# Some statistics: Version 51

## 1  Introduction

This page gives some statistics concerning this release of the tiling system, having 2857 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 *632 (p6m) 522 The symmetry group of the tiling is *8• (d8) 17 The symmetry group of the tiling is *2222 (pmm) 184 The symmetry group of the tiling is 2*22 (cmm) 290 The symmetry group of the tiling is *442 (p4m) 1025 The symmetry group of the tiling is 3*3 (p31m) 35 The symmetry group of the tiling is *5• (d5) 13 The symmetry group of the tiling is 6• (c6) 19 The symmetry group of the tiling is *6• (d6) 6 The symmetry group of the tiling is *4• (d4) 16 The symmetry group of the tiling is *10• (d10) 11 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 *333 (p3m1) 21 The symmetry group of the tiling is 22X (pgg) 17 The symmetry group of the tiling is 4*2 (p4g) 142 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 2• (c2) 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 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 *16• (d16) 1 The symmetry group of the tiling is 2*∞ (pmg) 2 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 261 Can be coloured with two colours 845 Can be coloured with two colours (straight cross-overs) 1299
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 775 1912 regular pentagon 261 3292 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 72.0 109 1049 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 1 1 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 90.0 579 1467 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 72.0 139 248 10 85.5 1 1 10 90.0 2 2 10 98.0 1 1 10 108.0 96 293 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 36.00 284 45.00 665 60.00 210 90.00 259 120.00 17

## 7  Does the pattern satisfy the two polygon condition?

 Property Number False 2651 True 184

## 8  The interlace condition

 Finite interlaces Infinite interlaces Total -1 0 64 0 0 1479 0 1 233 0 2 173 0 3 44 0 4 22 0 5 5 0 6 3 0 8 2 1 0 135 1 1 156 1 2 55 1 3 10 1 4 1 1 5 3 1 7 1 2 0 142 2 1 71 2 2 16 2 3 9 2 4 1 2 5 1 3 0 69 3 1 22 3 2 5 4 0 25 4 1 19 4 2 11 4 3 2 4 4 1 5 0 18 5 1 11 5 2 2 5 3 1 5 9 1 6 0 9 6 1 7 7 0 9 7 1 2 7 4 1 8 0 5 8 1 1 8 2 2 9 0 3 10 1 1 12 0 1 12 2 1 13 0 1 15 3 1

## 9  Polygonal tiles used

This excludes the regular polygons and star polygons.
 Reflective tiles Reflective pairs No mirror image Number 0 0 0 247 0 0 1 153 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 95 1 0 0 419 1 0 1 21 1 0 2 1 1 0 4 2 1 0 6 1 1 1 0 49 1 2 0 2 1 3 0 3 2 0 0 285 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 0 317 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 0 156 5 0 2 1 5 1 0 26 5 2 0 3 5 3 0 2 6 0 0 102 6 1 0 28 6 2 0 4 6 4 0 1 7 0 0 87 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 13 9 2 0 4 9 5 0 1 10 0 0 33 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 2 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 3 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 0 True 601

## 11  Publications

 Publication Number hill 51 aslanapa 25 wadei 674 ogel 4 erdmann 14 pc 721 paccard 92 necipoglu 28 cromwell1 1 cromwell4 30 wahhab 36 betsch 1 ww 179 broug2 59 bonner 241 cromwell2 2 hirsch 2 klaassen 1 pope 23 iran 172 smith2 98 bourgoin 179 okane2 45 bain 5 jones 49 schatt1 2 racinet 18 backhouse 7 wade 58 field2 14 pajares 25 grafton 28 balmelle 185 hessemer 55 vami 143 golomb1 29 d-avennes 41 cahier 66 rempel 21 field1 10 dawes 173 field4 26 etting 4 james 4 castera 47 fernandez 16 gands 114 abas 176 wich2 122 neal 7 dye 122 stevens 23 sarre 12 bour0 7 calvert 16 elsaid 49 broug 14 degeorge2 48 creswell 16 shafai 72 humbert 5 akber 17 murphy 5 hill2 10 briggs 12 smith1 1 critchlow 24 stronge 20 wilson 13 seherr 17 wilkinson 4 collin 38 useinov 3 grube 1 dury 3 bulut 24 lings 1 hankin2 1 booth 8 gands2 2 lee 14 gink 4 herzfeld1 1 denny 2 clevenot 14 scerrato 6 sourdel 3 dussaud 1 blair 3 rice 1 carey 5 escher 2 singer 8 rigby1 55 stierlin 3 ajlouni 1 cromwell3 1 orton 1 makov 4 martin 1 golomb2 3 reid 4 stock 6 myers 47 myers2 43 ransome 2 elsaid2 4 hedgecoe 1 herzfeld2 3 pavon 10 landau 2 burckhardt 6 schneider 1 hattstein 3 wilber 4 gluck 1 calvert2 1 jones2 1 hrbas 3 gailiunas 9 pope2 1 wich3 2 klarner 3 copple 1 volwah 2 glassner 1 marshall 17 frettloeh 18 orazi 5 rogers 2 migeon 2 muller 1 meinecke 1 berchem 1 guy 9 bakirer 1 hankin1 2 ex1995 7 betts 1 ray 13 viollet 1 hutt 2 sutton 7 otto 1 wurfel 1 gomez 1 williams 1 barry 2 stierlin2 1 siculo 1 lowry 1 arik 1 golombek 1 maussion 1 pickett 1 reuther 1 mols 1 knobloch 1 ekhtiar 1 sakkal 26 sakkal2 22 wild 1 pugatch 2 okane 1 herzfeld3 1 volait 7 blair2 1 burckhard2 1 marcais 1 herzfeld4 1 day 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 1663 of these.