Some statistics: Version 22

Brian Wichmann

1  Introduction

This page gives some statistics concerning this release of the tiling system, having 1941 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) 175
*X (cm) The tiling has no rotation symmetry, but one glide reflection and one reflection 10
2*22 (cmm) The tiling has a rotation symmetry of order two, two glide reflections and two reflections 162
O (p1) The tiling has no rotations, reflections or glide reflections 5
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 9
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 17
442 (p4) The tiling has rotations of order four, but no reflections or glide reflections 90
4*2 (p4g) The tiling has a rotation of order two another of order four, two glide reflections 110
*442 (p4m) The tiling has a rotation of order four, and is its own mirror image 665
632 (p6) The tiling has rotations of order six, but no reflections or glide reflections 85
*632 (p6m) The tiling has a rotation of order six, and is its own mirror image 376
XX (pg) The tiling has two glide reflections 19
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 16
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 99
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 254
Cannot be coloured with two colours 601
Can be coloured with two colours 1086
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 193 522
square 492 1158
regular pentagon 147 1124
regular hexagon 208 329
regular heptagon 9 16
regular octagon 136 189
regular enneagon 5 5
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 12 32
2 30.0 16 24
2 34.3 1 1
2 36.0 8 12
2 45.0 100 516
2 48.0 1 1
2 50.0 1 1
2 51.4 3 7
2 51.4 1 1
2 52.5 1 3
2 58.5 1 1
2 60.0 114 151
2 63.0 1 1
2 66.0 1 1
2 70.0 2 2
2 72.0 1 7
2 72.0 43 471
2 75.0 4 10
2 77.1 6 19
2 78.0 2 3
2 80.0 5 7
2 90.0 2 2
2 100.0 1 1
2 105.0 1 1
2 108.0 1 20
2 117.0 1 1
2 135.0 3 14
3 15.0 7 8
3 18.0 4 4
3 20.0 2 2
3 22.0 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 4 4
3 60.0 1 1
3 90.0 15 31
3 100.0 1 1
3 102.0 1 1
3 105.0 2 2
3 108.0 1 1
3 112.5 2 2
3 135.0 3 3
3 150.0 2 2
4 (undef) 1 1
4 18.0 2 3
4 22.0 2 3
4 24.0 1 1
4 30.0 8 8
4 31.5 1 1
4 36.0 1 1
4 45.0 58 90
4 48.0 1 1
4 51.4 1 3
4 52.5 1 1
4 54.0 2 2
4 60.0 22 23
4 63.0 1 1
4 64.3 5 6
4 67.5 5 5
4 70.0 1 1
4 75.0 3 3
4 90.0 1 1
4 120.0 15 20
4 126.0 4 5
5 36.0 27 198
5 48.0 1 1
5 72.0 15 54
6 (undef) 5 6
6 15.0 1 1
6 18.0 1 1
6 20.0 1 1
6 22.0 1 3
6 30.0 17 18
6 36.0 1 1
6 40.0 2 2
6 45.0 1 1
6 48.0 3 3
6 60.0 145 157
6 72.0 7 7
6 73.3 1 1
6 75.0 5 5
6 76.0 1 1
6 78.0 1 1
6 78.8 1 1
6 80.0 3 3
6 84.0 1 1
6 85.0 2 2
6 90.0 21 22
6 94.3 4 4
6 95.0 1 1
6 105.0 1 1
6 108.0 1 1
6 114.0 1 1
6 120.0 1 1
6 135.0 1 1
6 150.0 1 1
7 (undef) 1 1
7 77.1 7 8
8 (undef) 3 3
8 15.0 9 10
8 18.0 1 1
8 45.0 90 168
8 50.0 1 1
8 52.5 1 1
8 60.0 2 2
8 65.0 1 1
8 67.5 1 2
8 70.0 1 1
8 71.3 3 3
8 72.0 3 3
8 73.1 1 1
8 75.0 3 3
8 76.5 1 1
8 80.0 1 1
8 90.0 370 999
8 100.0 2 4
8 105.0 3 3
8 109.3 1 1
8 111.0 1 1
8 112.5 8 8
8 120.0 2 2
8 121.5 1 1
9 (undef) 1 1
9 20.0 4 4
9 32.0 1 1
9 40.0 2 2
9 70.0 3 3
9 72.0 1 1
9 72.5 1 1
9 80.0 10 10
9 92.0 1 1
9 100.0 1 1
9 105.0 1 1
9 110.0 3 3
9 120.0 1 1
10 (undef) 2 2
10 36.0 2 2
10 72.0 57 90
10 85.5 1 1
10 98.0 1 1
10 108.0 53 116
11 (undef) 1 1
12 (undef) 2 2
12 15.0 1 1
12 30.0 16 16
12 45.0 1 1
12 51.0 1 1
12 52.5 3 3
12 60.0 99 113
12 65.0 2 2
12 66.0 1 1
12 70.0 3 3
12 71.3 2 2
12 72.0 5 5
12 72.5 1 1
12 75.0 10 10
12 80.0 3 3
12 84.0 1 1
12 90.0 29 30
12 97.5 1 1
12 100.0 2 2
12 105.0 5 5
12 120.0 5 5
12 124.3 1 1
12 127.5 1 1
13 (undef) 1 1
14 51.4 5 5
14 70.7 1 1
14 77.1 2 2
14 102.9 2 2
15 51.0 1 1
16 (undef) 3 3
16 45.0 47 48
16 52.5 2 2
16 58.5 1 1
16 59.0 1 1
16 62.5 2 2
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 4 4
20 36.0 2 2
20 54.0 1 1
20 60.0 1 1
24 (undef) 1 1
24 30.0 2 2
24 45.0 2 2

