List of geodesic polyhedra and Goldberg polyhedra

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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 and . T is the triangulation number, which is equal to .

Icosahedral[edit]

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 Subdivided triangle 01 00.svg {3,5}
{3,5+}1,0
I Icosahedron.svg {5,3}
{5+,3}1,0
GP5(1,0)
D Dodecahedron.svg
2 0 4 I 42 120 80 Subdivided triangle 02 00.svg {3,5+}2,0 uI
dcdI
Pentakis icosidodecahedron.png {5+,3}2,0
GP5(2,0)
cD
cD
Truncated rhombic triacontahedron.png
3 0 9 I 92 270 180 Subdivided triangle 03 00.svg {3,5+}3,0 xI
ktI
Conway polyhedron K6k5tI.png {5+,3}3,0
GP5(3,0)
yD
tkD
Conway polyhedron Dk6k5tI.png
4 0 16 I 162 480 320 Subdivided triangle 04 00.svg {3,5+}4,0 uuI
dccD
Conway polyhedron k6k5at5daD.png {5+,3}4,0
GP5(4,0)
c2D Conway polyhedron dk6k5at5daD.png
5 0 25 I 252 750 500 Subdivided triangle 05 00.svg {3,5+}5,0 u5I

u5I

Icosahedron subdivision5.png {5+,3}5,0
GP5(5,0)
c5D
du5I
Goldberg polyhedron 5 0.png
6 0 36 I 362 1080 720 Subdivided triangle 06 00.svg {3,5+}6,0 uxI
dctkdI
Conway polyhedron kdkt5daD.png {5+,3}6,0
GP5(6,0)
cyD
ctkD
Conway polyhedron tkt5daD.png
7 0 49 I 492 1470 980 Subdivided triangle 07 00.svg {3,5+}7,0 vvI
dwrwdI
Conway dwrwD.png {5+,3}7,0
GP5(7,0)
wwD
wrwD
Goldberg polyhedron 7 0.png
8 0 64 I 642 1920 1280 Subdivided triangle 08 00.svg {3,5+}8,0 u3I
dcccdI
Conway dcccD.png {5+,3}8,0
GP5(8,0)
cccD Conway polyhedron dk6k5adk6k5at5daD.png
9 0 81 I 812 2430 1620 Subdivided triangle 09 00.svg {3,5+}9,0 xxI
ktktI
Conway ktktI.png {5+,3}9,0
GP5(9,0)
yyD
tktkD
Conway tdtdtkD.png
10 0 100 I 1002 3000 2000 Subdivided triangle 10 00.svg {3,5+}10,0 uu5I

uu5I

10-subdivided icosahedron.png {5+,3}10,0
GP5(10,0)
cc5D Goldberg polyhedron 10 0.png
11 0 121 I 1212 3630 2420 Subdivided triangle 11 00.svg {3,5+}11,0 u11I

u11I

11-subdivided icosahedron.png {5+,3}11,0
GP5(11,0)
c11D
12 0 144 I 1442 4320 2880 Subdivided triangle 12 00.svg {3,5+}12,0 uuxD
dcctkD
Conway dcctkD.png {5+,3}12,0
GP5(12,0)
ccyD
cctkD
Conway polyhedron dk5k6adk5k6adktI.png
13 0 169 I 1692 5070 3380 Subdivided triangle 13 00.svg {3,5+}13,0 u13I

