List of geodesic polyhedra and Goldberg polyhedra

Short description: none

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≥0. T is the triangulation number, which is equal to T=m2+mn+n2.

## Contents

* 1 Icosahedral
* 2 Octahedral
* 3 Tetrahedral
* 4 References

## Icosahedral

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 |  | {3,5}
{3,5+}1,0 | I | 60px | {5,3}
{5+,3}1,0
GP5(1,0) | D |
2 | 0 | 4 | I | 42 | 120 | 80 |  | {3,5+}2,0 | uI
dcdI | 60px | {5+,3}2,0
GP5(2,0) | cD
cD |
3 | 0 | 9 | I | 92 | 270 | 180 |  | {3,5+}3,0 | xI
ktI | 60px | {5+,3}3,0
GP5(3,0) | yD
tkD |
4 | 0 | 16 | I | 162 | 480 | 320 |  | {3,5+}4,0 | uuI
dccD | 60px | {5+,3}4,0
GP5(4,0) | c2D | Error creating thumbnail: Unable to save thumbnail to destination
5 | 0 | 25 | I | 252 | 750 | 500 |  | {3,5+}5,0 | u5I

u5I

|  | {5+,3}5,0
GP5(5,0) | c5D
du5I |
6 | 0 | 36 | I | 362 | 1080 | 720 |  | {3,5+}6,0 | uxI
dctkdI | 60px | {5+,3}6,0
GP5(6,0) | cyD
ctkD |
7 | 0 | 49 | I | 492 | 1470 | 980 |  | {3,5+}7,0 | vvI
dwrwdI | 60px | {5+,3}7,0
GP5(7,0) | wwD
wrwD |
8 | 0 | 64 | I | 642 | 1920 | 1280 |  | {3,5+}8,0 | u3I
dcccdI | 60px | {5+,3}8,0
GP5(8,0) | cccD |
9 | 0 | 81 | I | 812 | 2430 | 1620 |  | {3,5+}9,0 | xxI
ktktI | 60px | {5+,3}9,0
GP5(9,0) | yyD
tktkD |
10 | 0 | 100 | I | 1002 | 3000 | 2000 |  | {3,5+}10,0 | uu5I

uu5I

|  | {5+,3}10,0
GP5(10,0) | cc5D |
11 | 0 | 121 | I | 1212 | 3630 | 2420 |  | {3,5+}11,0 | u11I

u11I

|  | {5+,3}11,0
GP5(11,0) | c11D |
12 | 0 | 144 | I | 1442 | 4320 | 2880 |  | {3,5+}12,0 | uuxD
dcctkD | 60px | {5+,3}12,0
GP5(12,0) | ccyD
cctkD |
13 | 0 | 169 | I | 1692 | 5070 | 3380 |  | {3,5+}13,0 | u13I

