About Enumeration of Isogonal Polyhedral Families

Year: 2002 Authors: Vladimir Bulatov

Core claim

Isogonal polyhedra can be systematically enumerated into continuous families, and many thousands of new families were found computationally.

Topics

polyhedral enumeration, symmetry groups, continuous families, VRML models

Domains

polyhedra, discrete geometry, group symmetry, topology, computer visualization, geometric forms, interactive 3D models

Methods

computer search algorithm, symmetry transformation, face-chain construction, topological grouping

Media

unit sphere, VRML

Paper text

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BRIDGES Mathematical Connections in Art, Music, and Science

About Enumeration of Isogonal Polyhedral Families

Vladimir Bulatov

Department of Physics, Oregon State University. 301 Weniger Hall, Corvallis, OR, 97331, USA e-mail: bulatov@dots.physics.orst.edu

Isogonal polyhedron is a polyhedron whose vertices are transitive under transformation of polyhedron’s symmetry group. The well-known 76 uniform polyhedra are isogonal polyhedra with additional restriction - polyhedral faces may be only regular polygons. Isogonal polyhedra are much more numerous and in fact usually they form a continuous families of topologically equivalent shapes. To date there were few systematic attempts to classify or enumerate all such families. Gruenbaum in [1] gives a description of several new classes of isogonal prismatoids. Author in [2] gives a partial enumeration of special subclass kaleidoscopical isogonal polyhedra. Our current attempt is to enumerate general isogonal families of polyhedra. The author wrote a computer program, which searches for all possible ways to compose isogonal polyhedron with given symmetry. All polyhedra found are separated into topologically equivalent sets or families. The following algorithm is used. For given polyhedral symmetry some generic arbitrary location of first vertex on a unit sphere is chosen. Symmetry transformations applied to first vertex make all the vertices of polyhedron. All possible planes passing through first vertex and at least two others vertices are calculated. All possible polygons in every such plane are calculated. These polygons are candidates to be a polyhedron faces. After that the program tries to make complete chain of faces around first vertex. The adjacent faces in the chain should share exactly one edge. Due to processing power limitation only chains no longer then some fixed length are considered. Different location of the first vertex will generally lead to different polyhedron. However all such polyhedra with different first vertex location but with identical way to form faces will form one isogonal family. There is continuous transition between all polyhedra in one family. So far many thousands of new isogonal polyhedral families were found. Interactive VRML models of these families are made. Five representatives of one isogonal polyhedral family with full icosahedral symmetry $I_h$ are illustrated. The ongoing task is to study these sets and to find appropriate classification.

References.

[1] B.Gruenbaum, Isogonal Prismatoids. Discrete Comput. Geom.18, 13-52, 1997. [2] V.Bulatov Kaleidoscopical Polyhedra Families. Proceedings of Millennial Open Symposium on the Arts and Interdisciplinary Computing “Mosaic 2000”, p.103-113 (Seattle, USA). http://www.physics.orst.edu/~bulatov/polyhedra/mosaic2000/talk/index.