The Platonic Solids

An Exploration of the Five Regular 
Polyhedra and the Symmetries 
of Three-Dimensional Space Abstract 

The five Platonic solids (regular polyhedra) are the tetrahedron, cube, octahedron, icosahedron, and dodecahedron. 

The regular polyhedra are three dimensional shapes that maintain a certain level of equality; that is, congruent faces, equal length edges, and equal measure angles. In this paper we discuss some key ideas surrounding these shapes. We establish a historical context for the Platonic solids, show various properties of their features, and prove why there can be no more than five in total. We will also discuss the finite groups of symmetries on a line, in a plane, and in three dimensional space. Furthermore, we show how the Platonic solids can be used to visualize symmetries in $R^3$.

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