Polyhedral Models in Group Theory and Graph Theory

Raymond F. Tennant
Bridges: Mathematical Connections in Art, Music, and Science (2000)
Pages 293–300

Abstract

A longstanding method for understanding concepts in mathematics involves the creation of two or three-dimensional images which describe a particular mathematical idea. From our earliest learning experiences, we are taught mathematics by appealing to our strong visual and tactile intuition. For students studying mathematics at the college or university level, the use of polyhedral models and graph theoretic constructions may be a valuable tool for gaining insight into abstract areas such as group theory and topology. This investigation focuses on the use of Platonic and Archimedean solids to describe ideas in abstract algebra and to understand the concepts such as duality and symmetry subgroup. The reasoning behind several proofs of Euler's Formula are explored with the use of models. For the most part, planar graphs of polyhedra are used in place of actual three-dimensional models. This has the advantage of allowing for all of the vertices, edges, and faces to be viewed at the same time.

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