u13I

13-subdivided icosahedron.png {5+,3}13,0
GP5(13,0)
c13D
14 0 196 I 1962 5880 3920 Subdivided triangle 14 00.svg {3,5+}14,0 uvvI
dcwwdI
Conway dcwrwdI.png {5+,3}14,0
GP5(14,0)
cwrwD Conway cwrwdI.png
15 0 225 I 2252 6750 4500 Subdivided triangle 15 00.svg {3,5+}15,0 u5xI
u5ktI
15-subdivided icosahedron.png {5+,3}15,0
GP5(15,0)
c5yD
c5tkD
Goldberg polyhedron 15 0.png
16 0 256 I 2562 7680 5120 Subdivided triangle 16 00.svg {3,5+}16,0 dc4dI Conway dccccD.png {5+,3}16,0
GP5(16,0)
ccccD Conway polyhedron dk6k5adk6k5adk6k5at5daD.png
1 1 3 II 32 90 60 Subdivided triangle 01 01.svg {3,5+}1,1 nI
kD
Conway polyhedron kD.png {5+,3}1,1
GP5(1,1)
yD
tI
Uniform polyhedron-53-t12.png
2 2 12 II 122 360 240 Subdivided triangle 02 02.svg {3,5+}2,2 unI
=dctI
Conway polyhedron kt5daD.png {5+,3}2,2
GP5(2,2)
czD
cdkD
Conway polyhedron dkt5daD.png
3 3 27 II 272 810 540 Subdivided triangle 03 03.svg {3,5+}3,3 xnI
ktkD
Conway polyhedron kdktI.png {5+,3}3,3
GP5(3,3)
yzD
tkdkD
Conway polyhedron dkdktI.png
4 4 48 II 482 1440 960 Subdivided triangle 04 04.svg {3,5+}4,4 u2nI
dcctI
Conway polyhedron k5k6akdk5aD.png {5+,3}4,4
GP5(4,4)
c2zD
cctI
Conway polyhedron dadkt5daD.png
5 5 75 II 752 2250 1500 Subdivided triangle 05 05.svg {3,5+}5,5 u5nI Conway u5zI.png {5+,3}5,5
GP5(5,5)
c5zD Conway du5zI.png
6 6 108 II 1082 3240 2160 Subdivided triangle 06 06.svg {3,5+}6,6 uxnI
dctktI
Conway polyhedron kdkt5daD.png {5+,3}6,6
GP5(6,6)
cyzD
ctkdkD
Conway cyzD.png
7 7 147 II 1472 4410 2940 Subdivided triangle 07 07.svg {3,5+}7,7 vvnI
dwrwtI
Conway dwrwtI.png {5+,3}7,7
GP5(7,7)
wwzD
wrwdkD
Conway wrwdkD.png
8 8 192 II 1922 5760 3840 Subdivided triangle 08 08.svg {3,5+}8,8 u3nI
dccckD
Conway dccctI.png {5+,3}8,8
GP5(8,8)
c3zD
ccctI
Conway cccdkD.png
9 9 243 II 2432 7290 4860 Subdivided triangle 09 09.svg {3,5+}9,9 xxnI
ktktkD
Conway ktktkdI.png {5+,3}9,9
GP5(9,9)
yyzD
tktktI
Conway tktkdkD.png
12 12 432 II 4322 12960 8640 Subdivided triangle 12 12.svg {3,5+}12,12 uuxnI
dccdktkD
Conway dccdktkdI.png {5+,3}12,12
GP5(12,12)
ccyzD
cckttI
CctkdkD.png
14 14 588 II 5882 17640 11760 Subdivided triangle 14 14.svg {3,5+}14,14 uvvnI
dcwwkD
Conway dcwrwtI.png {5+,3}14,14
GP5(14,14)
cwwzD
cwrwtI
Conway cwwdkD.png
16 16 768 II 7682 23040 15360 Subdivided triangle 16 16.svg {3,5+}16,16 uuuunI
dcccctI
Conway dcccctI.png {5+,3}16,16
GP5(16,16)
cccczD
cccctI
Conway ccccdkD.png
2 1 7 III 72 210 140 Subdivided triangle 02 01.svg {3,5+}2,1 vI
dwD
Conway polyhedron K5sI.png {5+,3}2,1
GP5(2,1)
wD Conway polyhedron Dk5sI.png
3 1 13 III 132 390 260 Subdivided triangle 03 01.svg {3,5+}3,1 v3,1I Conway polyhedron u5I.png {5+,3}3,1
GP5(3,1)
w3,1D Goldberg polyhedron 3 1.png
3 2 19 III 192 570 380 Subdivided triangle 03 02.svg {3,5+}3,2 v3I Geodesic polyhedron 3 2.png {5+,3}3,2
GP5(3,2)
w3D Goldberg polyhedron 3 2.png
4 1 21 III 212 630 420 Subdivided triangle 04 01.svg {3,5+}4,1 dwtI Conway polyhedron K5k6st.png {5+,3}4,1
