Some statistics: Version 52

1  Introduction

This page gives some statistics concerning this release of the tiling system, having 2862 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

 The symmetry group of the tiling is *442 (p4m) 1025 The symmetry group of the tiling is 2*22 (cmm) 293 The symmetry group of the tiling is *632 (p6m) 522 The symmetry group of the tiling is *4• (d4) 16 The symmetry group of the tiling is *10• (d10) 12 The symmetry group of the tiling is *333 (p3m1) 21 The symmetry group of the tiling is *8• (d8) 17 The symmetry group of the tiling is *12• (d12) 5 The symmetry group of the tiling is 442 (p4) 124 The symmetry group of the tiling is 3*3 (p31m) 35 The symmetry group of the tiling is *6• (d6) 6 The symmetry group of the tiling is 6• (c6) 19 The symmetry group of the tiling is *5• (d5) 13 The symmetry group of the tiling is *2222 (pmm) 184 The symmetry group of the tiling is 22X (pgg) 17 The symmetry group of the tiling is 4*2 (p4g) 143 The symmetry group of the tiling is ** (pm) 21 The symmetry group of the tiling is 632 (p6) 98 The symmetry group of the tiling is *X (cm) 13 The symmetry group of the tiling is 22* (pmg) 28 The symmetry group of the tiling is 333 (p3) 9 The symmetry group of the tiling is *2• (d2) 13 The symmetry group of the tiling is 4• (c4) 32 The symmetry group of the tiling is 2222 (p2) 34 The symmetry group of the tiling is not symmetric and hence is not a repeat pattern 82 The symmetry group of the tiling is 2• (c2) 34 The symmetry group of the tiling is 5• (c5) 5 The symmetry group of the tiling is 3• (c3) 3 The symmetry group of the tiling is XX (pg) 18 The symmetry group of the tiling is O (p1) 7 The symmetry group of the tiling is 7• (c7) 1 The symmetry group of the tiling is 12• (c12) 1 The symmetry group of the tiling is *22∞ (p2mm) 6 The symmetry group of the tiling is *3• (d3) 1 The symmetry group of the tiling is 2*∞ (pmg) 2 The symmetry group of the tiling is *16• (d16) 1 The symmetry group of the tiling is *1• (d1) 1
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 452 Cannot be coloured with two colours 845 Can be coloured with two colours 263 Can be coloured with two colours (straight cross-overs) 1302
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 Tilings Total equilateral triangle 252 722 square 776 1914 regular pentagon 265 3398 regular hexagon 316 495 regular heptagon 28 87 regular octagon 220 285 regular enneagon 8 8 regular decagon 6 10 12-gon 12 12 16-gon 2 2 18-gon 1 1 24-gon 1 1

