[GAP Forum] Re: icosahedral group question

[Rudolf Zlabinger]

Dear Walter Becker,

attached to this message a GAP file generating a rotation group "fullerene".

Its the rotation group A5 (icosahedral group) acting on the 60 points of
your "buckyball".

This group supports a special labelling of the original icosahedron as
already outlined in my first message "icosahedron exercises". I repeat this
here:

Having a blueprint in mind of icosahedron, there is a top vertex, two
pentagons one on top of another, the second pentagon rotated by 36 degrees
clockwise against the first one, and a bottom vertex.

As we label the top vertex by 1, the two pentagons by 2,3,4,5,6 and
7,8,9,10,11 clockwise looking in bottom direction respectively and the
bottom vertex by 12, we have following picture:

The respective opposite vertices are: (1,12), (2,9), (3,10), (4,11), (5,7),
(6,8).

The edges are:

(1,2),(1,3),(1,4),(1,5),(1,6),
from the top vertex to first pentagon
(12,7),(12,8)(12,9)(12,10)(12,11)
from the bottom vertex to second pentagon
(2,3)(3,4)(4,5)(5,6)(6,2)
the top pentagon
(7,8)(8,9)(9,10)(10,11)(11,7)
the bottom pentagon
(2,11)(2,7)(3,7)(3,8)(4,8)(4,9)(5,9)(5,10)(6,10)(6,11)
between the pentagons

Following this labels of the vertices of the icosahedron you have to derive
the labels of the 60 vertex "buckyball" as follows:
label a pentagon by 1,2,3,4,5 and then the other pentagons by (x-1)*5 + y,
where x is the original vertex label of the icosahedron and y the numbers
1,2,3,4,5. The respective adjacent pentagons have to be labelled such as
after rotating a pentagon into an adjacent (or also another) pentagon the
respective target and image labels are congruent modulo 5.

Using the rotation group fullerene, you can derive the positions of the such
labelled "buckyball" vertices directly as result of a rotation permuation,
where the group contains all possible rotations of them.

best regards, Rudolf Zlabinger
PS.: by the way, I am not graduated, so to be correct, I have to tell it.

----- Original Message ----- 
From: "Walter Becker" <w_becker at hotmail.com>
To: <Rudolf.Zlabinger at chello.at>
Sent: Sunday, May 14, 2006 6:14 PM
Subject: icosahedral group question



dear Dr. Zlabinger:

You have been asking and getting several responses to questions dealing
withy the icosahedral group on the GAP forum. Do you have any interest or
knowledge about the uses of GAP in determining the symmetry adapted basis
functions that are used in various areas of chemistry and physics?  Here I
am esecially interested in using GAP as a method or tool in calculating
them--especially for teh higher order point groups e.g., the icosahedral
one. The applicarion of interest here is to the buckyball systems which
involve five-fold symmetres but the geometrical structure is a truncated
icosahedran--ie, take each vertex of the icosahedron and pass a plane betwen
it and the next series of vertices ---essentially converting each vertex in
five new vertices. The group for te buckyball is this 60 vertex object and
its vibratonsare determined by this point group.


Comments on interest ?????

I can give some references as to where the calclations are reported but not
much in the nitty-gritty details are gven ie computer routines or projection
operators used in the work.

Walter Becker

-------------- next part --------------
#   Thats the group covering a "canonical" label set of icosahedron
#   gencgroup:=GeneratorsOfGroup(cgroup);
#   [ (1,2,3)(4,6,7)(5,11,8)(9,10,12), (1,2)(3,6)(4,11)(5,7)(8,10)(9,12) ]

#   we claculate the generators as acting on numbers mod(5), where the mods 1 are the original
#   points of the icosahedron
 
    list0010:=[1,2,3,4,6,7,5,11,8,9,10,12];
    list0020:=[1,2,3,6,4,11,5,7,8,10,9,12];
    list0030:=[1,2,3,4,5];
    result0010:=List( list0030,c->List( list0010,d->( (d-1)*5+c) ) );
    result0020:=List( list0030,c->List( list0020,d->( (d-1)*5+c) ) );

	
#   result0010;
#   [ [ 1, 6, 11, 16, 26, 31, 21, 51, 36, 41, 46, 56 ], 
#   [ 2, 7, 12, 17, 27, 32, 22, 52, 37, 42, 47, 57 ], 
#   [ 3, 8, 13, 18, 28, 33, 23, 53, 38, 43, 48, 58 ], 
#   [ 4, 9, 14, 19, 29, 34, 24, 54, 39, 44, 49, 59 ], 
#   [ 5, 10, 15, 20, 30, 35, 25, 55, 40, 45, 50, 60 ] ]
#   result0020;
#   [ [ 1, 6, 11, 26, 16, 51, 21, 31, 36, 46, 41, 56 ], 
#   [ 2, 7, 12, 27, 17, 52, 22, 32, 37, 47, 42, 57 ], 
#   [ 3, 8, 13, 28, 18, 53, 23, 33, 38, 48, 43, 58 ], 
#   [ 4, 9, 14, 29, 19, 54, 24, 34, 39, 49, 44, 59 ], 
#   [ 5, 10, 15, 30, 20, 55, 25, 35, 40, 50, 45, 60 ] ]

#   So we get as generators of our icosahedron acting on 60 points of the Fullerene

    gen_0010:=  (1, 6, 11)  (16, 26, 31) (21, 51, 36) (41, 46, 56)  
                (2, 7, 12)  (17, 27, 32) (22, 52, 37) (42, 47, 57)  
                (3, 8, 13)  (18, 28, 33) (23, 53, 38) (43, 48, 58)  
                (4, 9, 14)  (19, 29, 34) (24, 54, 39) (44, 49, 59)  
                (5, 10, 15) (20, 30, 35) (25, 55, 40) (45, 50, 60);

    gen_0020:=  (1, 6)  (11, 26) (16, 51) (21, 31) (36, 46) (41, 56)  
                (2, 7)  (12, 27) (17, 52) (22, 32) (37, 47) (42, 57)  
                (3, 8)  (13, 28) (18, 53) (23, 33) (38, 48) (43, 58)  
                (4, 9)  (14, 29) (19, 54) (24, 34) (39, 49) (44, 59)  
                (5, 10) (15, 30) (20, 55) (25, 35) (40, 50) (45, 60);

    fullerene:=Group(gen_0010,gen_0020);

#    StructureDescription(fullerene);
#    "A5"
#    NrMovedPoints(fullerene);
#    60