EuroCG 2024 Additional Tables

Note

Table A: Archimedean solids

Archimedean solids |V| |E| |F| # MOPEs # edge unfoldings [HS13] # non-overlapping edge unfoldings Percentage (%)
Snub cube 24 60 38 72 89,904,012,853,248 85,967,688,920,076 95.62
Truncated dodecahedron 60 90 32 120 4,982,259,375,000,000,000 1,173,681,002,295,455,040 23.56
Truncated icosahedron 60 90 32 240 375,291,866,372,898,816,000 371,723,160,733,469,233,260 99.05

Table B: Johnson solids

Johnson solids |V| |E| |F| # MOPEs # edge unfoldings [HS13] # non-overlapping edge unfoldings Percentage (%)
Elongated pentagonal cupola (J20) 25 45 22 40 29,821,320,745 27,158,087,415 91.07
Elongated pentagonal rotunda (J21) 30 55 27 90 8,223,103,375,490 6,297,186,667,720 76.58
Gyroelongated pentagonal cupola (J24) 25 55 32 60 5,996,600,870,820 5,492,624,228,190 91.60
Pentagonal orthocupolarotunda (J32) 25 50 27 20 699,537,024,120 699,433,603,320 99.99
Pentagonal gyrocupolarotunda (J33) 25 50 27 10 745,208,449,920 745,105,029,120 99.99
Pentagonal orthobirotunda (J34) 30 60 32 10 193,003,269,869,040 190,653,702,525,040 98.78
Elongated pentagonal orthobicupola (J38) 30 60 32 80 270,745,016,304,350 214,085,775,357,270 79.07
Elongated pentagonal gyrobicupola (J39) 30 60 32 80 272,026,496,000,000 215,087,798,524,180 79.07
Gyroelongated triangular bicupola (J44) 18 42 26 24 5,295,528,588 5,231,781,954 98.80
Gyroelongated square bicupola (J45) 24 56 34 48 13,769,880,349,680 13,386,219,088,644 97.21
Augmented hexagonal prism (J54) 13 22 11 4 75,973 75,749 99.71
Parabiaugmented hexagonal prism (J55) 14 26 14 8 709,632 705,144 99.37
Metabiaugmented hexagonal prism (J56) 14 26 14 8 707,232 702,520 99.33
Triaugmented hexagonal prism (J57) 15 30 17 12 6,531,840 6,457,860 98.87
Augmented dodecahedron (J58) 21 35 16 5 92,724,962 92,219,782 99.46
Parabiaugmented dodecahedron (J59) 22 40 20 10 1,651,482,010 1,632,941,030 98.88
Metabiaugmented dodecahedron (J60) 22 40 20 10 1,641,317,568 1,621,738,522 98.81
Triaugmented dodecahedron (J61) 23 45 24 15 28,745,798,400 28,183,512,978 98.04
Augmented truncated cube (J66) 28 48 22 104 54,921,311,280 39,055,563,000 71.11
Biaugmented truncated cube (J67) 32 60 30 208 90,974,647,120,896 43,437,626,181,464 47.75

Table C: Archimedian prisms

Archimedian n-prism |V| |E| |F| # MOPEs # edge unfoldings [HS13] # non-overlapping edge unfoldings Percentage (%)
Archimedian 24-prism 48 72 26 96 639,620,518,118,400 611,750,144,604,960 95.64
Archimedian 25-prism 50 75 27 100 2,486,558,615,814,025 2,378,211,063,753,525 95.64
Archimedian 26-prism 52 78 28 208 9,651,161,613,824,796 9,120,749,762,911,540 94.50
Archimedian 27-prism 54 81 29 216 37,403,957,244,654,675 35,348,297,730,550,335 94.50
Archimedian 28-prism 56 84 30 336 144,763,597,316,784,768 136,369,030,045,792,768 94.20
Archimedian 29-prism 58 87 31 580 559,560,282,425,278,229 377,763,966,359,384,333 67.51
Archimedian 30-prism 60 90 32 720 2,160,318,004,043,512,500 1,457,228,998,699,944,660 67.45
Archimedian 31-prism 62 93 33 744 8,331,163,769,982,715,231 5,619,734,416,791,278,823 67.45
Archimedian 32-prism 64 96 34 768 32,095,304,749,163,937,792 21,649,687,090,073,296,384 67.45
Archimedian 33-prism 66 99 35 792 123,524,473,883,545,449,825 83,322,661,319,000,341,161 67.45
Archimedian 34-prism 68 102 36 952 474,969,297,739,230,927,564 320,315,730,957,505,974,740 67.43
Archimedian 35-prism 70 105 37 1,120 1,824,745,126,233,358,110,635 1,224,788,877,353,311,603,655 67.12
Archimedian 36-prism 72 108 38 1,440 7,004,614,136,879,907,849,600 4,397,626,384,555,854,813,048 62.78
Archimedian 37-prism 74 111 39 1,776 26,867,730,730,869,118,775,917 16,841,247,868,506,593,664,113 62.68
Archimedian 38-prism 76 114 40 2,128 102,981,783,095,242,242,871,908 64,245,596,838,412,691,619,868 62.39
Archimedian 39-prism 78 117 41 2,496 394,447,279,575,099,709,694,775 245,972,761,433,859,004,882,155 62.36
Archimedian 40-prism 80 120 42 2,880 1,509,843,372,596,510,348,221,440 898,435,929,860,914,751,335,120 59.51
Archimedian 41-prism 82 123 43 3,116 5,775,682,482,451,356,835,464,761 3,436,774,701,162,733,316,551,373 59.50
Archimedian 42-prism 84 126 44 3,360 22,080,875,606,379,223,850,418,300 13,138,720,470,258,404,605,154,004 59.50