GP5(4,1)
wkI Conway polyhedron Dk5k6st.png
4 2 28 III 282 840 560 Subdivided triangle 04 02.svg {3,5+}4,2 vnI
dwtI
Conway polyhedron dcwdI.png {5+,3}4,2
GP5(4,2)
wdkD Conway polyhedron dk6k5adk5sD.png
4 3 37 III 372 1110 740 Subdivided triangle 04 03.svg {3,5+}4,3 v4I {5+,3}4,3
GP5(4,3)
w4D Goldberg polyhedron 4 3.png
5 1 31 III 312 930 620 Subdivided triangle 05 01.svg {3,5+}5,1 u5,1I {5+,3}5,1
GP5(5,1)
w5,1D Goldberg polyhedron 5 1.png
5 2 39 III 392 1170 780 Subdivided triangle 05 02.svg {3,5+}5,2 u5,2I Geodesic polyhedron 5 2.png {5+,3}5,2
GP5(5,2)
w5,2D Goldberg polyhedron 5 2.png
5 3 49 III 492 1470 980 Subdivided triangle 05 03.svg {3,5+}5,3 vvI
dwwD
Conway dwwD.png {5+,3}5,3
GP5(5,3)
wwD Goldberg polyhedron 5 3.png
6 2 52 III 522 1560 1040 Subdivided triangle 06 02.svg {3,5+}6,2 v3,1uI Geodesic polyhedron 6 2.png {5+,3}6,2
GP5(6,2)
w3,1cD Goldberg polyhedron 6 2.png
6 3 63 III 632 1890 1260 Subdivided triangle 06 03.svg {3,5+}6,3 vxI
dwdktI
Conway polyhedron dtkwdI.png {5+,3}6,3
GP5(6,3)
wyD
wtkD
Conway polyhedron tkwD.png
8 2 84 III 842 2520 1680 Subdivided triangle 08 02.svg {3,5+}8,2 vunI
dwctI
{5+,3}8,2
GP5(8,2)
wczD
wcdkD
Conway dkcwD.png
8 4 112 III 1122 3360 2240 Subdivided triangle 08 04.svg {3,5+}8,4 vuuI
dwccD
{5+,3}8,4
GP5(8,4)
wccD Conway ccwD.png
11 2 147 III 1472 4410 2940 Subdivided triangle 11 02.svg {3,5+}11,2 vvnI
dwwtI
{5+,3}11,2
GP5(11,2)
wwzD Conway dkwwD.png
12 3 189 III 1892 5670 3780 Subdivided triangle 12 03.svg {3,5+}12,3 vxnI
dwtktktI
{5+,3}12,3
GP5(12,3)
wyzD
wtktI
Conway wtkdkD.png
10 6 196 III 1962 5880 3920 Subdivided triangle 10 06.svg {3,5+}10,6 vvuI
dwwcD
{5+,3}10,6
GP5(10,6)
wwcD Conway cwwD.png
12 6 252 III 2522 7560 5040 Subdivided triangle 12 06.svg {3,5+}12,6 vxuI
dwdktcI
{5+,3}12,6
GP5(12,6)
cywD
wctkD
Conway ctkwD.png
16 4 336 III 3362 10080 6720 Subdivided triangle 16 04.svg {3,5+}16,4 vuunI
dwdckD
{5+,3}16,4
GP5(16,4)
wcczD
wcctI
Conway ccwdkD.png
14 7 343 III 3432 10290 6860 Subdivided triangle 14 07.svg {3,5+}14,7 vvvI
dwrwwD
{5+,3}14,7
GP5(14,7)
wwwD
wrwwD
Conway wrwwD.png
15 9 441 III 4412 13230 8820 Subdivided triangle 15 09.svg {3,5+}15,9 vvxI
dwwtkD
{5+,3}15,9
GP5(15,9)
wwxD
wwtkD
Conway dkwdkwD.png
16 8 448 III 4482 13440 8960 Subdivided triangle 16 08.svg {3,5+}16,8 vuuuI
dwcccD
{5+,3}16,8
GP5(16,8)
wcccD Conway cccwD.png
18 1 343 III 3432 10290 6860 {3,5+}18,1 vvvI
dwwwD
{5+,3}18,1
GP5(18,1)
wwwD Conway wwwD.png
18 9 567 III 5672 17010 11340 {3,5+}18,9 vxxI
dwtktkD
{5+,3}18,9
GP5(18,9)
wyyD
wtktkD
Conway tktkwD.png
20 12 784 III 7842 23520 15680 {3,5+}20,12 vvuuI
dwwccD
{5+,3}20,12
GP5(20,12)
wwccD Conway cwcwD.png
20 17 1029 III 10292 30870 20580 {3,5+}20,17 vvvnI
dwwwtI
{5+,3}20,17
GP5(20,17)
wwwzD
wwwdkD
Conway wwwdkD.png
28 7 1029 III 10292 30870 20580 {3,5+}28,7 vvvnI
dwrwwdkD
{5+,3}28,7
GP5(28,7)
wwwzD
wrwwdkD
Conway wrwwdkD.png