5  Tilings containing regular star polygons

 Points Vertex angle Tiling count Total 2 (undef) 2 11 2 0.0 2 2 2 15.0 1 1 2 18.0 4 6 2 22.5 4 8 2 25.7 12 30 2 30.0 21 37 2 34.3 1 1 2 36.0 11 23 2 40.0 1 1 2 45.0 197 890 2 48.0 1 1 2 50.0 2 2 2 51.4 4 8 2 52.5 1 2 2 53.1 1 1 2 55.5 1 1 2 58.5 1 1 2 60.0 178 289 2 63.0 2 2 2 66.0 1 1 2 67.5 2 8 2 70.0 2 2 2 70.7 1 7 2 72.0 110 1072 2 73.1 2 16 2 75.0 8 13 2 77.1 10 18 2 78.0 2 2 2 80.0 6 7 2 82.5 1 1 2 87.4 1 1 2 99.2 1 1 3 15.0 7 8 3 18.0 4 4 3 20.0 3 3 3 22.0 3 4 3 25.7 1 1 3 30.0 27 34 3 34.3 5 6 3 37.5 1 1 3 40.0 3 3 3 45.0 4 4 3 60.0 9 9 3 80.0 1 1 3 90.0 36 42 3 100.0 1 1 3 102.0 1 1 3 105.0 12 12 3 108.0 1 1 3 112.5 3 3 3 120.0 1 1 3 150.0 3 4 3 165.0 1 1 4 0.0 1 1 4 18.0 2 3 4 22.0 2 3 4 24.0 1 1 4 30.0 13 15 4 31.5 1 1 4 36.0 1 1 4 40.0 1 1 4 45.0 81 103 4 48.0 1 1 4 51.4 1 1 4 52.5 1 1 4 54.0 5 5 4 56.3 1 1 4 60.0 42 44 4 63.0 1 1 4 64.3 5 6 4 65.0 1 1 4 67.5 7 7 4 68.0 1 1 4 70.0 2 2 4 75.0 3 3 4 90.0 4 4 4 98.0 1 1 4 120.0 28 31 4 126.0 3 3 4 135.0 10 10 5 (undef) 1 1 5 36.0 64 331 5 48.0 1 1 5 72.0 28 61 5 108.0 2 2 6 (undef) 1 1 6 0.0 2 2 6 15.0 1 1 6 18.0 1 1 6 20.0 1 1 6 22.0 1 3 6 30.0 19 20 6 36.0 1 1 6 40.0 3 3 6 45.0 2 2 6 48.0 3 3 6 60.0 230 264 6 65.0 1 1 6 72.0 7 7 6 73.3 1 1 6 75.0 9 9 6 76.0 1 1 6 77.1 1 1 6 78.0 2 2 6 78.8 1 1 6 80.0 4 4 6 84.0 1 1 6 85.0 3 3 6 90.0 51 51 6 94.3 4 4 6 95.0 1 1 6 100.0 3 3 6 102.9 1 1 6 105.0 3 3 6 108.0 2 2 6 114.0 1 1 6 120.0 14 14 6 135.0 1 1 6 150.0 2 2 7 (undef) 3 4 7 0.0 12 12 7 77.1 11 12 7 92.6 1 2 7 102.9 2 2 8 (undef) 2 2 8 0.0 11 11 8 15.0 9 10 8 18.0 1 1 8 25.0 1 1 8 35.0 1 1 8 45.0 159 235 8 50.0 1 1 8 52.5 1 1 8 55.0 1 1 8 60.0 2 2 8 63.0 1 1 8 65.0 2 2 8 67.5 4 5 8 69.0 1 1 8 70.0 3 3 8 71.3 3 3 8 72.0 6 7 8 73.1 3 3 8 75.0 8 8 8 76.5 1 1 8 80.0 2 2 8 82.0 1 1 8 90.0 580 1479 8 100.0 2 4 8 105.0 9 9 8 108.0 2 2 8 109.3 1 1 8 111.0 1 1 8 112.5 8 8 8 115.0 1 1 8 117.0 1 1 8 120.0 5 5 8 121.5 1 1 8 135.0 3 3 9 0.0 7 7 9 20.0 3 3 9 30.0 1 1 9 32.0 1 1 9 40.0 5 5 9 70.0 3 3 9 72.0 1 1 9 72.5 2 2 9 80.0 13 13 9 92.0 1 1 9 100.0 2 2 9 105.0 1 1 9 110.0 3 3 9 120.0 1 1 10 (undef) 2 2 10 0.0 3 3 10 36.0 5 5 10 54.0 1 1 10 72.0 140 250 10 85.5 1 1 10 90.0 2 2 10 98.0 1 1 10 108.0 98 309 10 126.0 1 1 11 (undef) 1 1 11 0.0 4 4 11 70.0 1 1 12 (undef) 2 2 12 0.0 10 10 12 15.0 1 1 12 30.0 20 20 12 45.0 1 1 12 51.0 1 1 12 52.5 3 3 12 60.0 151 186 12 65.0 3 3 12 66.0 2 2 12 67.5 1 1 12 70.0 5 5 12 71.3 2 2 12 72.0 8 8 12 72.5 2 2 12 75.0 15 15 12 78.0 1 1 12 80.0 8 8 12 82.5 1 1 12 84.0 1 1 12 85.0 1 1 12 90.0 55 56 12 97.5 4 4 12 100.0 2 2 12 105.0 10 10 12 120.0 8 8 12 124.3 1 1 12 127.5 1 1 13 0.0 1 1 13 90.0 1 1 14 0.0 1 1 14 51.4 8 8 14 70.7 3 3 14 77.1 4 4 14 102.9 16 19 15 51.0 2 2 16 0.0 4 4 16 22.5 1 1 16 45.0 86 88 16 52.5 4 4 16 58.5 1 1 16 59.0 1 1 16 60.0 2 2 16 62.5 1 1 16 67.5 4 4 16 73.1 3 3 16 75.0 1 1 16 80.0 1 1 16 90.0 6 6 16 100.0 1 1 18 40.0 2 2 18 44.0 1 1 18 60.0 2 2 18 80.0 4 4 20 (undef) 2 2 20 0.0 1 1 20 36.0 7 7 20 54.0 1 1 20 60.0 1 1 24 (undef) 1 1 24 0.0 5 5 24 30.0 13 13 24 40.0 1 1 24 45.0 3 3 32 0.0 1 1 32 22.5 1 1 48 0.0 4 4

