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

Icosahedral

m n T Class Vertices(geodesic)Faces(Goldberg) Edges Faces(geodesic)Vertices(Goldberg)FacetriangleGeodesicGoldberg Symbols Conway Image Symbols Conway Image101I123020 {3,5}{3,5+}1,0I {5,3}{5+,3}1,0GP5(1,0)D 204I4212080 {3,5+}2,0uIdcdI {5+,3}2,0GP5(2,0)cDcD309I92270180 {3,5+}3,0xIktI {5+,3}3,0GP5(3,0)yDtkD4016I162480320 {3,5+}4,0uuIdccD {5+,3}4,0GP5(4,0)c2D5025I252750500 {3,5+}5,0u5I u5I {5+,3}5,0GP5(5,0)c5Ddu5I6036I3621080720 {3,5+}6,0uxIdctkdI {5+,3}6,0GP5(6,0)cyDctkD7049I4921470980 {3,5+}7,0vIdwrwdI {5+,3}7,0GP5(7,0)wDwrwD8064I64219201280 {3,5+}8,0u3IdcccdI {5+,3}8,0GP5(8,0)cccD9081I81224301620 {3,5+}9,0xxIktktI {5+,3}9,0GP5(9,0)yyDtktkD100100I100230002000 {3,5+}10,0uu5I uu5I {5+,3}10,0GP5(10,0)cc5Dcdu5I110121I121236302420 {3,5+}11,0u11I u11I {5+,3}11,0GP5(11,0)c11D120144I144243202880 {3,5+}12,0uuxDdcctkD {5+,3}12,0GP5(12,0)ccyDcctkD130169I169250703380 {3,5+}13,0u13I u13I {5+,3}13,0GP5(13,0)c13D140196I196258803920 {3,5+}14,0uvIdcwdI {5+,3}14,0GP5(14,0)cwrwD150225I225267504500 {3,5+}15,0u5xIu5ktI {5+,3}15,0GP5(15,0)c5yDc5tkDdu5ktI160256I256276805120 {3,5+}16,0dc4dI {5+,3}16,0GP5(16,0)ccccD170289I289286705780{3,5+}17,0u17I{5+,3}17,0GP5(17,0)c17D180324I324297206480{3,5+}18,0u18I{5+,3}18,0GP5(18,0)c18D190361I3612108307220{3,5+}19,0u19I{5+,3}19,0GP5(19,0)c19D200400I4002120008000{3,5+}20,0u20I{5+,3}20,0GP5(20,0)c20D113II329060 {3,5+}1,1nIkD {5+,3}1,1GP5(1,1)yDtI2212II122360240 {3,5+}2,2unI=dctI {5+,3}2,2GP5(2,2)czDcdkD3327II272810540 {3,5+}3,3xnIktkD {5+,3}3,3GP5(3,3)yzDtkdkD4448II4821440960 {3,5+}4,4u2nIdcctI {5+,3}4,4GP5(4,4)c2zDcctI5575II75222501500 {3,5+}5,5u5nI {5+,3}5,5GP5(5,5)c5zD66108II108232402160 {3,5+}6,6uxnIdctktI {5+,3}6,6GP5(6,6)cyzDctkdkD77147II147244102940 {3,5+}7,7vnIdwrwtI {5+,3}7,7GP5(7,7)wzDwrwdkD88192II192257603840 {3,5+}8,8u3nIdccckD {5+,3}8,8GP5(8,8)c3zDccctI99243II243272904860 {3,5+}9,9xxnIktktkD {5+,3}9,9GP5(9,9)yyzDtktktI1212432II4322129608640 {3,5+}12,12uuxnIdccdktkD {5+,3}12,12GP5(12,12)ccyzDcckttI1414588II58821764011760 {3,5+}14,14uvnIdcwkD {5+,3}14,14GP5(14,14)cwzDcwrwtI1616768II76822304015360 {3,5+}16,16uuuunIdcccctI {5+,3}16,16GP5(16,16)cccczDcccctI217III72210140 {3,5+}2,1vIdwD{5+,3}2,1GP5(2,1)wD3113III132390260 {3,5+}3,1v3,1I {5+,3}3,1GP5(3,1)w3,1D3219III192570380 {3,5+}3,2v3I {5+,3}3,2GP5(3,2)w3D4121III212630420 {3,5+}4,1dwtI {5+,3}4,1GP5(4,1)wdkD4228III282840560 {3,5+}4,2vnIdwtI {5+,3}4,2GP5(4,2)wcD4337III3721110740 {3,5+}4,3v4I {5+,3}4,3GP5(4,3)w4D5131III312930620 {3,5+}5,1u5,1I {5+,3}5,1GP5(5,1)w5,1D5239III3921170780 {3,5+}5,2u5,2I {5+,3}5,2GP5(5,2)w5,2D5349III4921470980 {3,5+}5,3vvIdwwD {5+,3}5,3GP5(5,3)wwD5461III61218301220 {3,5+}5,4u5,4I {5+,3}5,4GP5(5,4)w5,4D6252III52215601040 {3,5+}6,2v3,1uI {5+,3}6,2GP5(6,2)w3,1cD6363III63218901260 {3,5+}6,3vxIdwdktI {5+,3}6,3GP5(6,3)wyDwtkD8284III84225201680 {3,5+}8,2vunIdwctI {5+,3}8,2GP5(8,2)wczDwcdkD84112III112233602240 {3,5+}8,4vuuIdwccD {5+,3}8,4GP5(8,4)wccD112147III147244102940 {3,5+}11,2vvnIdwwtI {5+,3}11,2GP5(11,2)wwzD123189III189256703780 {3,5+}12,3vxnIdwtktktI {5+,3}12,3GP5(12,3)wyzDwtktI106196III196258803920 {3,5+}10,6vvuIdwwcD {5+,3}10,6GP5(10,6)wwcD126252III252275605040 {3,5+}12,6vxuIdwctkD {5+,3}12,6GP5(12,6)cywDwctkD164336III3362100806720 {3,5+}16,4vuunIdwcctI {5+,3}16,4GP5(16,4)wcczDwcctI147343III3432102906860 {3,5+}14,7vvIdwrwwD {5+,3}14,7GP5(14,7)wwDwrwwD159441III4412132308820 {3,5+}15,9vvxIdwwtkD {5+,3}15,9GP5(15,9)wwxDwwtkD168448III4482134408960 {3,5+}16,8vuuuIdwcccD {5+,3}16,8GP5(16,8)wcccD181343III3432102906860{3,5+}18,1vvvIdwwwD {5+,3}18,1GP5(18,1)wwwD189567III56721701011340{3,5+}18,9vxxIdwtktkD {5+,3}18,9GP5(18,9)wyyDwtktkD2012784III78422352015680{3,5+}20,12vvuuIdwwccD {5+,3}20,12GP5(20,12)wwccD20171029III102923087020580{3,5+}20,17vvvnIdwwwtI {5+,3}20,17GP5(20,17)wwwzDwwwdkD2871029III102923087020580{3,5+}28,7vvnIdwrwwdkD {5+,3}28,7GP5(28,7)wwzDwrwwdkD