u13I

|  | {5+,3}13,0
GP5(13,0) | c13D |
14 | 0 | 196 | I | 1962 | 5880 | 3920 |  | {3,5+}14,0 | uvvI
dcwwdI | 60px | {5+,3}14,0
GP5(14,0) | cwrwD |
15 | 0 | 225 | I | 2252 | 6750 | 4500 |  | {3,5+}15,0 | u5xI
u5ktI | 60px | {5+,3}15,0
GP5(15,0) | c5yD
c5tkD |
16 | 0 | 256 | I | 2562 | 7680 | 5120 |  | {3,5+}16,0 | dc4dI | 60px | {5+,3}16,0
GP5(16,0) | ccccD |
1 | 1 | 3 | II | 32 | 90 | 60 |  | {3,5+}1,1 | nI
kD | 60px | {5+,3}1,1
GP5(1,1) | yD
tI |
2 | 2 | 12 | II | 122 | 360 | 240 |  | {3,5+}2,2 | unI
=dctI | 60px | {5+,3}2,2
GP5(2,2) | czD
cdkD |
3 | 3 | 27 | II | 272 | 810 | 540 |  | {3,5+}3,3 | xnI
ktkD | 60px | {5+,3}3,3
GP5(3,3) | yzD
tkdkD | Error creating thumbnail: Unable to save thumbnail to destination
4 | 4 | 48 | II | 482 | 1440 | 960 |  | {3,5+}4,4 | u2nI
dcctI | 60px | {5+,3}4,4
GP5(4,4) | c2zD
cctI |
5 | 5 | 75 | II | 752 | 2250 | 1500 |  | {3,5+}5,5 | u5nI | 60px | {5+,3}5,5
GP5(5,5) | c5zD |
6 | 6 | 108 | II | 1082 | 3240 | 2160 |  | {3,5+}6,6 | uxnI
dctktI | 60px | {5+,3}6,6
GP5(6,6) | cyzD
ctkdkD |
7 | 7 | 147 | II | 1472 | 4410 | 2940 |  | {3,5+}7,7 | vvnI
dwrwtI | 60px | {5+,3}7,7
GP5(7,7) | wwzD
wrwdkD |
8 | 8 | 192 | II | 1922 | 5760 | 3840 |  | {3,5+}8,8 | u3nI
dccckD | 60px | {5+,3}8,8
GP5(8,8) | c3zD
ccctI |
9 | 9 | 243 | II | 2432 | 7290 | 4860 |  | {3,5+}9,9 | xxnI
ktktkD | 60px | {5+,3}9,9
GP5(9,9) | yyzD
tktktI |
12 | 12 | 432 | II | 4322 | 12960 | 8640 |  | {3,5+}12,12 | uuxnI
dccdktkD | 60px | {5+,3}12,12
GP5(12,12) | ccyzD
cckttI |
14 | 14 | 588 | II | 5882 | 17640 | 11760 |  | {3,5+}14,14 | uvvnI
dcwwkD | 60px | {5+,3}14,14
GP5(14,14) | cwwzD
cwrwtI |
16 | 16 | 768 | II | 7682 | 23040 | 15360 |  | {3,5+}16,16 | uuuunI
dcccctI | 60px | {5+,3}16,16
GP5(16,16) | cccczD
cccctI |
2 | 1 | 7 | III | 72 | 210 | 140 |  | {3,5+}2,1 | vI
dwD |  | {5+,3}2,1
GP5(2,1) | wD |
3 | 1 | 13 | III | 132 | 390 | 260 |  | {3,5+}3,1 | v3,1I | 60px | {5+,3}3,1
GP5(3,1) | w3,1D |
3 | 2 | 19 | III | 192 | 570 | 380 |  | {3,5+}3,2 | v3I | 60px | {5+,3}3,2
GP5(3,2) | w3D |
4 | 1 | 21 | III | 212 | 630 | 420 |  | {3,5+}4,1 | dwtI | 60px | {5+,3}4,1
GP5(4,1) | wkI |
4 | 2 | 28 | III | 282 | 840 | 560 |  | {3,5+}4,2 | vnI
dwtI | 60px | {5+,3}4,2
GP5(4,2) | wdkD |