6  The angles of the tiling

 Angle Number - 103 0.38 1 0.50 7 1.00 7 1.07 1 1.25 8 1.50 8 1.67 1 1.88 3 2.00 11 2.14 1 2.50 28 2.81 3 3.00 10 3.21 4 3.75 17 4.00 1 4.29 4 4.50 11 5.00 40 5.63 1 6.00 14 6.43 2 7.50 97 8.57 4 9.00 14 10.00 14 11.25 21 12.00 11 12.86 7 15.00 259 18.00 38 20.00 20 22.50 276 25.71 34 30.00 341 36.00 288 45.00 666 60.00 210 90.00 259 120.00 17

7  Does the pattern satisfy the two polygon condition?

 Property Number False 2656 True 184

8  The interlace condition

 Finite interlaces Infinite interlaces Total -1 0 70 0 0 1480 0 1 233 0 2 173 0 3 44 0 4 22 0 5 5 0 6 3 0 8 3 1 0 135 1 1 155 1 2 55 1 3 8 1 4 1 1 5 3 1 7 1 2 0 138 2 1 71 2 2 16 2 3 9 2 4 1 2 5 1 3 0 72 3 1 23 3 2 5 4 0 25 4 1 18 4 2 10 4 3 2 4 4 1 5 0 17 5 1 11 5 2 2 5 3 1 5 9 1 6 0 10 6 1 7 7 0 9 7 1 2 7 4 1 8 0 6 8 1 1 8 2 2 9 0 3 10 1 1 12 0 1 12 2 1 13 0 1 15 3 2

9  Polygonal tiles used

This excludes the regular polygons and star polygons.
 Reflective tiles Reflective pairs No mirror image Number 0 0 0 239 0 0 1 154 0 0 2 57 0 0 3 11 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 0 101 1 0 0 420 1 0 1 21 1 0 2 1 1 0 4 2 1 0 6 1 1 1 0 45 1 2 0 2 1 3 0 3 2 0 0 280 2 0 1 5 2 0 2 3 2 0 3 1 2 0 6 1 2 1 0 26 2 1 1 1 2 2 0 4 2 3 0 1 2 4 0 2 2 5 0 2 2 6 0 1 3 0 0 320 3 0 1 2 3 0 2 3 3 0 5 2 3 1 0 29 3 2 0 3 3 3 0 3 3 4 0 1 3 5 0 2 4 0 0 217 4 0 5 3 4 1 0 19 4 2 0 7 4 3 0 2 4 6 0 1 5 0 0 158 5 0 2 1 5 1 0 26 5 2 0 3 5 3 0 2 6 0 0 103 6 1 0 28 6 2 0 4 6 4 0 1 7 0 0 88 7 0 2 1 7 1 0 22 7 2 0 3 7 3 0 1 7 4 0 1 8 0 0 58 8 1 0 15 8 2 0 7 8 3 0 3 9 0 0 40 9 0 2 1 9 1 0 14 9 2 0 4 9 5 0 1 10 0 0 34 10 1 0 7 10 2 0 6 10 3 0 1 10 4 0 3 11 0 0 20 11 1 0 9 11 2 0 4 12 0 0 15 12 1 0 9 12 2 0 5 12 2 2 1 12 3 0 1 12 4 0 1 13 0 0 9 13 1 0 4 13 2 0 6 13 3 0 1 13 4 0 1 14 0 0 8 14 1 0 6 14 2 0 7 14 3 2 1 15 0 0 3 15 1 0 7 15 2 0 5 15 4 0 3 16 0 0 5 16 0 22 1 16 1 0 4 16 2 0 3 16 3 0 1 17 0 0 4 17 1 0 1 17 2 0 3 17 3 0 4 18 0 0 7 18 1 0 4 18 3 0 1 18 8 0 1 19 1 0 3 19 2 0 2 20 1 0 1 20 3 0 1 21 1 0 1 21 2 0 1 22 0 13 1 22 4 0 1 22 5 0 1 23 0 0 1 23 1 0 1 23 4 0 2 24 7 0 1 26 4 0 1 26 5 0 1