Octahedral[edit]

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 Subdivided triangle 01 00.svg {3,4}
{3,4+}1,0
O Octahedron.svg {4,3}
{4+,3}1,0
GP4(1,0)
C Hexahedron.svg
2 0 4 I 18 48 32 Subdivided triangle 02 00.svg {3,4+}2,0 dcC
dcC
Tetrakis cuboctahedron.png {4+,3}2,0
GP4(2,0)
cC
cC
Truncated rhombic dodecahedron2.png
3 0 9 I 38 108 72 Subdivided triangle 03 00.svg {3,4+}3,0 ktO Octahedral geodesic polyhedron 03 00.svg {4+,3}3,0
GP4(3,0)
tkC Octahedral goldberg polyhedron 03 00.svg
4 0 16 I 66 192 128 Subdivided triangle 04 00.svg {3,4+}4,0 uuO
dccC
Octahedral geodesic polyhedron 04 00.svg {4+,3}4,0
GP4(4,0)
ccC Octahedral goldberg polyhedron 04 00.svg
5 0 25 I 102 300 200 Subdivided triangle 05 00.svg {3,4+}5,0 u5O Octahedral geodesic polyhedron 05 00.svg {4+,3}5,0
GP4(5,0)
c5C Octahedral goldberg polyhedron 05 00.svg
6 0 36 I 146 432 288 Subdivided triangle 06 00.svg {3,4+}6,0 uxO
dctkdO
Octahedral geodesic polyhedron 06 00.svg {4+,3}6,0
GP4(6,0)
cyC
ctkC
Octahedral goldberg polyhedron 06 00.svg
7 0 49 I 198 588 392 Subdivided triangle 07 00.svg {3,4+}7,0 dwrwO Octahedral geodesic polyhedron 07 00.svg {4+,3}7,0
GP4(7,0)
wrwO Octahedral goldberg polyhedron 07 00.svg
8 0 64 I 258 768 512 Subdivided triangle 08 00.svg {3,4+}8,0 uuuO
dcccC
Octahedral geodesic polyhedron 08 00.svg {4+,3}8,0
GP4(8,0)
cccC Octahedral goldberg polyhedron 08 00.svg
9 0 81 I 326 972 648 Subdivided triangle 09 00.svg {3,4+}9,0 xxO
ktktO
Octahedral geodesic polyhedron 09 00.svg {4+,3}9,0
GP4(9,0)
yyC
tktkC
Octahedral goldberg polyhedron 09 00.svg
1 1 3 II 14 36 24 Subdivided triangle 01 01.svg {3,4+}1,1 kC Tetrakishexahedron.jpg {4+,3}1,1
GP4(1,1)
tO Truncated octahedron.png
2 2 12 II 50 144 96 Subdivided triangle 02 02.svg {3,4+}2,2 ukC
dctO
{4+,3}2,2
GP4(2,2)
czC
ctO
Conway polyhedron dkt4daC.png
3 3 27 II 110 324 216 Subdivided triangle 03 03.svg {3,4+}3,3 ktkC {4+,3}3,3
GP4(3,3)
tktO Conway polyhedron tktO.png
4 4 48 II 194 576 384 Subdivided triangle 04 04.svg {3,4+}4,4 uunO
dcctO
{4+,3}4,4
GP4(4,4)
cczC
cctO
Conway polyhedron dk6k4adk6k4adkC.png
2 1 7 III 30 84 56 Subdivided triangle 02 01.svg {3,4+}2,1 vO
dwC
Conway polyhedron dwC.png {4+,3}2,1
GP4(2,1)
wC Conway polyhedron wC.png

