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" <[EMAIL PROTECTED]> To: "MCKAY john" <[EMAIL PROTECTED]> 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" <[EMAIL PROTECTED]> To: "Rudolf Zlabinger" <[EMAIL PROTECTED]> Cc: "Walter Becker" <[EMAIL PROTECTED]> 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" <[EMAIL PROTECTED]> > To: <[EMAIL PROTECTED]> > 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 > > _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