10  Edge-to-edge property

 Property Number False 601 True 0

11  Publications

 Publication Number abas 176 ajlouni 1 akber 17 arik 1 aslanapa 25 backhouse 7 bain 5 bakirer 1 balmelle 185 barry 2 berchem 1 betsch 1 betts 1 blair 3 blair2 1 bonner 241 booth 8 bour0 7 bourgoin 179 briggs 12 broug 14 broug2 59 bulut 24 burckhard2 1 burckhardt 6 cahier 66 calvert 16 calvert2 1 carey 5 castera 47 clevenot 14 collin 38 copple 1 creswell 16 critchlow 24 cromwell1 1 cromwell2 2 cromwell3 1 cromwell4 30 d-avennes 41 dawes 173 day 1 degeorge2 48 denny 2 dury 3 dussaud 1 dye 122 ekhtiar 1 elsaid 49 elsaid2 4 erdmann 14 escher 2 etting 4 ex1995 7 fernandez 16 field1 10 field2 14 field4 26 frettloeh 18 gailiunas 9 gands 114 gands2 2 gink 4 glassner 1 gluck 1 golomb1 29 golomb2 3 golombek 1 gomez 1 grafton 28 grube 1 guy 9 hankin1 2 hankin2 1 hattstein 3 hedgecoe 1 herzfeld1 1 herzfeld2 3 herzfeld3 1 herzfeld4 1 hessemer 55 hill 51 hill2 10 hirsch 2 hrbas 3 humbert 5 hutt 2 iran 172 james 4 jones 50 jones2 2 klaassen 1 klarner 3 knobloch 1 landau 2 lee 14 lings 1 lowry 1 makov 4 marcais 1 marshall 17 martin 1 maussion 1 meinecke 1 migeon 2 mols 1 muller 1 murphy 5 myers 47 myers2 43 neal 7 necipoglu 28 ogel 4 okane 1 okane2 45 orazi 5 orton 1 otto 1 paccard 92 pajares 25 pavon 10 pc 739 pickett 1 pope 23 pope2 1 pugatch 2 racinet 18 ransome 2 ray 13 reid 4 rempel 21 reuther 1 rice 1 riefstahl 1 rigby1 55 rogers 2 sakkal 26 sakkal2 22 sarre 12 scerrato 6 schatt1 2 schneider 1 seherr 17 shafai 72 siculo 1 singer 8 smith1 1 smith2 98 sourdel 3 stevens 23 stierlin 3 stierlin2 1 stock 6 stronge 20 sutton 7 useinov 3 vami 143 viollet 1 volait 7 volwah 2 wade 58 wadei 674 wahhab 36 wich2 122 wich3 2 wilber 4 wild 1 wilkinson 4 williams 1 wilson 13 wurfel 1 ww 179

12  Islamic Tilings

Those tilings which are referenced at least once in books about Islamic art can be counted as Islamic patterns. There are 1663 of these.