The fullerene C70

## The fullerene C70

Besides C60 the most prominent fullerene is C70, which has D5h 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