# List of geodesic polyhedra and Goldberg polyhedra

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 $m>0$ and $n\geq 0$. T is the triangulation number, which is equal to $T=m^{2}+mn+n^{2}$.

## Icosahedral

| m | n | T | Class | Vertices (geodesic) Faces (Goldberg) | Edges | Faces (geodesic) Vertices (Goldberg) | Face triangle | Geodesic | Geodesic | Geodesic | Goldberg | Goldberg | Goldberg |
| m | n | T | Class | Vertices (geodesic) Faces (Goldberg) | Edges | Faces (geodesic) Vertices (Goldberg) | Face triangle | Symbols | Conway | Image | Symbols | Conway | Image |
| 1 | 0 | 1 | I | 12 | 30 | 20 | | {3,5} {3,5+}$_{1,0}$ | I | | {5,3} {5+,3}$_{1,0}$ GP$_5$(1,0) | D | |
| 2 | 0 | 4 | I | 42 | 120 | 80 | | {3,5+}$_{2,0}$ | uI dcdI | | {5+,3}$_{2,0}$ GP$_5$(2,0) | cD cD | |
| 3 | 0 | 9 | I | 92 | 270 | 180 | | {3,5+}$_{3,0}$ | xI ktI | | {5+,3}$_{3,0}$ GP$_5$(3,0) | yD tkD | |
| 4 | 0 | 16 | I | 162 | 480 | 320 | | {3,5+}$_{4,0}$ | uuI dccD | | {5+,3}$_{4,0}$ GP$_5$(4,0) | c$^2$D | |
| 5 | 0 | 25 | I | 252 | 750 | 500 | | {3,5+}$_{5,0}$ | u5I u5I | | {5+,3}$_{5,0}$ GP$_5$(5,0) | c5D du5I | |
| 6 | 0 | 36 | I | 362 | 1080 | 720 | | {3,5+}$_{6,0}$ | uxI dctkdI | | {5+,3}$_{6,0}$ GP$_5$(6,0) | cyD ctkD | |
| 7 | 0 | 49 | I | 492 | 1470 | 980 | | {3,5+}$_{7,0}$ | vvI dwrwdI | | {5+,3}$_{7,0}$ GP$_5$(7,0) | wwD wrwD | |
| 8 | 0 | 64 | I | 642 | 1920 | 1280 | | {3,5+}$_{8,0}$ | u$^3$I dcccdI | | {5+,3}$_{8,0}$ GP$_5$(8,0) | cccD | |
| 9 | 0 | 81 | I | 812 | 2430 | 1620 | | {3,5+}$_{9,0}$ | xxI ktktI | | {5+,3}$_{9,0}$ GP$_5$(9,0) | yyD tktkD | |
| 10 | 0 | 100 | I | 1002 | 3000 | 2000 | | {3,5+}$_{10,0}$ | uu5I uu5I | | {5+,3}$_{10,0}$ GP$_5$(10,0) | cc5D | |
| 11 | 0 | 121 | I | 1212 | 3630 | 2420 | | {3,5+}$_{11,0}$ | u11I u11I | | {5+,3}$_{11,0}$ GP$_5$(11,0) | c11D | |
| 12 | 0 | 144 | I | 1442 | 4320 | 2880 | | {3,5+}$_{12,0}$ | uuxD dcctkD | | {5+,3}$_{12,0}$ GP$_5$(12,0) | ccyD cctkD | |
| 13 | 0 | 169 | I | 1692 | 5070 | 3380 | | {3,5+}$_{13,0}$ | u13I u13I | | {5+,3}$_{13,0}$ GP$_5$(13,0) | c13D | |
