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: G_III(n,m), G_IV(n,m), and G_V(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 G_III(m,n)	4T	6T	2T + 2	4 {3} and 2(T − 1) {6}
Octahedral G_IV(m,n)	8T	12T	4T + 2	6 {4} and 4(T − 1) {6}
Icosahedral G_V(m,n)	20T	30T	10T + 2	12 {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(0,6) T = 36	G(0,8) T = 64	G(0,9) T = 81	G(0,12) T = 144	G(0,16) T = 256
Tetrahedral G_III(0,n)	(T)	(cT)	(tkT)	(ccT)	(ctkT)	(cccT)	(tktkT)	(cctkT)	(ccccT)
Octahedral G_IV(0,n)	(C)	(cC)	(tkC)	(ccC)	(ctkC)	(cccC)	(tktkC)	(cctkC)	(ccccC)
Icosahedral G_V(0,n)	(D)	(cD)	(tkD)	(ccD)	(ctkD)	(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 G_III(n,n)	(tT)	(ctT)	(tktT)	(cctT)	(ctktT)	(ccctT)	(tktktT)
Octahedral G_IV(n,n)	(tO)	(ctO)	(tktO)	(cctO)	(ctktO)	(ccctO)	(tktktO)
Icosahedral G_V(n,n)	(tI)	(ctI)	(tktI)	(cctI)	(ctktI)	(ccctI)	(tktktI)

Class 3, G(m,n)
System	G(2,1) T = 7	G(4,1) T = 21	G(4,2) T = 28
Tetrahedral G_III(m,n)
Octahedral G_IV(m,n)
Icosahedral G_V(m,n)	(wD)	(tk5sD)	(cwD)

Icosahedral G(0,n) polyhedra

Goldberg polyhedra of the form G(0,n) have full icosahedral symmetry, I_h, [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	dktI=tdtI	ccD		tkt5daD
Vertices	20	80	180	320	500	720
Edges	30	120	270	480	750	1080
Hexagons	0	30	80	150	240	350

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		cccD	tdtdtkD			ccccD
Vertices	980	1280	1620	2000	2880	5120	20n²
Edges	1470	1920	2430	3000	4320	7650	30n²
Hexagons	480	630	800	990	1430	2550	10(n² − 1)

Icosahedral G(n,n) polyhedra

Goldberg polyhedra of the form G(n,n) have full icosahedral symmetry, I_h, [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(n,n)
Image
Conway notation	tI	ctI	tktI	cctI
Vertices	60	240	540	960	60n²
Edges	90	360	810	1440	90n²
Hexagons	20	110	260	470	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	dk5sD=t5gD		tk5sD
Vertices	20	60	140	260	420	620	20(n² + n + 1)
Edges	30	90	210	390	630	930	30(n² + n + 1)
Hexagons	0	20	60	120	200	300	10n(n + 1)

Index	G(2,0)	G(2,1)	G(2,2)	G(2,3)	G(2,4)	G(2,n)
Image
Conway notation	cD	dk5sD=t5gD	dkt5daD		cdk5sD
Vertices	80	140	240	380	560	20(n² + 2n + 4)
Edges	120	210	360	570	840	30(n² + 2n + 4)
Hexagons	30	60	110	180	270	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
Vertices	180	260	380	540	740	980	20(n² + 3n + 9)
Edges	270	390	570	810	1110	1470	30(n² + 3n + 9)
Hexagons	80	120	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
Vertices	320	420	560	740	960	1220	1520	20(n² + 4n + 16)
Edges	480	630	840	1110	1440	1830	2280	30(n² + 4n + 16)
Hexagons	150	200	270	360	470	600	750	10(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	500	620	780	980	1220	1500	1820	20(n² + 5n + 25)
Edges	750	930	1170	1470	1830	2250	2730	30(n² + 5n + 25)
Hexagons	240	300	380	480	600	740	900	10(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
Vertices	720	860	1040	1260	1520	1820	2160	20(n² + 6n + 36)
Edges	1080	1290	1560	1890	2280	2730	3240	30(n² + 6n + 36)
Hexagons	350	420	510	620	750	900	1070	10(n² + 6n + 35)

Notes

↑ Clinton’s Equal Central Angle Conjecture, JOSEPH D. CLINTON

References

Goldberg, Michael (1937). "A class of multi-symmetric polyhedra". Tohoku Mathematical Journal.
Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture
Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie. Shaping Space (2nd ed.). Springer. pp. 125–138. doi:10.1007/978-0-387-92714-5_9.
Hart, George (June 18, 2013). "Mathematical Impressions: Goldberg Polyhedra". Simons Science News.
Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses, Stan Schein and James Maurice Gaye, PNAS, Early Edition doi: 10.1073/pnas.1310939111

External links

Dual Geodesic Icosahedra
Goldberg variations: New shapes for molecular cages Flat hexagons and pentagons come together in new twist on old polyhedral, by Dana Mackenzie, February 14, 2014

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