Vladimir Bulatov     abstract creations

Isogonal Kaleidoscopical Polyhedra

These polyhedra represent a special case of isogonal polyhedra. They are in some sense closest among isogonal to uniform polyhedra. The procedure of building these polyhedra is natural generalization of Wythoff's procedure used for making all (except one) uniform polyhedra.

Wythoff method works for symmetry group formed by reflections in a set of planes of symmetry. The intersections of symmetry planes with the surface of unit sphere form a grid of big circles on the surface. Any spherical triangle formed by these circles ( Schwarz triangle ) is selected. Each pair of symmetry planes forming sides of Schwarz triangle assumes a role of two mirrors of simple kaleidoscope, making 3 kaleidoscopes alltogether. Arbitrary point (generator vertex) on the sphere inside of the Schwarz triangle is used for creating 3 polygons (non spherical, but flat) with vertices generated from generator vertex by series of consequent reflections in the mirrors of each kaleidoscope. The sides (edges) of resulting polygons are orthogonal to corresponding symmetry planes and generator vertex is their common vertex. Final polyhedron is created by applying all the operations of symmetry group to these polygons and keeping only non identical polygons.

The requirement of regularity of polygonal faces limits possible generator vertex locations to some discrete set of places (vertices, sides and centers of Schwarz triagles) yelding 75(???) uniform polyhedra. However, if we will not limit ourselves with regular polygonal faces and consequentially will allow arbitrary generator vertex location - we will deal with isogonal polyhedral families instead of discrete set of polyhedra. Location of generator vertex is the 2 dimensional parameter of the family.

The described Wythoff procedure of generating polyhedra may be generalized in the following direction:

  • Generator point may be positionad anywhere on the sphere.
  • Schwarz triangle may be replaced by spherical polygon with arbitrary number of sides. The sides of the polygons are formed by the same grid of intersections of the spherical surface and symmetry planes.
We will call polyhedra created as result of these generalisation kaleidoscopical polyhedra. Procedure of building these polyhedra is similar to described above Wythoff procedure. The difference is that Schwartz triangle is replaced by spherical polygon, which makes N kaleidoscopes instead of 3.

It is more naturally to use a ordered set of symmetry planes instead of spherical polygon, because there may be several spherical polygons, which create identical set of kaleidoscopes (for example - any side of polygon may be replaced by dual segment of the same big circle). If we limit our cosideration by aploic polyhedra only (polyhedra, with no collinear faces) all vertices of spherical polygon (or lines of intersections of symmetry planes) should be different. Additional limitation of having polyhedron without collinear edges forces all symmetry planes in the set to be different.

We will call the number of planes in the set the order of polyhedron, which is equal in fact to the number of polygonal faces meeting at every vertex. Number of different nonequivalent isogonal kaleidoscopical families is rather small for small order and became unmanageably huge for polyhedra with icosahedral symmetry and order greater then 6.

Below is a link to the computer generated tables of kaleidoscopical isogonal polyhedra. The following notations is used for kaleidoscopical polyhedra family:

(Q1.Q2. ... .Qn)m
where Qi means that the that angle between sequential symmetry planes equals Pi/Qi, and m (if it is presented) means an ordinal number of given family in a series with the identical set of angles. This index resolves unfortunate ambiguity in notations. The less unambiguous notation for polyhedral family would be an explicit enumeration of all symmetry planes, which in turn is non unique, because symmetry operations applied to the set of planes would change notation, but would not change the family.

Kaleidoscopic polyhedra tables