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

## 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 {5,3}
{5+,3}1,0
GP5(1,0)
D
2 0 4 I 42 120 80 {3,5+}2,0 uI
dcdI
{5+,3}2,0
GP5(2,0)
cD
cD
3 0 9 I 92 270 180 {3,5+}3,0 xI
ktI
{5+,3}3,0
GP5(3,0)
yD
tkD
4 0 16 I 162 480 320 {3,5+}4,0 uuI
dccD
{5+,3}4,0
GP5(4,0)
c2D
5 0 25 I 252 750 500 {3,5+}5,0 u5I {5+,3}5,0
GP5(5,0)
c5D
du5I
6 0 36 I 362 1080 720 {3,5+}6,0 uxI
dctkdI
{5+,3}6,0
GP5(6,0)
cyD
ctkD
7 0 49 I 492 1470 980 {3,5+}7,0 vvI
dwrwdI
{5+,3}7,0
GP5(7,0)
wwD
wrwD
8 0 64 I 642 1920 1280 {3,5+}8,0 u3I
dcccdI
{5+,3}8,0
GP5(8,0)
cccD
9 0 81 I 812 2430 1620 {3,5+}9,0 xxI
ktktI
{5+,3}9,0
GP5(9,0)
yyD
tktkD
10 0 100 I 1002 3000 2000 {3,5+}10,0 uu5I {5+,3}10,0
GP5(10,0)
cc5D
11 0 121 I 1212 3630 2420 {3,5+}11,0 u11I {5+,3}11,0
GP5(11,0)
c11D
12 0 144 I 1442 4320 2880 {3,5+}12,0 uuxD
dcctkD
{5+,3}12,0
GP5(12,0)
ccyD
cctkD
13 0 169 I 1692 5070 3380 {3,5+}13,0 u13I {5+,3}13,0
GP5(13,0)
c13D
14 0 196 I 1962 5880 3920 {3,5+}14,0 uvvI
dcwwdI
{5+,3}14,0
GP5(14,0)
cwrwD
15 0 225 I 2252 6750 4500 {3,5+}15,0 u5xI
u5ktI
{5+,3}15,0
GP5(15,0)
c5yD
c5tkD
16 0 256 I 2562 7680 5120 {3,5+}16,0 dc4dI {5+,3}16,0
GP5(16,0)
ccccD
1 1 3 II 32 90 60 {3,5+}1,1 nI
kD
{5+,3}1,1
GP5(1,1)
yD
tI
2 2 12 II 122 360 240 {3,5+}2,2 unI
=dctI
{5+,3}2,2
GP5(2,2)
czD
cdkD
3 3 27 II 272 810 540 {3,5+}3,3 xnI
ktkD
{5+,3}3,3
GP5(3,3)
yzD
tkdkD
4 4 48 II 482 1440 960 {3,5+}4,4 u2nI
dcctI
{5+,3}4,4
GP5(4,4)
c2zD
cctI
5 5 75 II 752 2250 1500 {3,5+}5,5 u5nI {5+,3}5,5
GP5(5,5)
c5zD
6 6 108 II 1082 3240 2160 {3,5+}6,6 uxnI
dctktI
{5+,3}6,6
GP5(6,6)
cyzD
ctkdkD
7 7 147 II 1472 4410 2940 {3,5+}7,7 vvnI
dwrwtI
{5+,3}7,7
GP5(7,7)
wwzD
wrwdkD
8 8 192 II 1922 5760 3840 {3,5+}8,8 u3nI
dccckD
{5+,3}8,8
GP5(8,8)
c3zD
ccctI
9 9 243 II 2432 7290 4860 {3,5+}9,9 xxnI
ktktkD
{5+,3}9,9
GP5(9,9)
yyzD
tktktI
12 12 432 II 4322 12960 8640 {3,5+}12,12 uuxnI
dccdktkD
{5+,3}12,12
GP5(12,12)
ccyzD
cckttI
14 14 588 II 5882 17640 11760 {3,5+}14,14 uvvnI
dcwwkD
{5+,3}14,14
GP5(14,14)
cwwzD
cwrwtI
16 16 768 II 7682 23040 15360 {3,5+}16,16 uuuunI
dcccctI
{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 {5+,3}3,1
GP5(3,1)
w3,1D