| 14 | 0 | 196 | I | 1962 | 5880 | 3920 | | {3,5+}$_{14,0}$ | uvvI dcwwdI | | {5+,3}$_{14,0}$ GP$_5$(14,0) | cwrwD | |
| 15 | 0 | 225 | I | 2252 | 6750 | 4500 | | {3,5+}$_{15,0}$ | u5xI u5ktI | | {5+,3}$_{15,0}$ GP$_5$(15,0) | c5yD c5tkD | |
| 16 | 0 | 256 | I | 2562 | 7680 | 5120 | | {3,5+}$_{16,0}$ | dc$^4$dI | | {5+,3}$_{16,0}$ GP$_5$(16,0) | ccccD | |
| 1 | 1 | 3 | II | 32 | 90 | 60 | | {3,5+}$_{1,1}$ | nI kD | | {5+,3}$_{1,1}$ GP$_5$(1,1) | yD tI | |
| 2 | 2 | 12 | II | 122 | 360 | 240 | | {3,5+}$_{2,2}$ | unI =dctI | | {5+,3}$_{2,2}$ GP$_5$(2,2) | czD cdkD | |
| 3 | 3 | 27 | II | 272 | 810 | 540 | | {3,5+}$_{3,3}$ | xnI ktkD | | {5+,3}$_{3,3}$ GP$_5$(3,3) | yzD tkdkD | |
| 4 | 4 | 48 | II | 482 | 1440 | 960 | | {3,5+}$_{4,4}$ | u$^2$nI dcctI | | {5+,3}$_{4,4}$ GP$_5$(4,4) | c$^2$zD cctI | |
| 5 | 5 | 75 | II | 752 | 2250 | 1500 | | {3,5+}$_{5,5}$ | u5nI | | {5+,3}$_{5,5}$ GP$_5$(5,5) | c5zD | |
| 6 | 6 | 108 | II | 1082 | 3240 | 2160 | | {3,5+}$_{6,6}$ | uxnI dctktI | | {5+,3}$_{6,6}$ GP$_5$(6,6) | cyzD ctkdkD | |
| 7 | 7 | 147 | II | 1472 | 4410 | 2940 | | {3,5+}$_{7,7}$ | vvnI dwrwtI | | {5+,3}$_{7,7}$ GP$_5$(7,7) | wwzD wrwdkD | |
| 8 | 8 | 192 | II | 1922 | 5760 | 3840 | | {3,5+}$_{8,8}$ | u$^3$nI dccckD | | {5+,3}$_{8,8}$ GP$_5$(8,8) | c$^3$zD ccctI | |
| 9 | 9 | 243 | II | 2432 | 7290 | 4860 | | {3,5+}$_{9,9}$ | xxnI ktktkD | | {5+,3}$_{9,9}$ GP$_5$(9,9) | yyzD tktktI | |
| 12 | 12 | 432 | II | 4322 | 12960 | 8640 | | {3,5+}$_{12,12}$ | uuxnI dccdktkD | | {5+,3}$_{12,12}$ GP$_5$(12,12) | ccyzD cckttI | |
| 14 | 14 | 588 | II | 5882 | 17640 | 11760 | | {3,5+}$_{14,14}$ | uvvnI dcwwkD | | {5+,3}$_{14,14}$ GP$_5$(14,14) | cwwzD cwrwtI | |
| 16 | 16 | 768 | II | 7682 | 23040 | 15360 | | {3,5+}$_{16,16}$ | uuuunI dcccctI | | {5+,3}$_{16,16}$ GP$_5$(16,16) | cccczD cccctI | |
| 2 | 1 | 7 | III | 72 | 210 | 140 | | {3,5+}$_{2,1}$ | vI dwD | | {5+,3}$_{2,1}$ GP$_5$(2,1) | wD | |
| 3 | 1 | 13 | III | 132 | 390 | 260 | | {3,5+}$_{3,1}$ | v3,1I | | {5+,3}$_{3,1}$ GP$_5$(3,1) | w3,1D | |
| 3 | 2 | 19 | III | 192 | 570 | 380 | | {3,5+}$_{3,2}$ | v3I | | {5+,3}$_{3,2}$ GP$_5$(3,2) | w3D | |