4 | 3 | 37 | III | 372 | 1110 | 740 |  | {3,5+}4,3 | v4I |  | {5+,3}4,3
GP5(4,3) | w4D |
5 | 1 | 31 | III | 312 | 930 | 620 |  | {3,5+}5,1 | u5,1I |  | {5+,3}5,1
GP5(5,1) | w5,1D |
5 | 2 | 39 | III | 392 | 1170 | 780 |  | {3,5+}5,2 | u5,2I | 60px | {5+,3}5,2
GP5(5,2) | w5,2D |
5 | 3 | 49 | III | 492 | 1470 | 980 |  | {3,5+}5,3 | vvI
dwwD | 60px | {5+,3}5,3
GP5(5,3) | wwD |
6 | 2 | 52 | III | 522 | 1560 | 1040 |  | {3,5+}6,2 | v3,1uI | 60px | {5+,3}6,2
GP5(6,2) | w3,1cD |
6 | 3 | 63 | III | 632 | 1890 | 1260 |  | {3,5+}6,3 | vxI
dwdktI | 60px | {5+,3}6,3
GP5(6,3) | wyD
wtkD |
8 | 2 | 84 | III | 842 | 2520 | 1680 |  | {3,5+}8,2 | vunI
dwctI |  | {5+,3}8,2
GP5(8,2) | wczD
wcdkD |
8 | 4 | 112 | III | 1122 | 3360 | 2240 |  | {3,5+}8,4 | vuuI
dwccD |  | {5+,3}8,4
GP5(8,4) | wccD |
11 | 2 | 147 | III | 1472 | 4410 | 2940 |  | {3,5+}11,2 | vvnI
dwwtI |  | {5+,3}11,2
GP5(11,2) | wwzD |
12 | 3 | 189 | III | 1892 | 5670 | 3780 |  | {3,5+}12,3 | vxnI
dwtktktI |  | {5+,3}12,3
GP5(12,3) | wyzD
wtktI |
10 | 6 | 196 | III | 1962 | 5880 | 3920 |  | {3,5+}10,6 | vvuI
dwwcD |  | {5+,3}10,6
GP5(10,6) | wwcD |
12 | 6 | 252 | III | 2522 | 7560 | 5040 |  | {3,5+}12,6 | vxuI
dwctkD |  | {5+,3}12,6
GP5(12,6) | cywD
wctkD |
16 | 4 | 336 | III | 3362 | 10080 | 6720 |  | {3,5+}16,4 | vuunI
dwcctI |  | {5+,3}16,4
GP5(16,4) | wcczD
wcctI |
14 | 7 | 343 | III | 3432 | 10290 | 6860 |  | {3,5+}14,7 | vvvI
dwrwwD |  | {5+,3}14,7
GP5(14,7) | wwwD
wrwwD |
15 | 9 | 441 | III | 4412 | 13230 | 8820 |  | {3,5+}15,9 | vvxI
dwwtkD |  | {5+,3}15,9
GP5(15,9) | wwxD
wwtkD |
16 | 8 | 448 | III | 4482 | 13440 | 8960 |  | {3,5+}16,8 | vuuuI
dwcccD |  | {5+,3}16,8
GP5(16,8) | wcccD |
18 | 1 | 343 | III | 3432 | 10290 | 6860 |  | {3,5+}18,1 | vvvI
dwwwD |  | {5+,3}18,1
GP5(18,1) | wwwD |
18 | 9 | 567 | III | 5672 | 17010 | 11340 |  | {3,5+}18,9 | vxxI
dwtktkD |  | {5+,3}18,9
GP5(18,9) | wyyD
wtktkD |
20 | 12 | 784 | III | 7842 | 23520 | 15680 |  | {3,5+}20,12 | vvuuI
dwwccD |  | {5+,3}20,12
GP5(20,12) | wwccD |
20 | 17 | 1029 | III | 10292 | 30870 | 20580 |  | {3,5+}20,17 | vvvnI
dwwwtI |  | {5+,3}20,17
GP5(20,17) | wwwzD
wwwdkD |
28 | 7 | 1029 | III | 10292 | 30870 | 20580 |  | {3,5+}28,7 | vvvnI
dwrwwdkD |  | {5+,3}28,7
GP5(28,7) | wwwzD
wrwwdkD |

