This is a list of selected geodesic polyhedra and Goldberg polyhedra, two infinite classes of polyhedra. Geodesic polyhedra and Goldberg polyhedra are duals of each other. The geodesic and Goldberg polyhedra are parameterized by integers m and n, with
m>0
n\ge0
T=m2+mn+n2
| m | n | T | Class | Vertices (geodesic) Faces (Goldberg) | Edges | Faces (geodesic) Vertices (Goldberg) | Face triangle | Geodesic | Goldberg | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symbols | Conway | Image | Symbols | Conway | Image | |||||||||
| 1 | 0 | 1 | I | 12 | 30 | 20 | 1,0 | 1,0 GP5(1,0) | ||||||
| 2 | 0 | 4 | I | 42 | 120 | 80 | 2,0 | uI dcdI | 2,0 GP5(2,0) | cD cD | ||||
| 3 | 0 | 9 | I | 92 | 270 | 180 | 3,0 | xI ktI | 3,0 GP5(3,0) | yD tkD | ||||
| 4 | 0 | 16 | I | 162 | 480 | 320 | 4,0 | uuI dccD | 4,0 GP5(4,0) | c2D | ||||
| 5 | 0 | 25 | I | 252 | 750 | 500 | 5,0 | u5Iu5I | 5,0 GP5(5,0) | c5D du5I | ||||
| 6 | 0 | 36 | I | 362 | 1080 | 720 | 6,0 | uxI dctkdI | 6,0 GP5(6,0) | cyD ctkD | ||||
| 7 | 0 | 49 | I | 492 | 1470 | 980 | 7,0 | vI dwrwdI | 7,0 GP5(7,0) | wD wrwD | ||||
| 8 | 0 | 64 | I | 642 | 1920 | 1280 | 8,0 | u3I dcccdI | 8,0 GP5(8,0) | cccD | ||||
| 9 | 0 | 81 | I | 812 | 2430 | 1620 | 9,0 | xxI ktktI | 9,0 GP5(9,0) | yyD tktkD | ||||
| 10 | 0 | 100 | I | 1002 | 3000 | 2000 | 10,0 | uu5Iuu5I | 10,0 GP5(10,0) | cc5D | ||||
| 11 | 0 | 121 | I | 1212 | 3630 | 2420 | 11,0 | u11Iu11I | 11,0 GP5(11,0) | c11D | ||||
| 12 | 0 | 144 | I | 1442 | 4320 | 2880 | 12,0 | uuxD dcctkD | 12,0 GP5(12,0) | ccyD cctkD | ||||
| 13 | 0 | 169 | I | 1692 | 5070 | 3380 | 13,0 | u13Iu13I | 13,0 GP5(13,0) | c13D | ||||
| 14 | 0 | 196 | I | 1962 | 5880 | 3920 | 14,0 | uvI dcwdI | 14,0 GP5(14,0) | cwrwD | ||||
| 15 | 0 | 225 | I | 2252 | 6750 | 4500 | 15,0 | u5xI u5ktI | 15,0 GP5(15,0) | c5yD c5tkD | ||||
| 16 | 0 | 256 | I | 2562 | 7680 | 5120 | 16,0 | dc4dI | 16,0 GP5(16,0) | ccccD | ||||
| 1 | 1 | 3 | II | 32 | 90 | 60 | 1,1 | nI kD | 1,1 GP5(1,1) | yD tI | ||||
| 2 | 2 | 12 | II | 122 | 360 | 240 | 2,2 | unI =dctI | 2,2 GP5(2,2) | czD cdkD | ||||
| 3 | 3 | 27 | II | 272 | 810 | 540 | 3,3 | xnI ktkD | 3,3 GP5(3,3) | yzD tkdkD | ||||
| 4 | 4 | 48 | II | 482 | 1440 | 960 | 4,4 | u2nI dcctI | 4,4 GP5(4,4) | c2zD cctI | ||||
| 5 | 5 | 75 | II | 752 | 2250 | 1500 | 5,5 | u5nI | 5,5 GP5(5,5) | c5zD | ||||
| 6 | 6 | 108 | II | 1082 | 3240 | 2160 | 6,6 | uxnI dctktI | 6,6 GP5(6,6) | cyzD ctkdkD | ||||
| 7 | 7 | 147 | II | 1472 | 4410 | 2940 | 7,7 | vnI dwrwtI | 7,7 GP5(7,7) | wzD wrwdkD | ||||
| 8 | 8 | 192 | II | 1922 | 5760 | 3840 | 8,8 | u3nI dccckD | 8,8 GP5(8,8) | c3zD ccctI | ||||