| 4 | 1 | 21 | III | 212 | 630 | 420 | | {3,5+}$_{4,1}$ | dwtI | | {5+,3}$_{4,1}$ GP$_5$(4,1) | wkI | |
| 4 | 2 | 28 | III | 282 | 840 | 560 | | {3,5+}$_{4,2}$ | vnI dwtI | | {5+,3}$_{4,2}$ GP$_5$(4,2) | wdkD | |
| 4 | 3 | 37 | III | 372 | 1110 | 740 | | {3,5+}$_{4,3}$ | v4I | | {5+,3}$_{4,3}$ GP$_5$(4,3) | w4D | |
| 5 | 1 | 31 | III | 312 | 930 | 620 | | {3,5+}$_{5,1}$ | u5,1I | | {5+,3}$_{5,1}$ GP$_5$(5,1) | w5,1D | |
| 5 | 2 | 39 | III | 392 | 1170 | 780 | | {3,5+}$_{5,2}$ | u5,2I | | {5+,3}$_{5,2}$ GP$_5$(5,2) | w5,2D | |
| 5 | 3 | 49 | III | 492 | 1470 | 980 | | {3,5+}$_{5,3}$ | vvI dwwD | | {5+,3}$_{5,3}$ GP$_5$(5,3) | wwD | |
| 6 | 2 | 52 | III | 522 | 1560 | 1040 | | {3,5+}$_{6,2}$ | v3,1uI | | {5+,3}$_{6,2}$ GP$_5$(6,2) | w3,1cD | |
| 6 | 3 | 63 | III | 632 | 1890 | 1260 | | {3,5+}$_{6,3}$ | vxI dwdktI | | {5+,3}$_{6,3}$ GP$_5$(6,3) | wyD wtkD | |
| 8 | 2 | 84 | III | 842 | 2520 | 1680 | | {3,5+}$_{8,2}$ | vunI dwctI | | {5+,3}$_{8,2}$ GP$_5$(8,2) | wczD wcdkD | |
| 8 | 4 | 112 | III | 1122 | 3360 | 2240 | | {3,5+}$_{8,4}$ | vuuI dwccD | | {5+,3}$_{8,4}$ GP$_5$(8,4) | wccD | |
| 11 | 2 | 147 | III | 1472 | 4410 | 2940 | | {3,5+}$_{11,2}$ | vvnI dwwtI | | {5+,3}$_{11,2}$ GP$_5$(11,2) | wwzD | |
| 12 | 3 | 189 | III | 1892 | 5670 | 3780 | | {3,5+}$_{12,3}$ | vxnI dwtktktI | | {5+,3}$_{12,3}$ GP$_5$(12,3) | wyzD wtktI | |
| 10 | 6 | 196 | III | 1962 | 5880 | 3920 | | {3,5+}$_{10,6}$ | vvuI dwwcD | | {5+,3}$_{10,6}$ GP$_5$(10,6) | wwcD | |
| 12 | 6 | 252 | III | 2522 | 7560 | 5040 | | {3,5+}$_{12,6}$ | vxuI dwctkD | | {5+,3}$_{12,6}$ GP$_5$(12,6) | cywD wctkD | |
| 16 | 4 | 336 | III | 3362 | 10080 | 6720 | | {3,5+}$_{16,4}$ | vuunI dwcctI | | {5+,3}$_{16,4}$ GP$_5$(16,4) | wcczD wcctI | |
| 14 | 7 | 343 | III | 3432 | 10290 | 6860 | | {3,5+}$_{14,7}$ | vvvI dwrwwD | | {5+,3}$_{14,7}$ GP$_5$(14,7) | wwwD wrwwD | |
| 15 | 9 | 441 | III | 4412 | 13230 | 8820 | | {3,5+}$_{15,9}$ | vvxI dwwtkD | | {5+,3}$_{15,9}$ GP$_5$(15,9) | wwxD wwtkD | |
| 16 | 8 | 448 | III | 4482 | 13440 | 8960 | | {3,5+}$_{16,8}$ | vuuuI dwcccD | | {5+,3}$_{16,8}$ GP$_5$(16,8) | wcccD | |
