 79
12 (undef) 5 5
12 15.0 1 1
12 30.0 15 15
12 45.0 1 1
12 51.0 1 1
12 52.5 2 2
12 60.0 82 92
12 65.0 1 1
12 68.0 1 1
12 70.0 2 2
12 71.3 2 2
12 72.0 4 4
12 72.5 1 1
12 75.0 5 5
12 80.0 1 1
12 84.0 1 1
12 90.0 25 26
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 5 5
14 70.7 1 1
14 77.1 2 2
14 102.9 1 1
14 102.9 1 1
15 51.0 1 1
16 (undef) 3 3
16 45.0 38 40
16 52.5 2 2
16 58.5 1 1
16 59.0 1 1
16 67.5 1 1
16 72.0 1 1
16 73.1 1 1
16 75.0 1 1
16 90.0 2 2
18 40.0 3 3
18 44.0 1 1
18 60.0 1 1
18 80.0 2 2
20 36.0 1 1
20 54.0 1 1
20 60.0 1 1
24 (undef) 1 1
24 30.0 1 1
24 45.0 1 1

6  The angles of the tiling

Angle Number
- 138
0.1 3
0.3 1
0.5 3
1.0 3
1.5 4
2.0 2
2.5 7
3.0 5
4.5 2
5.0 11
6.0 10
7.5 44
9.0 8
10.0 9
12.0 9
15.0 154
18.0 18
20.0 10
22.5 129
30.0 225
36.0 127
40.0 1
45.0 420
60.0 117
72.0 2
90.0 212
120.0 21

7  Does the pattern satisfy the two polygon condition?

Property Number
False 1517
True 178

8  The interlace condition

Finite interlaces Infinite interlaces Total
0 0 989
0 1 134
0 2 123
0 3 32
0 4 6
1 0 89
1 1 78
1 2 30
1 3 2
1 4 2
2 0 62
2 1 32
2 2 5
2 5 1
3 0 22
3 1 6
3 2 4
3 3 34
4 0 12
4 1 2
5 0 15
5 1 3
5 4 1
6 0 7
7 0 1
8 2 1
10 0 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 0 232
0 0 1 97
0 0 2 44
0 0 3 7
0 0 4 2
0 0 6 1
0 0 7 2
0 0 8 35
0 0 9 2
0 0 11 2
0 1 0 82
1 0 0 297
1 0 1 18
1 0 2 1
1 1 0 22
2 0 0 159
2 0 1 3
2 0 2 2
2 0 6 1
2 1 0 21
2 1 1 1
2 2 0 4
2 3 0 1
2 4 0 1
2 6 0 1
3 0 0 182
3 0 1 1
3 0 2 1
3 1 0 13
3 3 0 2
3 4 0 1
3 5 0 2
4 0 0 92
4 1 0 10
4 2 0 6
5 0 0 65
5 0 2 1
5 1 0 15
5 2 0 2
5 3 0 1
6 0 0 44
6 1 0 13
6 2 0 1
6 4 0 1
7 0 0 38
7 0 2 1
7 1 0 5
7 2 0 2
7 4 0 1
8 0 0 30
8 1 0 11
8 2 0 3
8 3 0 2
9 0 0 16
9 0 2 1
9 1 0 7
9 2 0 2
10 0 0 6
10 1 0 1
10 4 0 3
11 0 0 7
11 1 0 7
11 2 0 3
12 0 0 6
12 1 0 4
12 2 0 3
13 0 0 8
13 1 0 4
13 2 0 5
13 4 0 1
14 0 0 2
14 1 0 2
15 0 0 1
15 1 0 5
15 2 0 2
16 0 0 4
16 1 0 1
17 0 0 2
17 2 0 2
17 3 0 1
18 0 0 1
18 1 0 1
19 1 0 1
19 2 0 1
24 1 0 1

10  Edge-to-edge property

Property Number
False 401
True 1294

11  Publications

Publication Number
abas 179
akber 18
aslanapa 13
backhouse 10
bain 5
bakirer 1
berchem 1
bourgoin 203
briggs 11
burckhardt 6
cahier 66
calvert 18
calvert2 1
castera 40
clevenot 16
copple 1
creswell 12
critchlow 24
d-avennes 36
dawes 177
degeorge 37
dury 3
dye 121
elsaid 50
elsaid2 4
escher 2
ex1995 7
fernandez 15
field1 10
field2 14
field4 23
frettloeh 19
gailiunas 9
gands 80
gands2 2
glassner 1
gluck 1
golomb1 29
golomb2 3
grafton 27
grube 1
guy 9
hankin1 1
hankin2 1
herzfeld1 1
hessemer 53
hill 31
hill2 9
hrbas 2
humbert 5
iran 94
jones 51
jones2 1
klarner 3
landau 2
lee 16
makov 4
marshall 17
martin 1
meinecke 1
migeon 2
muller 1
murphy 6
myers 45
neal 5
necipoglu 17
ogel 2
orazi 4
orton 1
paccard 71
pajares 25
pavon 6
pc 306
pope 21
pope2 1
racinet 17
reid 4
rempel 1
rice 1
rigby1 55
rogers 1
scerrato 6
schatt1 2
seherr 12
smith1 1
stevens 22
stierlin 3
stock 6
stronge 17
useinov 3
volwah 2
wade 58
wahhab 18
wich2 121
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 811 of these.

13  Photographic links

There are 276 tiling patterns whose records link to a photograph. The total number of links to photographs is 368.

14  Version records

Version Date TilingsComment
16 2010-05-07 1695See.
15 2010-01-29 1646See.
14 2009-12-09 1601See.
13 2009-10-09 1563See.
12 2009-06-20 1499See.
11 2009-03-05 1442See.
10 2009-01-03 1403See.
9 2008-11-16 1353See.
8 2008-09-29 1319See.
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