Geometric, Algebraic, and Topological Connections in the Historical Sphere of the Platonic Solids

Abstract

The Platonic solids have made prominent appearances in the history of pure mathematics: in Euclid's <italic>Elements</italic>, which contains early geometric constructions of the solids and demonstrates the special manner in which each is comprehended by a sphere; in William Hamilton's geometric interpretation of the icosians, a non-abelian group that describes certain "passages" between faces of the Platonic solids; and in Henri Poincar&eacute's <italic>Analysis</italic> <italic>situs</italic>, in which 0- and 1-dimensional homology are determined. Incidentally, Poincar&eacute's non-polyhedral construction of a homology 3-sphere with non-trivial fundamental group revealed the same icosahedral relations that Hamilton had invented. Topologists later realized that this important construction could be obtained as a polyhedral manifold using the dodecahedron. On the basis of these historical observations, a salient pattern emerges: The recurring uses of the Platonic solids in key episodes of mathematical development that have connected Euclidean geometry, non-commutative algebra and topology in important ways.