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
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:
We will call polyhedra created as result of these generalisation
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.
- 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
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
(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