# Symplectic Toric Geometry and the Regular Dodecahedron

Authors: Elisa Prato
Journal: Journal of Mathematics
 (2015)

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## 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.


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## Body

## 1. Introduction

According to a well-known theorem by Delzant [1], there exists a one-to-one correspondence between symplectic toric manifolds and simple rational convex polytopes satisfying a special integrality condition. One of the important features of Delzant’s theorem is that it provides an explicit procedure for computing the symplectic manifold that corresponds to each given polytope. As it turns out, there are many important examples of simple convex polytopes that do not fall into this class, either because they do not satisfy Delzant’s integrality condition, or, even worse, because they are not rational. The regular dodecahedron is one of the most remarkable examples of a simple polytope that is not rational. In this paper, we apply a generalization of Delzant’s construction [2] to simple nonrational convex polytopes and we associate to the regular dodecahedron asymplectic toric quasifold. Quasifolds are a natural generalization of manifolds and orbifolds introduced by the author in [2]; they are not necessarily Hausdorff spaces and they are locally modeled by quotients of manifolds by the action of discrete groups.This paper is structured as follows. In Section2, we recall the generalized Delzant construction. In Section 3, we apply this construction to the regular dodecahedron and we describe the corresponding symplectic toric quasifold.


## 2. The Generalized Delzant Construction

We begin by recalling a few useful definitions.Definition 1 (simple polytope).
A dimensionn convex polytope Δ⊂(Rn)∗ is said to besimple if each of its vertices is contained in exactly n facets.Definition 2 (quasilattice).
Aquasilattice in Rn is the Z-span of a set of R-spanning vectors, Y1,…,Yd, of Rn.Notice thatSpanZ{Y1,…,Yd} is an ordinary lattice if and only if it admits a set of generators which is a basis of Rn.Consider now a dimensionn convex polytope Δ⊂(Rn)∗ having d facets. Then, there exist elements X1,…,Xd in Rn and λ1,…,λd in R such that(1)Δ=⋂j=1dμ∈Rn∗∣μ,Xj≥λj.Definition 3 (quasirational polytope).
LetQ be a quasilattice in Rn. A convex polytope Δ⊂(Rn)∗ is said to bequasirational with respect to Q if the vectors X1,…,Xd in (1) can be chosen in Q.We remark that each polytope in(Rn)∗ is quasirational with respect to some quasilattice Q: just take the quasilattice that is generated by the elements X1,…,Xd in (1). Notice that if X1,…,Xd can be chosen inside an ordinary lattice, then the polytope is rational in the usual sense.Definition 4 (quasitorus).
LetQ⊂Rn be a quasilattice. We callquasitorus of dimension n the group and quasifold Dn=Rn/Q.For the definition and main properties of symplectic quasifolds and of Hamiltonian actions of quasitori on symplectic quasifolds, we refer the reader to [2, 3]. On the other hand, the basic facts on polytopes that are needed can be found in Ziegler’s book [4]. We are now ready to recall from [2] the generalized Delzant construction. For the purposes of this paper, we will restrict our attention to the special case n=3.Theorem 5.
LetQ be a quasilattice in R3 and let Δ⊂(R3)∗ be a simple convex polytope that is quasirational with respect to Q. Then, there exists a 6-dimensional compact connected symplectic quasifold M and an effective Hamiltonian action of the quasitorus D3=R3/Q on M such that the image of the corresponding moment mapping is Δ.Proof.
Let us consider the spaceCd endowed with the standard symplectic form ω0=1/2πi∑j=1ddzj∧dz¯j and the action of the torus Td=Rd/Zd given by (2)τ:Td×Cd⟶Cde2πiθ1,…,e2πiθd,z_⟼e2πiθ1z1,…,e2πiθdzd.This is an effective Hamiltonian action with moment mapping given by (3)J:Cd⟶Rd∗z_⟼∑j=1dzj2ej∗+λ,λ∈Rd∗constant. The mapping J is proper and its image is given by the cone Cλ=λ+C, where C denotes the positive orthant of (Rd)∗. Take now vectors X1,…,Xd∈Q and real numbers λ1,…,λd as in (1). Consider the surjective linear mapping (4)π:Rd⟶R3,ej⟼Xj. Consider the dimension 3 quasitorus D3=R3/Q. Then, the linear mapping π induces a quasitorus epimorphism Π:Td→D3. Define now N to be the kernel of the mapping Π and choose λ=∑j=1d‍λjej∗. Denote by i the Lie algebra inclusion Lie(N)→Rd and notice that Ψ=i∗∘J is a moment mapping for the induced action of N on Cd. Then, according to [2, Theorem 3.1], the quotient M=Ψ-1(0)/N is a symplectic quasifold which is endowed with the Hamiltonian action of the quasitorus Td/N. Since π∗ is injective and J is proper, the quasifold M is compact. If we identify the quasitori D3 and Td/N via the epimorphism Π, we get a Hamiltonian action of the quasitorus D3 whose moment mapping Φ has image equal to (π∗)-1(Cλ∩keri∗)=(π∗)-1(Cλ∩imπ∗)=(π∗)-1(π∗(Δ)) which is exactly Δ. This action is effective since the level set Ψ-1(0) contains points of the form z_∈Cd, zj≠0, j=1,…,d, where the Td-action is free. Notice finally that dimM=2d-2dimN=2d-2(d-3)=6.Remark 6.
We will say thatM is asymplectic toric quasifold associated to the polytope Δ. The quasifold M depends on our choice of quasilattice Q with respect to which the polytope is quasirational and on our choice of vectors X1,…,Xd. Note that the case where the polytope is simple and rational, but does not necessarily satisfy Delzant’s integrality condition, was treated by Lerman and Tolman in [5]. They allowed orbifold singularities and introduced the notion of symplectic toric orbifold.


