Goldberg polyhedron

Icosahedral Goldberg polyhedra with pentagons in red

G(1,4)
G(4,4)

G(7,0)
G(5,3)

A Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described by Michael Goldberg (1902–1990) in 1937. They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. G(5,3) and G(3,5) are enantiomorphs of each other. A consequence of Euler's polyhedron formula is that there will be exactly twelve pentagons.

Icosahedral symmetry ensures that the pentagons are always regular, although many of the hexagons may not be. Typically all of the vertices lie on a sphere.

It is a dual polyhedron of a geodesic sphere, with all triangle faces and 6 triangles per vertex, except for 12 vertices with 5 triangles.

Simple examples of Goldberg polyhedra include the dodecahedron and truncated icosahedron. Other forms can be described by taking a chess knight move from one pentagon to the next: first take m steps in one direction, then turn 60° to the left and take n steps. Such a polyhedron is denoted G(m,n). A dodecahedron is G(1,0) and a truncated icosahedron is G(1,1).

A similar technique can be applied to construct polyhedra with tetrahedral symmetry and octahedral symmetry. These polyhedra will have triangles or squares rather than pentagons. These variations are given Roman numeral subscripts: GIII(n,m), GIV(n,m), and GV(n,m).

Polyhedral elements

The number of vertices, edges, and faces of G(m,n) can be computed from m and n, with T = m2 + mn + n2 = (m + n)2 − mn, depending on one of three symmetry systems:[1]

System Vertices Edges Faces Faces by type
Tetrahedral
GIII(m,n)		 4T6T2T + 24 {3} and 2(T − 1) {6}
Octahedral
GIV(m,n)		 8T12T4T + 26 {4} and 4(T − 1) {6}
Icosahedral
GV(m,n)		 20T30T10T + 212 {5} and 10(T − 1) {6}

Small examples by symmetry family

A few polyhedra are given with Conway polyhedron notation starting with (T)etrahedron, (C)ube, (O)ctahedron, and (D)odecahedron, (I)cosahedron seeds. The operator dk (dual kis) generates G(1,1). The chamfer operator, c, replaces all edges by hexagons and transforms G(m,n) to G(2m,2n). In addition, the tk operator, transforms G(m,n) to G(3m,3n).

Class 1 G(0,n)
System G(0,1)
T = 1		 G(0,2)
T = 4		 G(0,3)
T = 9		 G(0,4)
T = 16		 G(0,5)
T = 25		 G(0,6)
T = 36		 G(0,7)
T = 49		 G(0,8)
T = 64		 G(0,9)
T = 81		 G(0,12)
T = 144		 G(0,16)
T = 256
Tetrahedral
GIII(0,n)
(T)
(cT)
(tkT)		 (ccT) (ctkT) (wrwT) (cccT)
(tktkT)		 (cctkT) (ccccT)
Octahedral
GIV(0,n)
(C)
(cC)
(tkC)
(ccC)		 (ctkC) (wrwC)
(cccC)
(tktkC)		 (cctkC) (ccccC)
Icosahedral
GV(0,n)
(D)
(cD)
(tkD)
(ccD)
 
(ctkD)
(wrwD)
(cccD)
(tktkD)
(cctkD)
(ccccD)
Class 2, G(n,n)
System G(1,1)
T = 3		 G(2,2)
T = 12		 G(3,3)
T = 27		 G(4,4)
T = 48		 G(6,6)
T = 108		 G(8,8)
T = 192		 G(9,9)
T = 243
Tetrahedral
GIII(n,n)
(tT)		 (ctT)
(tktT)		 (cctT) (ctktT) (ccctT) (tktktT)
Octahedral
GIV(n,n)
(tO)
(ctO)
(tktO)
(cctO)		 (ctktO) (ccctO) (tktktO)
Icosahedral
GV(n,n)
(tI)
(ctI)
(tktI)
(cctI)		 (ctktI) (ccctI) (tktktI)
Class 3, G(m,n)
System G(2,1)
T = 7		 G(3,1)
T = 13		 G(3,2)
T = 19		 G(4,1)
T = 21		 G(4,2)
T = 28		 G(4,3)
T = 37		 G(5,1)
T = 31		 G(5,2)
T = 39		 G(5,3)
T = 49		 G(5,4)
T = 61
Tetrahedral
GIII(m,n)
Octahedral
GIV(m,n)
Icosahedral
GV(m,n)
(wD)


(tk5sD)
(cwD)

Icosahedral G(0,n) polyhedra

Goldberg polyhedra of the form G(0,n) have full icosahedral symmetry, Ih, [5,3], (*532). G(0,n) has 10(n2 − 1) hexagons.

