Dozenal Wiki
Advertisement
n total number of n-ominoes number of n-ominoes with hole number of n-ominoes without hole number of n-ominoes that can tile the plane number of n-ominoes that can tile the plane by translation number of n-ominoes that can tile the plane by 200-dozenaldegree rotation (Conway criterion) number of n-ominoes that can tile the plane by both translation and 200-dozenaldegree rotation (Conway criterion) number of n-ominoes that can tile the plane by translation but not 200-dozenaldegree rotation (Conway criterion) number of n-ominoes that can tile the plane by 200-dozenaldegree rotation (Conway criterion) but not translation number of n-ominoes that can tile the plane by either translation or 200-dozenaldegree rotation (Conway criterion) number of n-ominoes that can tile the plane but not by translation number of n-ominoes that can tile the plane but not by 200-dozenaldegree rotation (Conway criterion) number of n-ominoes that can tile the plane but not by either translation or 200-dozenaldegree rotation (Conway criterion) number of n-ominoes that cannot tile the plane number of n-ominoes without hole that cannot tile the plane
1 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0
2 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0
3 2 0 2 2 2 2 2 0 0 2 0 0 0 0 0
4 5 0 5 5 5 5 5 0 0 5 0 0 0 0 0
5 10 0 10 10 9 10 9 0 3 10 3 0 0 0 0
6 2E 0 2E 2E 20 2E 20 0 E 2E E 0 0 0 0
7 90 1 8E 88 35 85 35 0 50 85 53 3 3 4 3
8 269 6 263 247 X1 228 X1 0 147 228 166 1E 1E 22 18
9 8E1 31 880 736 159 680 158 1 524 681 599 76 75 177 146
X 283E 143 26E8 193X 376 1691 374 2 1319 1693 1584 269 267 E01 97X
E 9X69 697 9392 39E8 553 3698 545 X 3153 36X6 3465 320 312 6071 5596
10 30980 2847 2X135 116X1 16X0 E536 15X9 E3 9E49 E629 10001 2167 2074 1E29E 18654
11 E60X7 10516 X5791 1X465 1X0E 1825X 1911 EX 16549 18358 18656 2207 2109 97842 87328
12 375E83 47X14 32X16E 53180 5028 464E3 4815 413 4189X 46906 4X154 8889 8476 322X03 296EXE
13 1192E80 186261 100891E 115967 9751 E280X 80XX 1663 X6720 E4271 108216 23159 216E6 1079215 XE2E74
14 446901E 752204 3916X17 2245X5 1979E 1X3767 16710 308E 189057 1X6836 206X06 40X3X 3996E 4244636 36E2432
15 14945771 27E2724 12153049 33X737 24126 311736 20167 3E7E 2E158E 3156E5 316611 29001 25042 14607036 11X14512
16 54613118 E3EEX6X 4521326X 1035358 75604 9X934X 588X8 18918 950662 X06066 E7E954 24800X 22E2E2 53599980 44199E12
17 188853488 3EXE166E 148961X19 1170804 8E093 1107197 72XE8 18197 105429E 1123372 10X1731 65629 49452 1876X2884 1477E1215
18 6814705X6 1493E4E14 534077692 408990E 237928 3325X10 17X164 79784 3167868 33X3594 3X51EX3 963XEE 8X6337 6793X2897 52EEX9983
19 21X51104E2 5X198005X 1803350454 7411X60 363389 6224X24 251868 111721 5E93178 6336545 706X693 11X9038 1097517 21998EX652 17E7E3X5E4
1X 8454X48020 2041148563 64138EE679 104888E6 70X442 E05803X 52E441 19E001 X7287E9 E23703E E97X474 1430878 1251877 844457E326 6403432983
1E 2869X729928 84EE0052E2 2019E724636 15X38E84 X584EE 15344X33 7X331E 2751X0 14761714 155EX013 14EX0685 6E4151 43XE71 286848E0964 201858X7672
20 X6E3X0X866E 2X99201E437 78168089234 61E948E5 2E2X607 4E705075 194X836 119E991 4997643E 508X4X46 5E0662XX 1248E840 112XEX6E X6X98113976 781060E453E
n total number of n-ominoes number of n-ominoes with four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4 [D4] number of n-ominoes with two axes of reflection symmetry, both aligned with the gridlines [D2, 100 dozenal degrees] number of n-ominoes with two axes of reflection symmetry, both aligned with the diagonals [D2, 60 dozenal degrees] number of n-ominoes with an axis of reflection symmetry aligned with the gridlines [mirror, 100 dozenal degrees] number of n-ominoes with an axis of reflection symmetry aligned with the diagonals [mirror, 60 dozenal degrees] number of n-ominoes with rotational symmetry of order 4 [C4] number of n-ominoes with rotational symmetry of order 2 [C2] number of n-ominoes with no symmetry [none]
1 1 1 0 0 0 0 0 0 0
2 1 0 1 0 0 0 0 0 0
3 2 0 1 0 0 1 0 0 0
4 5 1 1 0 1 0 0 1 1
5 10 1 1 0 2 2 0 1 5
6 2E 0 2 0 6 2 0 5 18
7 90 0 3 1 9 7 0 4 70
8 269 1 4 1 1E 5 1 16 224
9 8E1 2 4 0 32 22 0 17 838
X 283E 0 8 1 76 1X 0 61 26E9
E 9X69 0 X 2 103 77 0 61 983X
10 30980 3 13 3 245 67 3 1E2 3047X
11 E60X7 2 15 3 3E0 232 2 1E7 E546X
12 375E83 0 26 5 8EX 211 0 758 3744X9
13 1192E80 0 2E 6 12E0 82X 0 76X 119047E
14 446901E 5 50 12 2X00 791 10 2479 4463116
15 14945771 4 54 9 48XE 2628 7 2507 14937X9E
16 54613118 0 99 18 X930 2530 0 9283 545E4700
17 188853488 0 X8 18 16185 93E3 0 93X1 188822627
18 6814705X6 10 178 48 350E3 9209 38 2E7E8 6813EX45X
19 21X51104E2 7 181 28 59799 2X9X6 21 2EE8X 21X5013948
1X 8454X48020 0 323 68 111441 2X935 0 E6650 8454811047
1E 2869X729928 0 338 54 1X4428 XXX56 0 E74E7 2869X35X99E
20 X6E3X0X866E 18 661 168 42509E E0236 119 38E59X X6E39403036