## 3. The Regular Dodecahedron from a Symplectic Viewpoint

LetΔ be the regular dodecahedron centered at the origin and having vertices(5)±1,±1,±10,±ϕ,±1ϕ±1ϕ,0,±ϕ±ϕ,±1ϕ,0, where ϕ=1+5/2 is thegolden ratio and satisfies ϕ=1+1/ϕ (see Figure 1). It is a well-known fact that the polytope Δ is simple but not rational. However, consider the quasilattice P that is generated by the following vectors in R3: (6)Y1=1ϕ,1,0Y2=0,1ϕ,1Y3=1,0,1ϕY4=-1ϕ,1,0Y5=0,-1ϕ,1Y6=1,0,-1ϕ. We remark that these six vectors and their opposites point to the twelve vertices of a regular icosahedron that is inscribed in the sphere of radius 3-ϕ (see Figures 2 and 3). The quasilattice P is known in physics as thesimple icosahedral lattice [6]. Now, an easy computation shows that (7)Δ=⋂j=112μ∈R3∗∣μ,Xj≥-ϕ, where Xi=Yi and X6+i=-Yi, for i=1,…,6. Therefore, Δ is quasirational with respect to P. Let us perform the generalized Delzant construction with respect to P and the vectors X1,…,X12. Following the proof of Theorem 5, we consider the surjective linear mapping (8)π:R12⟶R3ei⟼Xi. It is easy to see that the following relations (9)Y4Y5Y6=1ϕ1ϕ-1-11ϕ1ϕ1ϕ-11ϕY1Y2Y3 imply that the kernel of π, n, is the 9-dimensional subspace of R12 that is spanned by the vectors (10)e1+e7e2+e8e3+e9e4+e10e5+e11e6+e12e1+e2-ϕe3+e4e2+e3-ϕe1+e5e1+e3-ϕe2+e6. Since the vectors Xi, i=1,…,12, generate the quasilattice Q, the group N is connected and given by the group exp(n). Moreover, the moment mapping for the induced N-actionΨ:C12→(n)∗ has the 9 components below: (11)z12+z72-2ϕz22+z82-2ϕz32+z92-2ϕz42+z102-2ϕz52+z112-2ϕz62+z122-2ϕz12+z22-ϕz32+z42+2z22+z32-ϕz12+z52+2z12+z32-ϕz22+z62+2.The level set Ψ-1(0) is described by the 9 equations that are obtained by setting these components to 0. Finally, the symplectic toric quasifold M is given by the compact 6-dimensional quasifold Ψ-1(0)/N. The quasitorus D3=R3/P acts on M in a Hamiltonian fashion, with image of the corresponding moment mapping given exactly by the dodecahedron Δ. The action of D3 has 20 fixed points, and the moment mapping sends each of them to a different vertex of the dodecahedron. The quasifold M has an atlas made of 20 charts, each of which is centered around a different fixed point. To give an idea of the local behavior of M, we will describe the chart around the fixed point that maps to the vertex (-1,-1,-1). Consider the open neighborhood U~ of 0 in C3 defined by the inequalities below: (12)z12<2ϕz22<2ϕz32<2ϕ-2<z12+z22-ϕz32<2ϕ-2<z22+z32-ϕz12<2ϕ-2<z12+z32-ϕz22<2ϕand consider the following slice of Ψ-1(0) that is transversal to the