Index G(0,1) G(0,2) G(0,3) G(0,4) G(0,5) G(0,6)
Image
Conway notation D cD[2] dktI=tdtI[3] ccD[4] cdktI[5]
Vertices 2080180320500720
Edges 301202704807501080
Hexagons 03080150240350
Index G(0,7) G(0,8) G(0,9) G(0,10) G(0,12) G(0,16) G(0,n)
Image
Conway notation wrwD[6] cccD[7] tdtdtkD[8] ccccD[9]
Vertices 9801280162020002880512020n2
Edges 14701920243030004320765030n2
Hexagons 4806308009901430255010(n2 − 1)

Icosahedral G(n,n) polyhedra

Goldberg polyhedra of the form G(n,n) have full icosahedral symmetry, Ih, [5,3], (*532). G(n,n) has 10(3n2 − 1) hexagons.

Index G(1,1) G(2,2) G(3,3) G(4,4) G(5,5) G(6,6) G(n,n)
Image
Conway notation tI[10] ctI[11] tktI[12] cctI[13] ctktI[14]
Vertices 602405409601500216060n2
Edges 9036081014402250 3240 90n2
Hexagons 20110260470740 1070 10(3n2 − 1)

Icosahedral G(m,n) polyhedra

General Goldberg polyhedra (m > 0 and n > 0) with m ≠ n have chiral (rotational) icosahedral symmetry, I, [5,3] + , (532). In such cases G(n,m) and G(m,n) are mirror images.

Index G(1,0) G(1,1) G(1,2) G(1,3) G(1,4) G(1,5) G(1,n)
Image ...
Conway notation D dkD[15] wD[16] wdkD[17]
Vertices 206014026042062020(n2 + n + 1)
Edges 309021039063093030(n2 + n + 1)
Hexagons 0206012020030010n(n + 1)
Index G(2,0) G(2,1) G(2,2) G(2,3) G(2,4) G(2,5) G(2,n)
Image
Conway notation cD[18] dk5sD=t5gD=wD[19] dkt5daD[20] wcD = wcD[21]
Vertices 80140240380560 20(n2 + 2n + 4)
Edges 120210360570840 30(n2 + 2n + 4)
Hexagons 3060110180270 10(n2 + 2n + 3)
Index G(3,0) G(3,1) G(3,2) G(3,3) G(3,4) G(3,5) G(3,n)
Image
Conway notation tkD[22] wwD[23]
Vertices 180260 380 540 740 980 20(n2 + 3n + 9)
Edges 270390 570 810 1110 1470 30(n2 + 3n + 9)
Hexagons 80120 180 260 360 480 10(n2 + 3n + 8)
Index G(4,0) G(4,1) G(4,2) G(4,3) G(4,4) G(4,5) G(4,6) G(4,n)
Image
Conway notation ccD[24]dkwD[25] wcD[26] cdkt5daD[27]
Vertices 3204205607409601220152020(n2 + 4n + 16)
Edges 480630840111014401830228030(n2 + 4n + 16)
Hexagons 15020027036047060075010(n2 + 4n + 15)
Index G(5,0) G(5,1) G(5,2) G(5,3) G(5,4) G(5,5) G(5,6) G(5,n)
Image
Vertices 50062078098012201500182020(n2 + 5n + 25)
Edges 7509301170147018302250273030(n2 + 5n + 25)
Hexagons 24030038048060074090010(n2 + 5n + 24)
Index G(6,0) G(6,1) G(6,2) G(6,3) G(6,4) G(6,5) G(6,6) G(6,n)
Image
Conway notation tkt5daD=ctkD[28]
Vertices 7208601040126015201820216020(n2 + 6n + 36)
Edges 108012901560189022802730324030(n2 + 6n + 36)
Hexagons 350420510620750900107010(n2 + 6n + 35)

See also

Notes

  1. ↑ Clinton’s Equal Central Angle Conjecture, JOSEPH D. CLINTON
  2. ↑ cD
  3. ↑ tdtI
  4. ↑ ccD
  5. ↑ cdktI
  6. ↑ wrwD
  7. ↑ cccD
  8. ↑ tdtdtkD
  9. ↑ ccccD
  10. ↑ tI
  11. ↑ ctI
  12. ↑ tktI
  13. ↑ cctI
  14. ↑ ctktI
  15. ↑ dkD
  16. ↑ wD
  17. ↑ wdkD
  18. ↑ cD
  19. ↑ wD
  20. ↑ wD
  21. ↑ wcD
  22. ↑ tkD
  23. ↑ wwD
  24. ↑ ccD
  25. ↑ dkwD
  26. ↑ wcD
  27. ↑ cdkt5daD
  28. ↑ ctkD

References

External links

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