Table D: Archimedian antiprisms

Archimedian m-antiprism |V| |E| |F| # MOPEs # edge unfoldings [HS13] # non-overlapping edge unfoldings Percentage (%)
Archimedian 12-antiprism 24 48 26 48 51,599,794,176 49,743,531,024 96.40
Archimedian 13-antiprism 26 52 28 52 383,142,771,674 369,359,503,344 96.40
Archimedian 14-antiprism 28 56 30 56 2,828,107,288,188 2,726,368,290,352 96.40
Archimedian 15-antiprism 30 60 32 60 20,768,716,848,000 20,021,578,135,380 96.40
Archimedian 16-antiprism 32 64 34 64 151,840,963,183,392 146,378,600,602,880 96.40
Archimedian 17-antiprism 34 68 36 204 1,105,779,284,582,146 989,008,190,008,480 89.44
Archimedian 18-antiprism 36 72 38 432 8,024,954,790,380,544 1,517,682,139,108,200 18.91
Archimedian 19-antiprism 38 76 40 456 58,059,628,319,357,318 10,550,126,657,845,736 18.17
Archimedian 20-antiprism 40 80 42 560 418,891,171,182,561,000 72,542,787,706,846,320 17.31
Archimedian 21-antiprism 42 84 44 672 3,014,678,940,049,375,872 500,034,989,831,068,818 16.59
Archimedian 22-antiprism 44 88 46 704 21,646,865,272,061,272,716 3,449,844,625,120,946,448 15.94
Archimedian 23-antiprism 46 92 48 736 155,113,904,634,576,144,814 23,752,014,262,731,255,118 15.31
Archimedian 24-antiprism 48 96 50 960 1,109,391,149,998,449,819,648 162,941,846,136,285,049,392 14.69
Archimedian 25-antiprism 50 100 52 1,000 7,920,708,398,483,722,531,250 1,117,782,108,867,439,830,950 14.11
Archimedian 26-antiprism 52 104 54 1,040 56,460,916,728,463,179,389,652 7,655,723,643,342,875,568,936 13.56
Archimedian 27-antiprism 54 108 56 1,404 401,873,068,071,158,383,691,136 40,561,091,359,603,932,081,708 10.09
Archimedian 28-antiprism 56 112 58 1,792 2,856,496,726,273,368,888,420,984 264,478,642,931,290,919,674,648 9.26
Archimedian 29-antiprism 58 116 60 2,204 20,277,959,821,998,087,658,569,178 1,739,241,575,214,473,110,359,470 8.58
Archimedian 30-antiprism 60 120 62 2,520 143,779,866,504,299,168,102,784,000 11,718,403,001,480,040,992,138,460 8.15
Archimedian 31-antiprism 62 124 64 2,852 1,018,331,261,238,041,888,906,149,982 79,227,778,390,260,949,681,648,022 7.78
Archimedian 32-antiprism 64 128 66 2,944 7,204,899,406,395,028,729,775,662,656 535,088,808,567,353,165,252,109,504 7.42

Reference

[HS13] Takashi Horiyama and Wataru Shoji. The number of different unfoldings of polyhedra. In 24th International Symposium on Algorithms and Computation, volume 8283 of LNCS, pages 623-633. Springer, 2013.