N-orbits(13)U~⟶τ~w_∈Ψ-10∣wi≠0,i=4,…,12z1,z2,z3⟼z1,z2,z3,τ4z_,τ5z_,τ6z_,τ7z_,τ8z_,τ9z_,τ10z_,τ11z_,τ12z_, where z_=(z1,z2,z3)∈C3, w_=(w1,w2,w3,w4,w5,w6,w7,w8,w9,w10,w11,w12)∈C12, and (14)τ4z_=1ϕz12+z22-ϕz32+2τ5z_=1ϕz22+z32-ϕz12+2τ6z_=1ϕz12+z32-ϕz22+2τ7z_=2ϕ-z12τ8z_=2ϕ-z22τ9z_=2ϕ-z32τ10z_=z32-1ϕz12+z22+2τ11z_=z12-1ϕz22+z32+2τ12z_=z22-1ϕz12+z32+2. The mapping τ~ induces a homeomorphism(15)U~Γ⟶τUz1,z2,z3⟼τ~z1,z2,z3, where the open subset U of M is the quotient (16)w_∈Ψ-10∣wi≠0,i=4,…,12N and the discrete group Γ is given by (17)e2πiϕh+l,e2πiϕh+k,e2πiϕk+l∈T3∣h,k,l∈Z. The triple (U,τ,U~/Γ) defines a chart around the fixed point corresponding to the vertex (-1,-1,-1). The other charts can be described in a similar way.Figure 1
The regular dodecahedron.Figure 2
The vectors±Y1,…,±Y6.Figure 3
The regular icosahedron.Remark 7.
According to joint work with Battaglia [7], to any nonrational simple polytope one can also associate acomplex toric quasifold. If we apply the explicit construction given in [7] to the regular dodecahedron, similarly to the symplectic case, we get a complex 3-quasifold MC endowed with an action of the complex quasitorus DC3=C3/P. This action is holomorphic and has a dense open orbit. Moreover, by [7, Theorem 3.2], the quasifold MC is D3-equivariantly diffeomorphic to M and the induced symplectic form on MC is Kähler.Remark 8.
The only other platonic solids that are simple are the cube and the regular tetrahedron. They both satisfy the hypotheses of the Delzant theorem. By applying the Delzant procedure to the cube, with respect to the standard latticeZ3, we get the symplectic toric manifold S2×S2×S2; by applying it to the tetrahedron, with respect to the sublattice of Z3 that is generated by the 8 vectors (±1,±1,±1), we get the symplectic toric manifold CP3. On the other hand, the remaining platonic solids, the regular octahedron and the regular icosahedron, are not simple. In these cases, the Delzant procedure and our generalization will not work. The octahedron is rational, thus, the standard toric geometry applies: the toric variety associated to the octahedron is described, for example, in [8, Section 1.5]. The icosahedron, on the other hand, is not rational. However, by work of Battaglia on arbitrary convex polytopes [9, 10], one can associate to the icosahedron, both in the symplectic and complex category, a space that is stratified by quasifolds. Incidentally, Battaglia’s approach can also be applied to the octahedron, yielding, in this case, a space that is stratified by manifolds.



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