The Platonic Letters—II

The Platonic Solids

The tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. From a first glance, one immediately notices that the Platonic Solids exhibit remarkable symmetry. They are the only convex polyhedra for which the same same regular polygon is used for each face, and the same number of faces meet at each vertex. Their symmetries are aesthetically pleasing, like those of stones cu...

Platonic Surfaces

If SO is a Riemann surface with a complete metric of finite area and constant curvature −1, let SC denote the conformal compactification of SO. We show that, under the assumption that the cusps of SO are large, there is a close relationship between the hyperbolic metrics on SO and SC . We use this relationship to show that lim infk→∞ λ1(Pk) ≥ 5/36, where the Platonic surface Pk is the conformal...

Platonic Dynamics

We develop a rigorous theory of observation for dynamical systems in Hilbert spaces. This work generalizes the results of Ott and Yorke [8] for dynamical systems in Rn.

Irony in the Platonic Dialogues

Platonic Triangles of Groups

Research supported by NSA and NSF. 1980 Mathematics Subject Classi cation (1985 Revision): 20E99.