Octahedral m n T Class Vertices(geodesic)Faces(Goldberg) Edges Faces(geodesic)Vertices(Goldberg)FacetriangleGeodesicGoldberg Symbols Conway Image Symbols Conway Image101I6128 {3,4}{3,4+}1,0O {4,3}{4+,3}1,0GP4(1,0)C 204I184832 {3,4+}2,0dcCdcC {4+,3}2,0GP4(2,0)cCcC309I3810872 {3,4+}3,0ktO {4+,3}3,0GP4(3,0)tkC4016I66192128 {3,4+}4,0uuOdccC {4+,3}4,0GP4(4,0)ccC5025I102300200 {3,4+}5,0u5O {4+,3}5,0GP4(5,0)c5C6036I146 432 288 {3,4+}6,0uxOdctkdO {4+,3}6,0GP4(6,0)cyCctkC7049I198588392 {3,4+}7,0dwrwO {4+,3}7,0GP4(7,0)wrwO8064I258768512 {3,4+}8,0uuuOdcccC {4+,3}8,0GP4(8,0)cccC9081I326972648 {3,4+}9,0xxOktktO {4+,3}9,0GP4(9,0)yyCtktkC113II143624 {3,4+}1,1kC {4+,3}1,1GP4(1,1)tO 2212II5014496 {3,4+}2,2ukCdctO {4+,3}2,2GP4(2,2)czCctO3327II110324216 {3,4+}3,3ktkC {4+,3}3,3GP4(3,3)tktO4448II194576384 {3,4+}4,4uunOdcctO {4+,3}4,4GP4(4,4)cczCcctO217III308456 {3,4+}2,1vOdwC {4+,3}2,1GP4(2,1)wC

Tetrahedral m n T Class Vertices(geodesic)Faces(Goldberg) Edges Faces(geodesic)Vertices(Goldberg)FacetriangleGeodesicGoldberg Symbols Conway Image Symbols Conway Image101I464 {3,3}{3,3+}1,0T {3,3}{3+,3}1,0GP3(1,0)T 113II81812 {3,3+}1,1kTkT {3+,3}1,1GP3(1,1)tTtT 204I102416 {3,3+}2,0dcTdcT {3+,3}2,0GP3(2,0)cTcT 309I205436 {3,3+}3,0ktT {3+,3}3,0GP3(3,0)tkT4016I349664 {3,3+}4,0uuTdccT {3+,3}4,0GP3(4,0)ccT5025I52150100 {3,3+}5,0u5T {3+,3}5,0GP3(5,0)c5T6036I74216144 {3,3+}6,0uxTdctkdT {3+,3}6,0GP3(6,0)cyTctkT7049I100294196 {3,3+}7,0vrvTdwrwT {3+,3}7,0GP3(7,0)wrwT8064I130384256 {3,3+}8,0u3TdcccdT {3+,3}8,0GP3(8,0)c3TcccT9081I164486324 {3,3+}9,0xxTktktT {3+,3}9,0GP3(9,0)yyTtktkT3327II56162108 {3,3+}3,3ktkT {3+,3}3,3GP3(3,3)tktT217III164228 {3,3+}2,1dwT {3+,3}2,1GP5(2,1)wT

References Reprinted by Dover 1999