Toric Geometry of the Regular Convex Polyhedra
Fiammetta Battaglia, Elisa Prato

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Combinatorial Mathematics · Geometric and Algebraic Topology
