A new class of isogonal polyhedra is considered. Polyhedra are constructed using a combination of reflections in several symmetry planes of a given symmetry group. The procedure is a generalization of a Wythoff construction used for building uniform polyhedra. Every valid combination of symmetry planes generates the whole family of isogonal polyhedra. There are seven families with tetrahedral symmetry, 284 with octahedral symmetry and few million with icosahedral symmetry.

1      Introduction

Polyhedra have always been some of the most visually appealing geometrical objects. They have fascinated people from ancient times when the simplest polyhedra - the tetrahedron, cube, octahedron, dodecahedron, and icosahedron ( the platonic solids) were known. Most people's knowledge of polyhedra is usually limited to above mentioned five figures. Twentieth century mathematics has not devoted much attention to polyhedra. Perhaps that is because the subject is very old and too simple from the point of view of modern mathematics. However, polyhedra are extremely interesting objects to investigate and there are many unsolved problems.

One of most most notable polyhedra-related mathematical works of recent times is the paper by Coxeter at all [2], which enumerates all 75 of the so called "uniform" polyhedra. These polyhedra have symmetrically equivalent vertices and regular faces. The requirement of regular faces limits the number of possible figures quite significantly. A much broader set of shapes form polyhedra with symmetrically equivalent vertices - isogonal polyhedra. Actually there are infinite many isogonal polyhedra. Not much is published about isogonal polyhedra: [1] describes so called "noble" polyhedra - polyhedra with equivalent vertices and faces, and [3] gives several interesting new classes of isogonal prismatoids, which are generalizations of prisms and antiprisms. No classification of general isogonal polyhedra has so far been published.

In this paper, we will attempt to describe a subset of all isogonal polyhedra which we call kaleidoscopical polyhedra. These polyhedra may be constructed using the procedure which is similar in some aspects to the process of creating images in a kaleidoscope. We have also attempted computer assisted enumerations of all possible kaleidoscopical polyhedra.

2      Kaleidoscopical polyhedra construction

We will limit our consideration to the following three groups of symmetry: full symmetry of the tetrahedron (Td), octahedron (Oh) and icosahedron (Ih). These are the only point symmetry groups in addition to the simpler dihedral group symmetry of the prism, which consist only of reflections in several symmetry planes and compositions of those reflections. The tetrahedral group Td has six symmetry planes, the octahedral group Oh has nine, and the icosahedral group Ih has fifteen symmetry planes. Isogonal polyhedra should have all vertices equivalent under the operation of a given symmetry group. Therefore all vertices of a polyhedron should lie on a sphere with the center at the origin. The intersection of the symmetry planes with this sphere will be a great circle on this sphere. All such intersections form a grid of great circles. Fig.1a-c show these grids for each symmetry group along with their corresponding platonic solids with that symmetry.

(a)
(b)
(c)

Fig.1 Symmetry planes of selected symmetry groups:
a) Tetrahedral T_d
b) Octahedral O_h
c) Icosahedral I_h

Next we choose a "kaleidoscope" - an ordered circular sequence of symmetry planes (i.e. mirrors): [P1, P2, ..., PN]. Let's call N the order of the isogonal polyhedron. We call the sequence circular, because we assume it to have no beginning and no end, in that the element following element PN is element P1 again and any circular transposition of a given sequence is considered to be the same sequence. This kaleidoscope resembles the typical real kaleidoscope with the difference that real kaleidoscope has non-penetrating one sided mirrors. Our kaleidoscope has ideal, two sided mirrors possibly penetrating each other.

We can visualize such a kaleidoscope as a spherical polygon on the sphere, whose sides are pieces of great circles. Fig.2 shows an example of such spherical polygon, which corresponds to kaleidoscope consisting of four mirrors.

Fig.2 A kaleidoscope is formed by a sequence of symmetry planes.

Fig.3 A generator vertex on the sphere.

Fig.4a The face formed by reflections in the symmetry planes P4 and P1.	Fig.4b The face formed by reflections in the symmetry planes P3 and P4.
Fig.4c The face formed by reflections in the symmetry planes P2 and P3.	Fig.4d The face formed by reflections in the symmetry planes P1 and P2.

Fig.5 All the faces containing the generator vertex.

Fig.6 Completed kaleidoscopical polyhedron (Polyhedron's 5-fold axis is turned to the viewer).

3        Kaleidoscopical polyhedra family

It is important to note that the kaleidoscopical construction described above has one arbitrary parameter - the position of the generator vertex. Every new position of this vertex will create a new polyhedron. All polyhedra constructed using one kaleidoscope and various generator vertex locations will have some common properties. In particular, they will all have the same number of faces, vertices and edges. Moreover, the type of all faces remains the same though the shape of the faces may be very different. Fig.7 shows the metamorphoses of two different faces types presented in a 4-mirrored kaleidoscope.

These drawings are similar to those found in [4].

The metamorphosis of the whole polyhedron is much more complex. Fig.8 attempts to show such a metamorphosis which happens when the generator vertex moves along a trajectory on the spherical surface.

Fig.8 Metamorphosis of a kaleidoscopical polyhedron.


Fig.8 Metamorphosis of a kaleidoscopical polyhedron (continued).

4      Enumeration of kaleidoscopical polyhedra families

How many kaleidoscopical polyhedra families are there? Not every combination of symmetry planes gives a kaleidoscope which can produce a valid polyhedron. The limitations are:
a) Mirrors may not be coincident. This prevents the creation of polyhedra with coincident edges.
b) The symmetry axes along which sequential mirrors of a kaleidoscope intersect must be unique.
This prevents the creation of polyhedra with coplanar faces.

Not all kaleidoscopes will produce unique families. If there exists a symmetry transformation which maps the mirrors of one kaleidoscope into mirrors of another, the corresponding polyhedral families will be identical. Therefore all equivalent kaleidoscopes can be represented by one arbitrary selected (canonical) representative.

We have conducted a brute force computer search for all possible canonical kaleidoscopes, however the search is not complete. The Table 1 shows the number of families for each symmetry group being investigated. Kaleidoscope with tetrahedral symmetry may have up to 6 mirrors, with octahedral symmetry - up to 9 and with icosahedral symmetry up to 15. Question marks in some cells of icosahedral symmetry mean that the algorithm currently used failed to give results in a reasonable time, however, we think that the nearly one million kaleidoscopes already enumerated give enough material to play with and enjoy. The list of all kaleidoscopical polyhedra is available on the web at http://www.bulatov.org/polyhedra/mosaic2000/kaleido_poly/index.html

5      Conclusion

6      Acknowledgments

        References

[1] M.Bruckner. Uber die gleicheckig-gleichflachigen, diskontinuierlichen und nichtkonvexen Polyheder. Nova Acta, Band LXXXVI, #1, Halle 1906]
[2] H.S.M.Coxeter, M.S.Longuet-Higgins and J.C.P.Miller, Uniform Polyhedra. Philos. Trans. Roy. Soc. London, Ser.A v.246, p.401-450, (1954).
[3] B.Grunbaum, Isogonal Prismatoids. Discrete & Computational Geometry. 18, p.13-52(1997).
[4] B.Grunbaum, Metamorphoses of Polygons. The lighter side of Mathematics. Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics & its History. Edited by Richard K. Guy & Robert E. Woodrow, Mathematical Association of America p.35-48(1994).