Fullerenes -- The fullerene C70

The fullerene C₇₀

The fullerene C₇₀

Besides C₆₀ the most prominent fullerene is C₇₀, which has D_5h as its symmetry group. The C₇₀ can be constructed from the C₆₀ by cutting the C₆₀ along the edges given by the following sequence of vertices: 21, 31, 32, 22, 23, 33, 34, 24, 25, 35, 36, 26, 27, 37, 38, 28, 29, 39, 40, 30, 21. Then we have two halves of the truncated icosahedron; the vertices in the first half will be labelled by 1, 2, ..., 40, the vertices of the second half by 21', 22', ..., 60'. Now lift the upper half, turn it by an angle of p/5 such that we get 5 new hexagons with labels

21'=32,	22,	23,	22'=33,	32',	31'
23'=34,	24,	25,	24'=35,	34',	33'
25'=36,	26,	27,	26'=37,	36',	35'
27'=38,	28,	29,	28'=39,	38',	37'
29'=40,	30,	21,	30'=31,	40',	39'

Since ten of the labels of the vertices in the first half coincide with ten of the labels in the second half there are only 70 vertices (but 80 labels). The group of rotational symmetries is given by one 5-fold rotation p₁ and five 2-fold rotations. Combining these rotations with one reflection s gives the group of all symmetries. Renaming the labels i' by i+10 for i³31, the generators for these two groups acting on the set of vertices are given by:

p₁= (66,67,68,69,70)(61,62,63,64,65)(52,54,56,58,60)(51,53,55,57,59)(42,44,46,48,50) (41,43,45,47,49)(32,34,36,38,40)(31,33,35,37,39)(22,24,26,28,30)(21,23,25,27,29) (12,14,16,18,20)(11,13,15,17,19)(6,7,8,9,10)(1,2,3,4,5)

p₂= (37,38)(36,39)(35,40)(32,33)(31,34)(30,44)(29,45)(28,46)(27,47)(26,48)(25 ,49) (24,50)(23,41)(22,42)(21,43)(20,54)(19,55)(18,56)(17,57)(16,58)(15,59)(14,60)(13,51) (12,52)(11,53)(10,63)(9,64)(8,65)(7,61)(6,62)(5,68)(4,69)(3,70)(2,66)(1,67)

s= (70)(67,68)(66,69)(65)(62,63)(61,64)(59,60)(54,55)(53,56)(52,57)(51,58)(49,50) (44,45)(43,46)(42,47)(41,48)(35,36)(34,37)(33,38)(32,39)(31,40)(25,26)(24,27)(23,28) (22,29)(21,30)(15,16)(14,17)(13,18)(12,19)(11,20)(8,9)(7,10)(6)(3,4)(2,5)(1)

Finally the 3-dimensional cycle indices for the action on the sets of vertices, edges and faces are

Z₃(R(C₇₀))=(1)/(10)( 4 v₅¹⁴ e₅²¹ f₁² f₅⁷ + 5 v₂³⁵ e₁ e₂⁵² f₁ f₂¹⁸ + v₁⁷⁰ e₁¹⁰⁵ f₁³⁷ )

Z₃(S(C₇₀))=(1)/(2)Z₃(R(C₇₀)) + (1)/(20)( 4 v₅² v₁₀⁶ e₅ e₁₀¹⁰ f₂ f₅ f₁₀³+ 5 v₁⁴ v₂³³ e₁⁹ e₂⁴⁸ f₁⁹ f₂¹⁴ + v₁¹⁰ v₂³⁰ e₁⁵ e₂⁵⁰ f₁⁵ f₂¹⁶ ).

harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001

The fullerene C₇₀