# Toric Geometry of the Regular Convex Polyhedra

Authors: Fiammetta Battaglia; Elisa Prato
Journal: Journal of Mathematics
 (2017)

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## 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.


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## Body

## 1. Introduction

Among the five regular convex polyhedra, the regular tetrahedron and the cube are examples of simple rational convex polytopes. To these, the standard smooth toric geometry applies, in both the symplectic and complex category. From the symplectic viewpoint, in fact, the regular tetrahedron and the cube satisfy the hypotheses of the Delzant theorem [1], and it is easily seen that they correspond, respectively, to S7/S1 and S2×S2×S2. From the complex viewpoint, on the other hand, the toric variety associated with the regular tetrahedron is CP3 while CP1×CP1×CP1 corresponds to the cube (see, e.g., [2, Section  1.5]); these can also be obtained as quotients by a complex version of the Delzant procedure described by Audin in [3, Chapter  VI].The regular octahedron is still rational, but it is no longer simple. The toric variety associated with the regular octahedron is however well known and is described, for example, by Fulton in [2, Section  1.5]; it can also be obtained as a complex quotient by applying the Cox construction [4, Theorem  2.1].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 [5] that, by applying her extension of the Delzant procedure to the case of general simple convex polytopes [6], one can associate with the regular dodecahedron asymplectic 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.In this article, we recall all of the above and we complete the picture, first of all, by associating with the regular dodecahedron acomplex toric quasifold. We do so by applying a generalization, given by the authors in [7], of the procedure described by Audin in [3, Chapter  VI] to the case of general simple convex polytopes. As in the smooth case, the symplectic and complex quotients can be identified [7, Theorem  3.2], endowing the corresponding toric quasifold with a Kähler structure.We go on to address the case of the regular icosahedron. From the toric viewpoint, this is certainly the most complicated of the five regular convex polyhedra, since it is neither simple nor rational. However, we can apply generalizations by Battaglia and Prato [8] and Battaglia [9, 10] of toric quasifolds [6, 7] and of the Cox construction [4] to arbitrary convex polytopes; this allows us to associate with the regular icosahedron, in both the symplectic and complex category, a space that is stratified by quasifolds. As for all 3-dimensional polytopes, here there are only zero-dimensional singular strata and an open dense regular stratum. Moreover, by [10, Theorem  3.3], the symplectic and complex quotients can be identified, endowing the regular stratum with the structure of a Kähler quasifold.Notice, finally, that we are still missing a symplectic toric space corresponding to the regular octahedron; this too can be found by applying Battaglia’s work on arbitrary convex polytopes. What we get here is a space that is stratified by symplectic manifolds (see [9, Remark  6.6]); moreover, by [10, Theorem  3.3], this symplectic quotient can be identified with the complex quotient, and the regular stratum is Kähler.The article is structured as follows: in Section2 we recall a few necessary facts on convex polytopes; in Section 3 we recall from [6, 7] how to construct symplectic and complex toric quasifolds from simple convex polytopes; in Section 4 we recall from [9, 10] the construction of the symplectic and complex toric spaces corresponding to arbitrary convex polytopes; finally, in Sections 5, 6, 7, and 8 we describe the symplectic and complex toric spaces corresponding to the five regular convex polyhedra.


## 2. Facts on Convex Polytopes

Consider a dimensionn convex polytope Δ⊂Rn∗.Definition 1 (simple polytope).
Δ is said to besimple if each of its vertices is contained in exactly n facets.Assume now thatΔ has d facets. Then there exist elements X1,…,Xd in Rn and real numbers λ1,…,λd such that(1)Δ=⋂j=1dμ∈Rn∗∣μ,Xj≥λj.Let us consider the open faces of Δ. They can be described as follows. For each such face F there exists a possibly empty subset IF⊂{1,…,d} such that(2)F=μ∈Δ∣μ,Xj=λjiffj∈IF.A partial order on the set of all open faces of Δ is defined by setting F≤F′ (we say F contained in F′) if F⊆F′¯. Notice that F≤F′ if and only if IF′⊂IF. The polytope Δ is the disjoint union of its open faces. Let rF=cardIF; we have the following definitions.Definition 2.
Ap-dimensional open face F of the polytope is said to besingular if rF>n-p.Definition 3.
Ap-dimensional open face F of the polytope is said to beregular if rF=n-p.Remark 4.
LetF be a p-dimensional singular face in Rn∗, then p<n-2. Therefore any polytope in R2∗ is simple and the singular faces of a nonsimple polytope in R3∗ are vertices.We refer the reader to Ziegler’s book [11] for additional basic facts on convex polytopes. We now go on to recall what is meant by quasilattice and quasirational polytope.Definition 5 (quasilattice).
Aquasilattice in Rn is the Z-span of a set of R-spanning vectors, Y1,…,Yq, of Rn.Notice thatSpanZY1,…,Yq is a lattice if and only if it is generated by a basis of Rn.Definition 6 (quasirational polytope).
LetQ be a quasilattice in Rn. A convex polytope Δ⊂(Rn)∗ is said to bequasirational with respect to the quasilattice Q if the vectors X1,…,Xd in (1) can be chosen in Q.Remark that each polytope inRn∗ is quasirational with respect to the quasilattice Q that is generated by the elements X1,…,Xd in (1). We note that if X1,…,Xd can be chosen inside a lattice, then the polytope is rational.


