Toric Geometry of the Regular Convex Polyhedra

TL;DR

This paper explores the toric geometry associated with the five regular convex polyhedra, highlighting how their rationality and simplicity affect the applicability of standard toric methods.

Contribution

It characterizes the symplectic and complex toric spaces linked to each regular polyhedron, emphasizing cases where standard toric geometry techniques are insufficient.

Findings

Tetrahedron and cube are rational and simple.

Octahedron is not simple.

Dodecahedron and icosahedron are not rational, with the icosahedron also not simple.

Abstract

In this article, we describe symplectic and complex toric spaces associated to the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational and the regular icosahedron is neither simple nor rational. We remark that the last two cases cannot be treated via standard toric geometry.