| 9 | 9 | 243 | II | 2432 | 7290 | 4860 | 9,9 | xxnI ktktkD | 9,9 GP5(9,9) | yyzD tktktI | ||||
| 12 | 12 | 432 | II | 4322 | 12960 | 8640 | 12,12 | uuxnI dccdktkD | 12,12 GP5(12,12) | ccyzD cckttI | ||||
| 14 | 14 | 588 | II | 5882 | 17640 | 11760 | 14,14 | uvnI dcwkD | 14,14 GP5(14,14) | cwzD cwrwtI | ||||
| 16 | 16 | 768 | II | 7682 | 23040 | 15360 | 16,16 | uuuunI dcccctI | 16,16 GP5(16,16) | cccczD cccctI | ||||
| 2 | 1 | 7 | III | 72 | 210 | 140 | 2,1 | vI dwD | 2,1 GP5(2,1) | wD | ||||
| 3 | 1 | 13 | III | 132 | 390 | 260 | 3,1 | v3,1I | 3,1 GP5(3,1) | w3,1D | ||||
| 3 | 2 | 19 | III | 192 | 570 | 380 | 3,2 | v3I | 3,2 GP5(3,2) | w3D | ||||
| 4 | 1 | 21 | III | 212 | 630 | 420 | 4,1 | vnI | dwtI | 4,1 GP5(4,1) | wkI | |||
| 4 | 2 | 28 | III | 282 | 840 | 560 | 4,2 | vnI dwtI | 4,2 GP5(4,2) | wdkD | ||||
| 4 | 3 | 37 | III | 372 | 1110 | 740 | 4,3 | v4I | 4,3 GP5(4,3) | w4D | ||||
| 5 | 1 | 31 | III | 312 | 930 | 620 | 5,1 | u5,1I | 5,1 GP5(5,1) | w5,1D | ||||
| 5 | 2 | 39 | III | 392 | 1170 | 780 | 5,2 | u5,2I | 5,2 GP5(5,2) | w5,2D | ||||
| 5 | 3 | 49 | III | 492 | 1470 | 980 | 5,3 | vvI dwwD | 5,3 GP5(5,3) | wwD | ||||
| 6 | 2 | 52 | III | 522 | 1560 | 1040 | 6,2 | v3,1uI | 6,2 GP5(6,2) | w3,1cD | ||||
| 6 | 3 | 63 | III | 632 | 1890 | 1260 | 6,3 | vxI dwdktI | 6,3 GP5(6,3) | wyD wtkD | ||||
| 8 | 2 | 84 | III | 842 | 2520 | 1680 | 8,2 | vunI dwctI | 8,2 GP5(8,2) | wczD wcdkD | ||||
| 8 | 4 | 112 | III | 1122 | 3360 | 2240 | 8,4 | vuuI dwccD | 8,4 GP5(8,4) | wccD | ||||
| 11 | 2 | 147 | III | 1472 | 4410 | 2940 | 11,2 | vvnI dwwtI | 11,2 GP5(11,2) | wwzD | ||||
| 12 | 3 | 189 | III | 1892 | 5670 | 3780 | 12,3 | vxnI dwtktktI | 12,3 GP5(12,3) | wyzD wtktI | ||||
| 10 | 6 | 196 | III | 1962 | 5880 | 3920 | 10,6 | vvuI dwwcD | 10,6 GP5(10,6) | wwcD | ||||
| 12 | 6 | 252 | III | 2522 | 7560 | 5040 | 12,6 | vxuI dwctkD | 12,6 GP5(12,6) | cywD wctkD | ||||
| 16 | 4 | 336 | III | 3362 | 10080 | 6720 | 16,4 | vuunI dwcctI | 16,4 GP5(16,4) | wcczD wcctI | ||||
| 14 | 7 | 343 | III | 3432 | 10290 | 6860 | 14,7 | vvI dwrwwD | 14,7 GP5(14,7) | wwD wrwwD | ||||
| 15 | 9 | 441 | III | 4412 | 13230 | 8820 | 15,9 | vvxI dwwtkD | 15,9 GP5(15,9) | wwxD wwtkD | ||||
| 16 | 8 | 448 | III | 4482 | 13440 | 8960 | 16,8 | vuuuI dwcccD | 16,8 GP5(16,8) | wcccD | ||||
| 18 | 1 | 343 | III | 3432 | 10290 | 6860 | 18,1 | vvvI dwwwD | 18,1 GP5(18,1) | wwwD | ||||
| 18 | 9 | 567 | III | 5672 | 17010 | 11340 | 18,9 | vxxI dwtktkD | 18,9 GP5(18,9) | wyyD wtktkD | ||||
| 20 | 12 | 784 | III | 7842 | 23520 | 15680 | 20,12 | vvuuI dwwccD | 20,12 GP5(20,12) | wwccD | ||||
| 20 | 17 | 1029 | III | 10292 | 30870 | 20580 | 20,17 | vvvnI dwwwtI | 20,17 GP5(20,17) | wwwzD wwwdkD | ||||