## 3. The Simple Case

LetΔ⊂Rn∗ be an n-dimensional simple convex polytope. We are now ready to recall from [6] and [7] the construction of the symplectic and complex toric quasifolds associated with Δ. For the definition and main properties of symplectic and complex quasifolds we refer the reader to [6, 12] and [7]. For the purposes of this article, we will restrict our attention to the special case n=3. We begin by remarking that both constructions rely on the notion of quasitorus, which we recall.Definition 7 (quasitorus).
LetQ be a quasilattice in R3. We callquasitorus of dimension 3 the group and quasifold D3=R3/Q.Notice that, if the quasilattice is a lattice, we obtain the classical notion of torus. The quasilatticeQ also acts naturally on C3: (3)Q×C3⟶C3A,X+iY⟼X+A+iY.Therefore, in the complex category we have the following.Definition 8 (complex quasitorus).
LetQ be a quasilattice in R3. We callcomplex quasitorus of dimension 3 the group and complex quasifold DC3=C3/Q.In analogy with the smooth case, we will say thatDC3 is thecomplexification of D3. Assume now that our polytope Δ is quasirational with respect to a quasilattice Q and write(4)Δ=⋂j=1dμ∈R3∗∣μ,Xj≥λjfor some elements X1,…,Xd∈Q and some real numbers λ1,…,λd; again, d here is the number of facets of Δ. Let {e1,…,ed} denote the standard basis of Rd and Cd. Consider the surjective linear mapping (5)π:Rd⟶R3ej⟼Xj,and its complexification (6)πC:Cd⟶C3ej⟼Xj.Consider the quasitori D3=R3/Q and DC3=C3/Q. The mappings π and πC each induce group epimorphisms(7)Π:Td=RdZd⟶D3,ΠC:TCd=CdZd⟶DC3.We define N to be the kernel of the mapping Π and NC to be the kernel of the mapping ΠC. Notice that neither N nor NC are honest tori unless Q is a honest lattice. The Lie algebras of N and NC are, respectively, n=ker⁡π and nC=ker⁡πC. The mappings Π and ΠC induce isomorphisms (8)TdN⟶D3,TCdNC⟶DC3.Let us begin with the symplectic construction. Consider the space Cd, endowed with the symplectic form(9)ω0=12πi∑j=1ddzj∧dz¯jand the action of the torus Td=Rd/Zd:(10)Td×Cd⟶Cde2πiθ1,…,e2πiθd,z_⟼e2πiθ1z1,…,e2πiθdzd.This action is effective and Hamiltonian, with moment mapping given by (11)J:Cd⟶Rd∗z_⟼∑j=1dzj2ej∗+λ,λ∈Rd∗constant.Choose now λ=∑j=1dλjej∗, with λ1,…,λd as in (4). Denote by i the Lie algebra inclusion n→Rd and notice that Ψ=i∗∘J is a moment mapping for the induced action of N on Cd. Consider now the orbit space MΔ=Ψ-10/N. Then we have, from [6, Theorem  3.3], the following.Theorem 9 (generalized Delzant construction).
LetQbe a quasilattice inR3 and let Δ⊂R3∗ be a 3-dimensional simple convex polytope that is quasirational with respect to Q. Assume that d is the number of facets of Δ and consider vectors X1,…,Xd in Q that satisfy (4). For each Δ,Q,X1,…,Xd, the orbit space MΔ is a compact, connected 6-dimensional symplectic quasifold endowed with an effective Hamiltonian action of the quasitorus D3=R3/Q such that, if Φ:M→R3∗ is the corresponding moment mapping, then ΦMΔ=Δ.We say that the quasifoldMΔ with the effective Hamiltonian action of D3 is thesymplectic toric quasifold associated with Δ,Q,X1,…,Xd.Let us now pass to the complex construction. Following the notation of the previous section, consider, for any open faceF of Δ, the TCd-orbit (12)CFd=z_∈Cd∣zj=0iffj∈IF.Consider the open subset of Cd given by(13)CΔd=⋃F∈Δz_∈Cd∣zj≠0ifj∉IF.Notice that(14)CΔd=⋃μ∈Δz_∈Cd∣zj≠0ifj∉Iμ,where μ ranges over all the vertices of the polytope Δ. Moreover, since the polytope is simple, we have that (15)CΔd=⋃FCFd.In fact, in this case, z_∈Cd∣zj≠0ifj∉IF=∪F′≥FCF′d. The group NC acts on the space CΔd. Consider the space of orbits XΔ=CΔd/NC. We then have, from [7, Theorem  2.2], the following.Theorem 10.
LetQ be a quasilattice in R3 and let Δ⊂R3∗ be a 3-dimensional simple convex polytope that is quasirational with respect to Q. Assume that d is the number of facets of Δ and consider vectors X1,…,Xd in Q that satisfy (4). For each Δ,Q,X1,…,Xd, the corresponding quotient XΔ is a complex quasifold of dimension 3, endowed with a holomorphic action of the complex quasitorus DC3=C3/Q having a dense open orbit.We say that the quasifoldXΔ with the holomorphic action of DC3 is thecomplex toric quasifold associated with Δ,Q,X1,…,Xd.Finally, we conclude this section by recalling that the natural embedding(16)Ψ-10↪CΔdinduces a mapping (17)χ:MΔ⟶XΔthat sends each N-orbit to the corresponding NC-orbit. This mapping is equivariant with respect to the actions of the quasitori D3 and DC3. Then, under the same assumptions of Theorems 9 and 10, we have, from [7, Theorem  3.2], the following.Theorem 11.
The mappingχ: MΔ→XΔ is an equivariant diffeomorphism of quasifolds. Moreover, the induced symplectic form on the complex quasifold XΔ is Kähler.For the smooth case see Audin [3, Proposition  3.1.1] but also Guillemin [13, Appendix  1, Theorem  1.4].