| 18 | 1 | 343 | III | 3432 | 10290 | 6860 | | {3,5+}$_{18,1}$ | vvvI dwwwD | | {5+,3}$_{18,1}$ GP$_5$(18,1) | wwwD | |
| 18 | 9 | 567 | III | 5672 | 17010 | 11340 | | {3,5+}$_{18,9}$ | vxxI dwtktkD | | {5+,3}$_{18,9}$ GP$_5$(18,9) | wyyD wtktkD | |
| 20 | 12 | 784 | III | 7842 | 23520 | 15680 | | {3,5+}$_{20,12}$ | vvuuI dwwccD | | {5+,3}$_{20,12}$ GP$_5$(20,12) | wwccD | |
| 20 | 17 | 1029 | III | 10292 | 30870 | 20580 | | {3,5+}$_{20,17}$ | vvvnI dwwwtI | | {5+,3}$_{20,17}$ GP$_5$(20,17) | wwwzD wwwdkD | |
| 28 | 7 | 1029 | III | 10292 | 30870 | 20580 | | {3,5+}$_{28,7}$ | vvvnI dwrwwdkD | | {5+,3}$_{28,7}$ GP$_5$(28,7) | wwwzD wrwwdkD | |

## Octahedral

| m | n | T | Class | Vertices (geodesic) Faces (Goldberg) | Edges | Faces (geodesic) Vertices (Goldberg) | Face triangle | Geodesic | Geodesic | Geodesic | Goldberg | Goldberg | Goldberg |
| m | n | T | Class | Vertices (geodesic) Faces (Goldberg) | Edges | Faces (geodesic) Vertices (Goldberg) | Face triangle | Symbols | Conway | Image | Symbols | Conway | Image |
| 1 | 0 | 1 | I | 6 | 12 | 8 | | {3,4} {3,4+}$_{1,0}$ | O | | {4,3} {4+,3}$_{1,0}$ GP$_4$(1,0) | C | |
| 2 | 0 | 4 | I | 18 | 48 | 32 | | {3,4+}$_{2,0}$ | dcC dcC | | {4+,3}$_{2,0}$ GP$_4$(2,0) | cC cC | |
| 3 | 0 | 9 | I | 38 | 108 | 72 | | {3,4+}$_{3,0}$ | ktO | | {4+,3}$_{3,0}$ GP$_4$(3,0) | tkC | |
| 4 | 0 | 16 | I | 66 | 192 | 128 | | {3,4+}$_{4,0}$ | uuO dccC | | {4+,3}$_{4,0}$ GP$_4$(4,0) | ccC | |
| 5 | 0 | 25 | I | 102 | 300 | 200 | | {3,4+}$_{5,0}$ | u5O | | {4+,3}$_{5,0}$ GP$_4$(5,0) | c5C | |
| 6 | 0 | 36 | I | 146 | 432 | 288 | | {3,4+}$_{6,0}$ | uxO dctkdO | | {4+,3}$_{6,0}$ GP$_4$(6,0) | cyC ctkC | |
| 7 | 0 | 49 | I | 198 | 588 | 392 | | {3,4+}$_{7,0}$ | dwrwO | | {4+,3}$_{7,0}$ GP$_4$(7,0) | wrwO | |
| 8 | 0 | 64 | I | 258 | 768 | 512 | | {3,4+}$_{8,0}$ | uuuO dcccC | | {4+,3}$_{8,0}$ GP$_4$(8,0) | cccC | |
| 9 | 0 | 81 | I | 326 | 972 | 648 | | {3,4+}$_{9,0}$ | xxO ktktO | | {4+,3}$_{9,0}$ GP$_4$(9,0) | yyC tktkC | |
| 1 | 1 | 3 | II | 14 | 36 | 24 | | {3,4+}$_{1,1}$ | kC | | {4+,3}$_{1,1}$ GP$_4$(1,1) | tO | |
