[GAP Forum] Indexing of Ikosaedrons vertices

[Mariano Suarez-Alvarez]

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


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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.
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