the polyhedral symmetries ( 2/2 - transitivity )

transitivity

When considering polyhedra, "regularity" and "symmetry" are closely related concepts; the first is a local condition and the second is a global viewpoint. They both try to express that a regular polyhedron looks the same when it is viewed from different directions. The notion of transitivity is a more precise way to explicit that faces or vertices "look all the same". Transitivity involves the symmetry group of the polyhedron; a polyhedron can be

face-transitive (or isohedral) if, for any pair of faces, there is a symmetry in the group which transforms the first face into the second (examples are Plato's and Catalan's polyhedra),
vertex-transitive (or isogonal) if any vertex can be moved in any other by a symmetry from the group (examples are Plato's and Archimedes' polyhedra),
edge-transitive (or isotoxal) if any edge can be carried to any other by a symmetry from the group; edge-transitivity is always associated with either face- or vertex-transitivity.

The five Platonic solids are totally transitive (face- vertex- and edge-); are there other totally transitive polyhedra?
Peter R. Cromwell proves that there are two methods to generate the totally transitive polyhedra from a list of nine polyhedra (the rhombic dodecahedron and triacontahedron, the five Plato's polyhedra, the cuboctahedron and the icosidodecahedron): stellating one of the first seven and facetting one of the last seven; both produce the same list.
Finally there are fourteen totally transitive polyhedra: the nine regular polyhedra (five Platonic solids and four Kepler-Poinsot) and five compounds (two, five and ten tetrahedra, five cubes and five octahedra).

vertex-transitive convex polyhedra

Transitivity makes it very easy to describe polyhedra. A vertex-transitive convex polyhedron is completely described with one of its vertices and its symmetry group. In reverse which polyhedra are created when we apply all the symmetries of a given group to one given point? We can move the seed point on a sphere centred at the centre of symmetry and change the symmetry group.

The mirror planes of the group intersect the sphere in great circles which build a tessellation of the sphere in spherical triangles. A n-fold axes intersect the sphere where n great circles converge. When the seed point lies inside one triangle we get one vertex in each triangle (the number of triangles is equal to the group's order). If the seed point is the incentre then all the faces of the convex polyhedron are regular, and if we move it inside the triangle (fundamental region ) the polyhedra are isomorphic (have the same global shape). Other classes of polyhedra appear if the seed point is moved on a great circle.

These interactive graphics show the changes and the variety of generated polyhedra; move the big point - the seed point - with you mouse in the fundamental triangle.
To see the polyhedra generated by the different groups you must open the pop-up windows (links below the graphics).
(Reminder: you can minimize the main window to see more easily the pop-up windows, and use the "f" key to hide/display the polyhedra faces.)

group T_d ( T_d and T_h and T )

group O_h ( O_h and O )

group I_h ( I_h and I )

group D_5h

group D_4v

references:

Polyhedra (pages 366-393) by Peter R. Cromwell (Cambridge University Press, 1997).
Point Groups and Space Groups in Geometric Algebra by David Hestenes.

groups (first page)

summary

December 2005
updated 31-12-2005