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 [D_{4}] | number of n-ominoes with two axes of reflection symmetry, both aligned with the gridlines [D_{2}, 100 dozenal degrees] | number of n-ominoes with two axes of reflection symmetry, both aligned with the diagonals [D_{2}, 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 [C_{4}] | number of n-ominoes with rotational symmetry of order 2 [C_{2}] | 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:
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"two axes of reflection symmetry, both aligned with the gridlines" without hole:
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"two axes of reflection symmetry, both aligned with the diagonals" without hole:
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"an axis of reflection symmetry aligned with the gridlines" without hole:
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"an axis of reflection symmetry aligned with the diagonals" without hole:
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"rotational symmetry of order 4" without hole:
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"rotational symmetry of order 2" without hole:
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"no symmetry" without hole:
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"four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4" with hole:
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"two axes of reflection symmetry, both aligned with the gridlines" with hole:
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"two axes of reflection symmetry, both aligned with the diagonals" with hole":
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"an axis of reflection symmetry aligned with the gridlines" with hole:
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"an axis of reflection symmetry aligned with the diagonals" with hole:
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"rotational symmetry of order 4" with hole:
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"rotational symmetry of order 2" with hole:
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"no symmetry" with hole:
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