| 2 | 2 | 12 | II | 50 | 144 | 96 | | {3,4+}$_{2,2}$ | ukC dctO | | {4+,3}$_{2,2}$ GP$_4$(2,2) | czC ctO | |
| 3 | 3 | 27 | II | 110 | 324 | 216 | | {3,4+}$_{3,3}$ | ktkC | | {4+,3}$_{3,3}$ GP$_4$(3,3) | tktO | |
| 4 | 4 | 48 | II | 194 | 576 | 384 | | {3,4+}$_{4,4}$ | uunO dcctO | | {4+,3}$_{4,4}$ GP$_4$(4,4) | cczC cctO | |
| 2 | 1 | 7 | III | 30 | 84 | 56 | | {3,4+}$_{2,1}$ | vO dwC | | {4+,3}$_{2,1}$ GP$_4$(2,1) | wC | |

## Tetrahedral

| m | n | T | Class | Vertices (geodesic) Faces (Goldberg) | Edges | Faces (geodesic) Vertices (Goldberg) | Face triangle | Geodesic | Geodesic | Geodesic | Goldberg | Goldberg | Goldberg |
| m | n | T | Class | Vertices (geodesic) Faces (Goldberg) | Edges | Faces (geodesic) Vertices (Goldberg) | Face triangle | Symbols | Conway | Image | Symbols | Conway | Image |
| 1 | 0 | 1 | I | 4 | 6 | 4 | | {3,3} {3,3+}$_{1,0}$ | T | | {3,3} {3+,3}$_{1,0}$ GP$_3$(1,0) | T | |
| 1 | 1 | 3 | II | 8 | 18 | 12 | | {3,3+}$_{1,1}$ | kT kT | | {3+,3}$_{1,1}$ GP$_3$(1,1) | tT tT | |
| 2 | 0 | 4 | I | 10 | 24 | 16 | | {3,3+}$_{2,0}$ | dcT dcT | | {3+,3}$_{2,0}$ GP$_3$(2,0) | cT cT | |
| 3 | 0 | 9 | I | 20 | 54 | 36 | | {3,3+}$_{3,0}$ | ktT | | {3+,3}$_{3,0}$ GP$_3$(3,0) | tkT | |
| 4 | 0 | 16 | I | 34 | 96 | 64 | | {3,3+}$_{4,0}$ | uuT dccT | | {3+,3}$_{4,0}$ GP$_3$(4,0) | ccT | |
| 5 | 0 | 25 | I | 52 | 150 | 100 | | {3,3+}$_{5,0}$ | u5T | | {3+,3}$_{5,0}$ GP$_3$(5,0) | c5T | |
| 6 | 0 | 36 | I | 74 | 216 | 144 | | {3,3+}$_{6,0}$ | uxT dctkdT | | {3+,3}$_{6,0}$ GP$_3$(6,0) | cyT ctkT | |
| 7 | 0 | 49 | I | 100 | 294 | 196 | | {3,3+}$_{7,0}$ | vrvT dwrwT | | {3+,3}$_{7,0}$ GP$_3$(7,0) | wrwT | |
| 8 | 0 | 64 | I | 130 | 384 | 256 | | {3,3+}$_{8,0}$ | u$^3$T dcccdT | | {3+,3}$_{8,0}$ GP$_3$(8,0) | c$^3$T cccT | |
| 9 | 0 | 81 | I | 164 | 486 | 324 | | {3,3+}$_{9,0}$ | xxT ktktT | | {3+,3}$_{9,0}$ GP$_3$(9,0) | yyT tktkT | |
| 3 | 3 | 27 | II | 56 | 162 | 108 | | {3,3+}$_{3,3}$ | ktkT | | {3+,3}$_{3,3}$ GP$_3$(3,3) | tktT | |
| 2 | 1 | 7 | III | 16 | 42 | 28 | | {3,3+}$_{2,1}$ | dwT | | {3+,3}$_{2,1}$ GP$_5$(2,1) | wT | |