The set of 10-ominoes (dozominoes) is the lowest polyomino set in which all eight possible symmetries (four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4; two axes of reflection symmetry, both aligned with the gridlines; two axes of reflection symmetry, both aligned with the diagonals; an axis of reflection symmetry aligned with the gridlines; an axis of reflection symmetry aligned with the diagonals; rotational symmetry of order 4; rotational symmetry of order 2; no symmetry) are realized for polyominoes without holes. (if we allow polyominoes with holes, it is the set of 8-ominoes (octominoes)) besides, the set of 10-ominoes (dozominoes) is also the lowest polyomino set in which all eight possible symmetries are realized for polyominoes which can tile the plane, besides, the set of 10-ominoes (dozominoes) is also the lowest polyomino set in which all eight possible symmetries are realized for polyominoes with holes, besides, if we allow polyominoes with holes, the number of 10-ominoes (dozominoes) with all eight possible symmetries get records for the number of n-ominoes with the same symmetry except “an axis of reflection symmetry aligned with the diagonals”. (there are totally 30980 10-ominoes (dozominoes) (including 2X135 without holes and 2847 with holes), 3 with “four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4”, 13 with “two axes of reflection symmetry, both aligned with the gridlines”, 3 with “two axes of reflection symmetry, both aligned with the diagonals”, 245 with “an axis of reflection symmetry aligned with the gridlines”, 67 with “an axis of reflection symmetry aligned with the diagonals”, 3 with “rotational symmetry of order 4”, 1E2 with “rotational symmetry of order 2”, 3047X with “no symmetry”, all get records except “an axis of reflection symmetry aligned with the diagonals”, the former records of these eight types of symmetry are 2 (by nonomino (9-omino)), X (by elpomino (E-omino)), 2 (by elpomino (E-omino)), 103 (by elpomino (E-omino)), 77 (by elpomino (E-omino)), 1 (by octomino (8-omino)), 61 (by both dekromino (X-omino) and elpomino (E-omino)), 983X (by elpomino (E-omino)), respectively, there are 77 elpominoes (E-ominoes) with this type of symmetry, note that the difference of 77 and 67 is exactly 10)

"four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4" without hole:

 oo
oooo
oooo
 oo

"two axes of reflection symmetry, both aligned with the gridlines" without hole:

oooo
oooo
oooo

"two axes of reflection symmetry, both aligned with the diagonals" without hole:

  o
  o
oooo
  oooo
   o
   o

"an axis of reflection symmetry aligned with the gridlines" without hole:

ooooooooooo
     o

"an axis of reflection symmetry aligned with the diagonals" without hole:

oooooo
oo
o
o
o
o

"rotational symmetry of order 4" without hole:

  o
  o
  oooo
oooo
   o
   o

"rotational symmetry of order 2" without hole:

oooooo
     oooooo

"no symmetry" without hole:

ooooooooooo
o

"four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4" with hole:

  o
 ooo
oo oo
 ooo
  o

"two axes of reflection symmetry, both aligned with the gridlines" with hole:

ooooo
 o o
ooooo

"two axes of reflection symmetry, both aligned with the diagonals" with hole":

 o
oooo
 o o
 oooo
   o

"an axis of reflection symmetry aligned with the gridlines" with hole:

ooo
o ooooo
ooo

"an axis of reflection symmetry aligned with the diagonals" with hole:

ooo
o o
ooooo
  o
  o

"rotational symmetry of order 4" with hole:

 o
 oooo
 o o
oooo
   o

"rotational symmetry of order 2" with hole:

oo
 ooo
 o o
 ooo
   oo

"no symmetry" with hole:

ooooooo
o o
ooo