## Octahedral

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 |  | {3,4}
{3,4+}1,0 | O | 60px | {4,3}
{4+,3}1,0
GP4(1,0) | C |
2 | 0 | 4 | I | 18 | 48 | 32 |  | {3,4+}2,0 | dcC
dcC | 60px | {4+,3}2,0
GP4(2,0) | cC
cC |
3 | 0 | 9 | I | 38 | 108 | 72 |  | {3,4+}3,0 | ktO | 60px | {4+,3}3,0
GP4(3,0) | tkC |
4 | 0 | 16 | I | 66 | 192 | 128 |  | {3,4+}4,0 | uuO
dccC | 60px | {4+,3}4,0
GP4(4,0) | ccC |
5 | 0 | 25 | I | 102 | 300 | 200 |  | {3,4+}5,0 | u5O | 60px | {4+,3}5,0
GP4(5,0) | c5C |
6 | 0 | 36 | I | 146 | 432 | 288 |  | {3,4+}6,0 | uxO
dctkdO | 60px | {4+,3}6,0
GP4(6,0) | cyC
ctkC |
7 | 0 | 49 | I | 198 | 588 | 392 |  | {3,4+}7,0 | dwrwO | 60px | {4+,3}7,0
GP4(7,0) | wrwO |
8 | 0 | 64 | I | 258 | 768 | 512 |  | {3,4+}8,0 | uuuO
dcccC | 60px | {4+,3}8,0
GP4(8,0) | cccC |
9 | 0 | 81 | I | 326 | 972 | 648 |  | {3,4+}9,0 | xxO
ktktO | 60px | {4+,3}9,0
GP4(9,0) | yyC
tktkC |
1 | 1 | 3 | II | 14 | 36 | 24 |  | {3,4+}1,1 | kC | 60px | {4+,3}1,1
GP4(1,1) | tO |
2 | 2 | 12 | II | 50 | 144 | 96 |  | {3,4+}2,2 | ukC
dctO |  | {4+,3}2,2
GP4(2,2) | czC
ctO |
3 | 3 | 27 | II | 110 | 324 | 216 |  | {3,4+}3,3 | ktkC |  | {4+,3}3,3
GP4(3,3) | tktO |
4 | 4 | 48 | II | 194 | 576 | 384 |  | {3,4+}4,4 | uunO
dcctO |  | {4+,3}4,4
GP4(4,4) | cczC
cctO |
2 | 1 | 7 | III | 30 | 84 | 56 |  | {3,4+}2,1 | vO
dwC | 60px | {4+,3}2,1
GP4(2,1) | wC |

## Tetrahedral

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 |  | {3,3}
{3,3+}1,0 | T | 60px | {3,3}
{3+,3}1,0
GP3(1,0) | T | 60px
1 | 1 | 3 | II | 8 | 18 | 12 |  | {3,3+}1,1 | kT
kT | 60px | {3+,3}1,1
GP3(1,1) | tT
tT |
2 | 0 | 4 | I | 10 | 24 | 16 |  | {3,3+}2,0 | dcT
dcT | 60px | {3+,3}2,0
GP3(2,0) | cT
cT |
3 | 0 | 9 | I | 20 | 54 | 36 |  | {3,3+}3,0 | ktT | 60px | {3+,3}3,0
GP3(3,0) | tkT |
4 | 0 | 16 | I | 34 | 96 | 64 |  | {3,3+}4,0 | uuT
dccT | 60px | {3+,3}4,0
GP3(4,0) | ccT |
5 | 0 | 25 | I | 52 | 150 | 100 |  | {3,3+}5,0 | u5T | 60px | {3+,3}5,0
GP3(5,0) | c5T |
6 | 0 | 36 | I | 74 | 216 | 144 |  | {3,3+}6,0 | uxT
dctkdT | 60px | {3+,3}6,0
GP3(6,0) | cyT
ctkT |
7 | 0 | 49 | I | 100 | 294 | 196 |  | {3,3+}7,0 | vrvT
dwrwT | 60px | {3+,3}7,0
GP3(7,0) | wrwT |
8 | 0 | 64 | I | 130 | 384 | 256 |  | {3,3+}8,0 | u3T
dcccdT | 60px | {3+,3}8,0
GP3(8,0) | c3T
cccT |
9 | 0 | 81 | I | 164 | 486 | 324 |  | {3,3+}9,0 | xxT
ktktT | 60px | {3+,3}9,0
GP3(9,0) | yyT
tktkT |
3 | 3 | 27 | II | 56 | 162 | 108 |  | {3,3+}3,3 | ktkT |  | {3+,3}3,3
GP3(3,3) | tktT |
2 | 1 | 7 | III | 16 | 42 | 28 |  | {3,3+}2,1 | dwT | 60px | {3+,3}2,1
GP5(2,1) | wT |

## References

* Wenninger, Magnus (1979), Spherical Models, Cambridge University Press, ISBN 978-0-521-29432-4, archived from the original on July 4, 2008, https://web.archive.org/web/20080704103400/http://www.employees.csbsju.edu/mwenninger/magnus2000.html Reprinted by Dover 1999 ISBN 978-0-486-40921-4

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