6  The angles of the tiling

Angle Number
- 85
0.15 1
0.23 1
0.38 1
0.50 3
1.00 4
1.25 6
1.50 6
1.67 1
1.88 3
2.00 11
2.50 13
2.81 1
3.00 6
3.21 2
3.75 9
4.29 4
4.50 2
5.00 24
5.63 1
6.00 13
6.43 1
7.50 47
8.57 4
9.00 9
10.00 10
11.25 10
12.00 10
12.86 3
15.00 175
18.00 24
20.00 13
22.50 163
25.71 16
30.00 250
36.00 157
40.00 1
45.00 476
60.00 137
72.00 2
90.00 215
120.00 21

7  Does the pattern satisfy the two polygon condition?

Property Number
False 1763
True 178

8  The interlace condition

Finite interlaces Infinite interlaces Total
0 0 1086
0 1 166
0 2 139
0 3 37
0 4 6
0 5 2
1 0 105
1 1 96
1 2 35
1 3 3
1 4 2
2 0 80
2 1 41
2 2 7
2 5 1
3 0 29
3 1 6
3 2 4
3 3 50
4 0 12
4 1 2
5 0 16
5 1 3
5 4 1
6 0 7
7 0 2
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 0 235
0 0 1 111
0 0 2 47
0 0 3 6
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 0 311
1 0 1 21
1 0 2 6
1 0 4 1
1 1 0 26
2 0 0 180
2 0 1 3
2 0 2 2
2 0 6 1
2 1 0 22
2 1 1 1
2 2 0 4
2 3 0 1
2 4 0 1
2 6 0 1
3 0 0 217
3 0 1 1
3 0 2 5
3 0 4 1
3 0 5 1
3 1 0 14
3 2 0 1
3 3 0 2
3 4 0 1
3 5 0 2
4 0 0 119
4 0 2 1
4 0 5 2
4 1 0 11
4 2 0 6
5 0 0 84
5 0 2 1
5 1 0 15
5 2 0 2
5 3 0 1
6 0 0 63
6 1 0 15
6 2 0 2
6 4 0 1
7 0 0 53
7 0 2 2
7 1 0 10
7 2 0 3
7 4 0 1
8 0 0 33
8 1 0 15
8 2 0 3
8 3 0 2
9 0 0 22
9 0 2 1
9 0 5 1
9 1 0 8
9 2 0 3
10 0 0 9
10 1 0 6
10 2 0 1
10 4 0 3
11 0 0 8
11 1 0 8
11 2 0 3
12 0 0 11
12 1 0 4
12 2 0 4
13 0 0 8
13 0 2 1
13 1 0 3
13 2 0 5
13 4 0 1
14 0 0 4
14 1 0 3
15 1 0 5
15 2 0 2
16 0 0 5
16 1 0 1
17 0 0 3
17 1 0 1
17 2 0 2
17 3 0 1
18 0 0 1
18 1 0 2
19 1 0 1
19 2 0 1
21 0 13 1
21 2 0 1
23 1 0 1

10  Edge-to-edge property

Property Number
False 429
True 1512

11  Publications

Publication Number
abas 179
akber 19
aslanapa 23
backhouse 10
bain 5
bakirer 1
barry 2
berchem 1
betts 1
blair 3
bour0 5
bourgoin 208
briggs 12
broug 15
burckhardt 6
cahier 66
calvert 17
calvert2 1
castera 44
clevenot 15
copple 1
creswell 16
critchlow 24
d-avennes 36
dawes 177
degeorge 39
denny 2
dury 3
dye 121
elsaid 49
elsaid2 4
erdmann 1
escher 2
ex1995 7
fernandez 16
field1 10
field2 14
field4 23
frettloeh 19
gailiunas 9
gands 79
gands2 2
gink 1
glassner 1
gluck 1
golomb1 29
golomb2 3
gomez 1
grafton 27
grube 2
guy 9
hankin1 1
hankin2 1
hattstein 3
hedgecoe 1
herzfeld1 1
herzfeld2 2
hessemer 54
hill 42
hill2 10
hrbas 2
humbert 5
hutt 2
iran 112
jones 51
jones2 1
klarner 3
landau 2
lee 16
lings 1
makov 4
marshall 17
martin 1
meinecke 1
migeon 2
muller 1
murphy 6
myers 45
neal 5
necipoglu 20
ogel 3
orazi 4
orton 1
otto 1
paccard 88
pajares 25
pavon 10
pc 360
pope 23
pope2 1
racinet 17
ray 13
reid 4
rempel 20
rice 1
rigby1 55
rogers 2
scerrato 6
schatt1 2
seherr 17
siculo 1
singer 7
smith1 1
smith2 16
stevens 22
stierlin 3
stierlin2 1
stock 6
stronge 18
sutton 7
useinov 3
viollet 1
volwah 3
wade 59
wahhab 23
wich2 121
wich3 2
wilber 4
williams 1
wilson 11
wurfel 1

12  Islamic Tilings

Those tilings which are referenced at least once in books about Islamic art can be counted as Islamic patterns. There are 1024 of these.

13  Photographic links

There are 464 tiling patterns whose records link to a photograph. The total number of links to photographs is 649.

14  Version records

Version Date TilingsComment
22 2011-12-17 1941See.
21 2011-09-19 1908See.
20 2011-06-21 1868See.
19 2011-02-28 1829See.
18 2010-11-15 1771See.
17 2010-08-14 1727See.
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