## 4. The Nonsimple Case

Consider now a nonsimple convex polytopeΔ⊂R3∗ and assume that Δ is quasirational with respect to a quasilattice Q. The idea here is to repeat the constructions of the previous section. If we do so, we again find the groups N and NC and the quasitori D3 and DC3, isomorphic to Td/N and TCd/NC, respectively. However, the symplectic construction produces spaces that arestratified by symplectic quasifolds, while the complex construction yields spaces that arestratified by complex quasifolds. For the exact definitions of these notions we refer the reader to [9, Section  2] and [10, Definition  1.5]. We remark that in the smooth case these definitions yield the classical definition of Goresky and MacPherson [14]. Many of the important features of these stratified structures will be clarified when addressing the relevant examples (see Sections 6 and 8).Let us consider the symplectic case first. The main difference with respect to the case of simple polytopes is that here there are points in the level setΨ-10 that have isotropy groups of positive dimension; therefore Ψ-10 is no longer a smooth manifold. From Proposition  3.3 and Theorems  5.3, 5.10, 5.11, 6.4 in [9] we have the following.Theorem 12 (generalized Delzant construction: nonsimple case).
LetQ be a quasilattice in R3 and let Δ⊂R3∗ be a 3-dimensional convex polytope that is quasirational with respect to Q. Assume that d is the number of facets of Δ and consider vectors X1,…,Xd in Q that satisfy (4). For each Δ,Q,X1,…,Xd, the quotient MΔ is a compact, connected 6-dimensional space stratified by symplectic quasifolds, endowed with an effective continuous action of the quasitorus D3=R3/Q. Moreover, there exists a continuous mapping Φ: M→R3∗ such that ΦMΔ=Δ. Finally, the restriction of the D3-action to each stratum is smooth and Hamiltonian, with moment mapping given by the restriction of Φ.When the polytope is rational, these quotients are examples of the symplectic stratified spaces described by Sjamaar and Lerman in [15]; in particular, the strata are either manifolds or orbifolds [9, Remark  6.6]. We remark that the nonsimple rational case was addressed also by Burns et al. in [16, 17]; they gave an in-depth treatment and a classification theorem in the case of isolated singularities.Let us now examine the complex case. Here, one still considers the open subsetCΔd as defined in (13) but while in the simple case the orbits of exp⁡in on CΔd were closed, here there are nonclosed orbits. We recall first that NC=NA, where A=exp⁡in; this actually happens also in the simple case. Then, from [10, Theorem  2.1], we have the following.Theorem 13 (closed orbits).
Letz_∈CΔd. Then A·z_, the A-orbit through z_, is closed if and only if there exists a face F such that z_ is in CFd. Moreover, if A·z_ is nonclosed, then its closure contains one, and only one, closed A-orbit.Therefore, in order to define a notion of quotient, one defines the following equivalence relation: two pointsz_ and w_ in CΔd are equivalent with respect to the action of the group NC if and only if (18)NA·z_¯∩A·w_¯≠∅,where the closure is meant in CΔd. The space XΔ is then defined to be the quotient CΔd//NC with respect to this equivalence relation. Notice that if the polytope is simple, CΔd=∪F∈ΔCFd; thus, by Theorem 13, A-orbits through points in CΔd are always closed and the quotient XΔ is just the orbit space endowed with the quotient topology. From [10, Proposition  3.1,Theorem  3.2] we have the following.Theorem 14.
