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Journal of Mathematics
Volume 2017, Article ID 2542796, 15 pages
https://doi.org/10.1155/2017/2542796
Research Article

Toric Geometry of the Regular Convex Polyhedra

Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy

Correspondence should be addressed to Elisa Prato; ti.ifinu@otarp.asile

Received 30 July 2016; Accepted 31 January 2017; Published 23 March 2017

Academic Editor: Frédéric Mynard

Copyright © 2017 Fiammetta Battaglia and Elisa Prato. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We describe symplectic and complex toric spaces associated with 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.