On Tue, 9 May 2006, Rudolf Zlabinger wrote: > Dear MCKAY john, > > Thank you for your hint to the finite rotation groups, you are right, the > icosahedron (dodecahedron) has 3 pole classes, with 2,3,5 cycle rotations, > resulting in 30,20,12 cosets. > > My theoretical problem was, to translate this structure to mere permutation > representations. Distributing numbers 1 to 12 to the vertices has to some > degree influence to the selection of distinct permutation groups > representing the icosahedron indexed by these numbers. > > As in our example indexing the icosahedrons vertices from top to down as 1, > (2,3,4,5,6),(7,8,9,10,11), 12 lead to the request, that a representing group > has to contain the permutation (2,3,4,5,6)(7,8,9,10,11) as the rotation > around the axis 1,12. > > The problem is, that there is no formal way to describe a special indexing > of the vertices, as to do it intuitively as above. So i am looking for a > method to derive the indexing(s) from the selected permutation group(s) in > reverse order if this is possible.
There is a very nice indexing of the vertices which very nicely interacts with the symmetry group, constructed as follows. One can partition, in exactly two ways, the vertices of the icosahedron in five sets of four vertices each, in such a way that each such subset is the set of vertices of a regular tetrahedron. Moreover, the two different partitions are related by a central inversion. Call the partitions A and B. Label each of the tetrahedra in partition A with 1, 2, 3, 4 and 5. Label also each tetrahedron in the B partition with the label of the corresponding tetrahedron in partition A (under inversion). Finally, label each vertex v in the icosahedron with the pair (i,j) with i (respectively j) being the label of the tetrahedron in the A partition (respectively, in the B partition) to which v belongs. Each vertex thus gets labeled and all the labels are different. It is very easy to see that A_5 acts on the set of vertices permuting the pairs of labels (i,j) coordinate-wise. Moreover, the vertex (i,j) is the antipode of (j,i). One can work out the combinatorial structure (incidence, flags, etc) of the solid from this labeling in a nice way. (Btw, I'd love to have a nice formula for the coordinate of the vertex (i,j) from i and j.) This construction is explained by Coxeter in his `Regular polytopes'; if I recall correctly, he provides no reference, so I guess he came up with the idea. The configuration of the tetrahedra is a bit difficult to visualize for most of us who are not Coxeter. There is a Mathematica notebook which constructs it at <http://mate.dm.uba.ar/~aldoc9/tmp/Dodecahedron.nb>; if you have Geomview available, you can play with the picture a little more comfortably. For maximal fun, you can construct the whole thing out of paper (no cuts, no glue...); it has quite a surprisng bewitching effect on people (both mathematicians and regular people). There's a pic of one of my attempts at building it at <http://mate.dm.uba.ar/~aldoc9/tetra.jpg>, and you can find instructions on how to build it by googling for "five intersecting tetrahedra"---you can even feel lucky about it. Cheers, -- m -- ------------------------------------------------------------------------------ Mariano Suárez-Alvarez Departamento de Matemática Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Ciudad Universitaria, Pabellón I. Buenos Aires (1428). Argentina. http://mate.dm.uba.ar/~aldoc9 Pienso en efecto que, salvo si se cree en milagros, sólo cabe esperar el progreso de la razón de una acción política racionalmente orientada hacia la defensa de las condiciones sociales del ejercicio de la razón, de una movilización permanente de todos los productores culturales con el propósito de defender, mediante intervenciones continuadas y modestas, las bases intelectuales de la actividad intelectual. Todo proyecto de desarrollo del espíritu humano que, olvidando el arraigo histórico de la razón, cuente con la única fuerza de la razón y de la prédica racional para hacer progresar las causas de la razón, y que no apele a la lucha política para tratar de dotar a la razón y a la libertdad de los intrumentos propiamente políticos que constituyen la condición de su realización en la história, continúa todavía prisionero de la ilusión escolástica. Pierre Bourdieu, Le point de vue scolastique, ``Raisons pratiques. Sur la théorie de l'action''. Points, vol. 331. París: Éditions du Seuil, 1994. ------------------------------------------------------------------------------ _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
