Fw: [GAP Forum] Re: icosahedral group question

[Rudolf Zlabinger]

Dear MCKAY john,

I did read now the publication you recommended to me, but there was no
explicit construction for the A5 acting on 60 points in permutation
representation based on a special labelling procedure. So I think, it was
helpful though for people looking for special samples.

best regards, Rudolf Zlabinger

----- Original Message ----- 
From: "Rudolf Zlabinger" <Rudolf.Zlabinger at chello.at>
To: "MCKAY john" <mckay at encs.concordia.ca>
Sent: Wednesday, May 17, 2006 5:58 PM
Subject: Re: [GAP Forum] Re: icosahedral group question


Dear MCKAY john,

Thank you for hint; I am learning by doing now, and often one spends more
time for looking for related publications, than redeveloping it yourself, if
its not too complicated.

best regards, Rudolf Zlabinger


----- Original Message ----- 
From: "MCKAY john" <mckay at encs.concordia.ca>
To: "Rudolf Zlabinger" <Rudolf.Zlabinger at chello.at>
Cc: "Walter Becker" <w_becker at hotmail.com>
Sent: Wednesday, May 17, 2006 5:26 PM
Subject: Re: [GAP Forum] Re: icosahedral group question





Read  the article(s) by Kostant in Notices of AMS about
C60.

The icosahedron is well understood.

Best,
John McKay


On Wed, 17 May 2006, Rudolf Zlabinger wrote:

> 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
>
>