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 center 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 incenter 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 Td ( Td - Th - T )
group Oh ( Oh - O )
group Ih ( Ih - I )
• Polyhedra (pages 366-393) by Peter R. Cromwell (Cambridge University Press, 1997)
• Point Groups and Space Groups in Geometric Algebra par David Hestenes
||convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects||December 2005 |