| 28 | 7 | 1029 | III | 10292 | 30870 | 20580 | 28,7 | vvnI dwrwwdkD | 28,7 GP5(28,7) | wwzD wrwwdkD |
| m | n | T | Class | Vertices (geodesic) Faces (Goldberg) | Edges | Faces (geodesic) Vertices (Goldberg) | Face triangle | Geodesic | Goldberg | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symbols | Conway | Image | Symbols | Conway | Image | |||||||||
| 1 | 0 | 1 | I | 6 | 12 | 8 | 1,0 | 1,0 GP4(1,0) | ||||||
| 2 | 0 | 4 | I | 18 | 48 | 32 | 2,0 | dcC dcC | 2,0 GP4(2,0) | cC cC | ||||
| 3 | 0 | 9 | I | 38 | 108 | 72 | 3,0 | ktO | 3,0 GP4(3,0) | tkC | ||||
| 4 | 0 | 16 | I | 66 | 192 | 128 | 4,0 | uuO dccC | 4,0 GP4(4,0) | ccC | ||||
| 5 | 0 | 25 | I | 102 | 300 | 200 | 5,0 | u5O | 5,0 GP4(5,0) | c5C | ||||
| 6 | 0 | 36 | I | 146 | 432 | 288 | 6,0 | uxO dctkdO | 6,0 GP4(6,0) | cyC ctkC | ||||
| 7 | 0 | 49 | I | 198 | 588 | 392 | 7,0 | dwrwO | 7,0 GP4(7,0) | wrwO | ||||
| 8 | 0 | 64 | I | 258 | 768 | 512 | 8,0 | uuuO dcccC | 8,0 GP4(8,0) | cccC | ||||
| 9 | 0 | 81 | I | 326 | 972 | 648 | 9,0 | xxO ktktO | 9,0 GP4(9,0) | yyC tktkC | ||||
| 1 | 1 | 3 | II | 14 | 36 | 24 | 1,1 | 1,1 GP4(1,1) | ||||||
| 2 | 2 | 12 | II | 50 | 144 | 96 | 2,2 | ukC dctO | 2,2 GP4(2,2) | czC ctO | ||||
| 3 | 3 | 27 | II | 110 | 324 | 216 | 3,3 | ktkC | 3,3 GP4(3,3) | tktO | ||||
| 4 | 4 | 48 | II | 194 | 576 | 384 | 4,4 | uunO dcctO | 4,4 GP4(4,4) | cczC cctO | ||||
| 2 | 1 | 7 | III | 30 | 84 | 56 | 2,1 | vO dwC | 2,1 GP4(2,1) | wC |
| m | n | T | Class | Vertices (geodesic) Faces (Goldberg) | Edges | Faces (geodesic) Vertices (Goldberg) | Face triangle | Geodesic | Goldberg | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symbols | Conway | Image | Symbols | Conway | Image | |||||||||
| 1 | 0 | 1 | I | 4 | 6 | 4 | 1,0 | 1,0 GP3(1,0) | ||||||
| 1 | 1 | 3 | II | 8 | 18 | 12 | 1,1 | kT kT | 1,1 GP3(1,1) | tT tT | ||||
| 2 | 0 | 4 | I | 10 | 24 | 16 | 2,0 | dcT dcT | 2,0 GP3(2,0) | cT cT | ||||
| 3 | 0 | 9 | I | 20 | 54 | 36 | 3,0 | ktT | 3,0 GP3(3,0) | tkT | ||||
| 4 | 0 | 16 | I | 34 | 96 | 64 | 4,0 | uuT dccT | 4,0 GP3(4,0) | ccT | ||||
| 5 | 0 | 25 | I | 52 | 150 | 100 | 5,0 | u5T | 5,0 GP3(5,0) | c5T | ||||
| 6 | 0 | 36 | I | 74 | 216 | 144 | 6,0 | uxT dctkdT | 6,0 GP3(6,0) | cyT ctkT | ||||
| 7 | 0 | 49 | I | 100 | 294 | 196 | 7,0 | vrvT dwrwT | 7,0 GP3(7,0) | wrwT | ||||
| 8 | 0 | 64 | I | 130 | 384 | 256 | 8,0 | u3T dcccdT | 8,0 GP3(8,0) | c3T cccT | ||||
| 9 | 0 | 81 | I | 164 | 486 | 324 | 9,0 | xxT ktktT | 9,0 GP3(9,0) | yyT tktkT | ||||
| 3 | 3 | 27 | II | 56 | 162 | 108 | 3,3 | ktkT | 3,3 GP3(3,3) | tktT | ||||
| 2 | 1 | 7 | III | 16 | 42 | 28 | 2,1 | dwT | 2,1 GP5(2,1) | wT |