Tetrahedral[edit]

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 Subdivided triangle 01 00.svg {3,3}
{3,3+}1,0
T Tetrahedron.svg {3,3}
{3+,3}1,0
GP3(1,0)
T Tetrahedron.svg
1 1 3 II 8 18 12 Subdivided triangle 01 01.svg {3,3+}1,1 kT
kT
Triakistetrahedron.jpg {3+,3}1,1
GP3(1,1)
tT
tT
Truncated tetrahedron.png
2 0 4 I 10 24 16 Subdivided triangle 02 00.svg {3,3+}2,0 dcT
dcT
Dual chamfered tetrahedron.png {3+,3}2,0
GP3(2,0)
cT
cT
Alternate truncated cube.png
3 0 9 I 20 54 36 Subdivided triangle 03 00.svg {3,3+}3,0 ktT Tetrahedral geodesic polyhedron 03 00.svg {3+,3}3,0
GP3(3,0)
tkT Tetrahedral Goldberg polyhedron 03 00.svg
4 0 16 I 34 96 64 Subdivided triangle 04 00.svg {3,3+}4,0 uuT
dccT
Tetrahedral geodesic polyhedron 04 00.svg {3+,3}4,0
GP3(4,0)
ccT Tetrahedral Goldberg polyhedron 04 00.svg
5 0 25 I 52 150 100 Subdivided triangle 05 00.svg {3,3+}5,0 u5T Tetrahedral geodesic polyhedron 05 00.svg {3+,3}5,0
GP3(5,0)
c5T Tetrahedral Goldberg polyhedron 05 00.svg
6 0 36 I 74 216 144 Subdivided triangle 06 00.svg {3,3+}6,0 uxT
dctkdT
Tetrahedral geodesic polyhedron 06 00.svg {3+,3}6,0
GP3(6,0)
cyT
ctkT
Tetrahedral Goldberg polyhedron 06 00.svg
7 0 49 I 100 294 196 Subdivided triangle 07 00.svg {3,3+}7,0 vrvT
dwrwT
Tetrahedral geodesic polyhedron 07 00.svg {3+,3}7,0
GP3(7,0)
wrwT Tetrahedral Goldberg polyhedron 07 00.svg
8 0 64 I 130 384 256 Subdivided triangle 08 00.svg {3,3+}8,0 u3T
dcccdT
Tetrahedral geodesic polyhedron 08 00.svg {3+,3}8,0
GP3(8,0)
c3T
cccT
Tetrahedral Goldberg polyhedron 08 00.svg
9 0 81 I 164 486 324 Subdivided triangle 09 00.svg {3,3+}9,0 xxT
ktktT
Tetrahedral geodesic polyhedron 09 00.svg {3+,3}9,0
GP3(9,0)
yyT
tktkT
Tetrahedral Goldberg polyhedron 09 00.svg
3 3 27 II 56 162 108 Subdivided triangle 03 03.svg {3,3+}3,3 ktkT {3+,3}3,3
GP3(3,3)
tktT Conway polyhedron tktT.png
2 1 7 III 16 42 28 Subdivided triangle 02 01.svg {3,3+}2,1 dwT Conway polyhedron dwT.png {3+,3}2,1
GP5(2,1)
wT Conway polyhedron wT.png

References[edit]

  • 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