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
π/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 π1 and five 2-fold
rotations. Combining these rotations with one reflection σ
gives the group of all symmetries. Renaming the labels i' by i+10
for i≥ 31, the generators for these two groups acting on the set
of vertices are given by:
π1=
(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)
π2=
(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)
σ=
(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 "at" uni-graz.at, May 10,
2016
