Symplectic Toric Geometry and the Regular Dodecahedron

Abstract

The regular dodecahedron is the only simple polytope among the platonic solids which is not rational. Therefore, it corresponds neither to a symplectic toric manifold nor to a symplectic toric orbifold. In this paper, we associate to the regular dodecahedron a highly singular space called symplectic toric quasifold.

Toric Geometry of the Regular Convex Polyhedra

“…, X 12 point to the twelve vertices of the regular icosahedron that is dual to ∆ (see Figures 5 and 6). We recall from [15] that, if we apply the generalized Delzant construction to ∆ with Figure 5: The vectors X 1 , . .…”

“…The quotient M ∆ = Ψ −1 (0)/N is a symplectic quasifold; it has an atlas made of 20 charts, each corresponding to a different fixed point of the D 3 -action; we refer the reader to [15] for a description of one of them. From the complex viewpoint, the complex toric quasifold corresponding to the dodecahedron, with the choice of the same vectors X j (j = 1, .…”

“…, X 12 point to the twelve vertices of the regular icosahedron that is dual to ∆ (see Figures 5 and 6). We recall from [15] that, if we apply the generalized Delzant construction to ∆ with The level set Ψ −1 (0) for the moment mapping Ψ with respect to the induced N -action on C 12 is therefore given by the compact subset of points z in C 12 such that…”

“…The quotient M ∆ = Ψ −1 (0)/N is a symplectic quasifold; it has an atlas made of 20 charts, each corresponding to a different fixed point of the D 3 -action; we refer the reader to [15] for a description of one of them.…”

“…The regular dodecahedron, on the other hand, is simple but it is the first of the five regular convex polyhedra that is not rational. It is shown by Prato in [15] that, by applying her extension of the Delzant procedure to the case of general simple convex polytopes [14], one can associate to the regular dodecahedron a symplectic toric quasifold. Quasifolds are a generalization of manifolds and orbifolds: they are not necessarily Hausdorff and they are locally modeled by the quotient of a manifold modulo the action of a discrete group.…”