3 2 19 III 192 570 380 {3,5+}3,2 v3I {5+,3}3,2
GP5(3,2)
w3D
4 1 21 III 212 630 420 {3,5+}4,1 dwtI {5+,3}4,1
GP5(4,1)
wkI
4 2 28 III 282 840 560 {3,5+}4,2 vnI
dwtI
{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 {5+,3}5,2
GP5(5,2)
w5,2D
5 3 49 III 492 1470 980 {3,5+}5,3 vvI
dwwD
{5+,3}5,3
GP5(5,3)
wwD
6 2 52 III 522 1560 1040 {3,5+}6,2 v3,1uI {5+,3}6,2
GP5(6,2)
w3,1cD
6 3 63 III 632 1890 1260 {3,5+}6,3 vxI
dwdktI
{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
dwdktcI
{5+,3}12,6
GP5(12,6)
cywD
wctkD
16 4 336 III 3362 10080 6720 {3,5+}16,4 vuunI
dwdckD
{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 {4,3}
{4+,3}1,0
GP4(1,0)
C
2 0 4 I 18 48 32 {3,4+}2,0 dcC
dcC
{4+,3}2,0
GP4(2,0)
cC
cC
3 0 9 I 38 108 72 {3,4+}3,0 ktO {4+,3}3,0
GP4(3,0)
tkC
4 0 16 I 66 192 128 {3,4+}4,0 uuO
dccC
{4+,3}4,0
GP4(4,0)
ccC
5 0 25 I 102 300 200 {3,4+}5,0 u5O {4+,3}5,0
GP4(5,0)
c5C
6 0 36 I 146 432 288 {3,4+}6,0 uxO
dctkdO
{4+,3}6,0
GP4(6,0)
cyC
ctkC
7 0 49 I 198 588 392 {3,4+}7,0 dwrwO {4+,3}7,0
GP4(7,0)
wrwO
8 0 64 I 258 768 512 {3,4+}8,0 uuuO
dcccC
{4+,3}8,0
GP4(8,0)
cccC
9 0 81 I 326 972 648 {3,4+}9,0 xxO
ktktO
{4+,3}9,0
GP4(9,0)
yyC
tktkC
1 1 3 II 14 36 24 {3,4+}1,1 kC {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
{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 {3,3}
{3+,3}1,0
GP3(1,0)
T
1 1 3 II 8 18 12 {3,3+}1,1 kT
kT
{3+,3}1,1
GP3(1,1)
tT
tT
2 0 4 I 10 24 16 {3,3+}2,0 dcT
dcT
{3+,3}2,0
GP3(2,0)
cT
cT
3 0 9 I 20 54 36 {3,3+}3,0 ktT {3+,3}3,0
GP3(3,0)
tkT
4 0 16 I 34 96 64 {3,3+}4,0 uuT
dccT
{3+,3}4,0
GP3(4,0)
ccT
5 0 25 I 52 150 100 {3,3+}5,0 u5T {3+,3}5,0
GP3(5,0)
c5T
6 0 36 I 74 216 144 {3,3+}6,0 uxT
dctkdT
{3+,3}6,0
GP3(6,0)
cyT
ctkT
7 0 49 I 100 294 196 {3,3+}7,0 vrvT
dwrwT
{3+,3}7,0
GP3(7,0)
wrwT
8 0 64 I 130 384 256 {3,3+}8,0 u3T
dcccdT
{3+,3}8,0
GP3(8,0)
c3T
cccT
9 0 81 I 164 486 324 {3,3+}9,0 xxT
ktktT
{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 {3+,3}2,1
GP5(2,1)
wT

## References

• Wenninger, Magnus (1979), Spherical Models, Cambridge University Press, ISBN 978-0-521-29432-4, MR 0552023, archived from the original on July 4, 2008 Reprinted by Dover 1999 ISBN 978-0-486-40921-4