LetQ be a quasilattice in R3 and let Δ⊂R3∗ be a 3-dimensional convex polytope that is quasirational with respect to Q. Assume that d is the number of facets of Δ and consider vectors X1,…,Xd in Q that satisfy (4). For each Δ,Q,X1,…,Xd, the corresponding quotient XΔ is endowed with a stratification by complex quasifolds of dimension 3. The complex quasitorus DC3 acts continuously on XΔ, with a dense open orbit. Moreover, the restriction of the DC3-action to each stratum is holomorphic.We remark that whenQ is a lattice and the vectors X1,…,Xd are primitive in Q, the quotient XΔ coincides with the Cox presentation [4] of the classical toric variety that corresponds to the fan normal to the polytope Δ. As for classical toric varieties, there is a one-to-one correspondence between p-dimensional orbits of the quasitorus DC3 and p-dimensional faces of the polytope. In particular, the dense open orbit corresponds to the interior of the polytope and the singular strata correspond to singular faces.We remark that, like in the simple case, the natural embeddingΨ-10↪CΔd induces an identification between symplectic and complex quotients. From [10, Theorem  3.3] we have the following.Theorem 15.
The mappingχΔ: MΔ→XΔ is a homeomorphism which is equivariant with respect to the actions of D3 and DC3, respectively. Moreover, the restriction of χΔ to each stratum is a diffeomorphism of quasifolds. Finally, the induced symplectic form on each stratum is compatible with its complex structure, so that each stratum is Kähler.We conclude by pointing out thatMΔ≃XΔ has two different kinds of singularities, namely, the stratification and the quasifold structure of the strata. The nonsimplicity of the polytope yields the decomposition in strata of the corresponding topological space, while its nonrationality produces the quasifold structure of the strata and also intervenes in the way the strata are glued to each other. This last feature can be observed only in spaces with strata of positive dimension; this led to a definition of stratification that naturally extends the usual one [9, Section  2].


## 5. Simple and Rational: The Regular Tetrahedron and the Cube

The regular tetrahedron and the cube are both simple and rational. Let us recall the construction of the corresponding symplectic and complex toric manifolds. We follow the notation of Section3, which also applies to the smooth case.Let us begin with the regular tetrahedronΔ (see Figure 1) having vertices (19)ν1=1,1,1,ν2=1,-1,-1,ν3=-1,1,-1,ν4=-1,-1,1.Consider the sublattice L of Z3 that is generated by the corresponding four vectors (20)Y1=1,1,1,Y2=1,-1,-1,Y3=-1,1,-1,Y4=-1,-1,1.Notice that Y1+Y2+Y3+Y4=0; therefore any three of these four vectors form a basis of L. Moreover, (21)Δ=⋂j=14μ∈R3∗∣μ,Xj≥-1,where Xj=Yj,j=1,…,4. Thus the regular tetrahedron Δ satisfies the hypotheses of Delzant’s theorem [1] with respect to L. From the symplectic viewpoint, it is readily verified here that N=e2πiθ,e2πiθ,e2πiθ,e2πiθ∣θ∈R=S1 and therefore that MΔ is given by (22)z_∈C4∣z12+z22+z32+z42=4S1=S7S1,where S7 denotes the 7-sphere of radius 2. From the complex viewpoint, it is easy to see that CΔ4=C4∖0, that NC=C∗, and therefore that (23)XΔ=C4∖0C∗=CP3.Consider now the cube Δ having vertices (±1,±1,±1) (see Figure 2). Notice that (24)Δ=⋂j=16μ∈R3∗∣μ,Xj≥-1,where X1=e1, X2=-e1, X3=e2, X4=-e2, X5=e3, and X6=-e3. We can again apply the Delzant procedure, this time relatively to the lattice Z3, and we get that the 3-dimensional group N is given by (25)e2πiθ1,e2πiθ1,e2πiθ2,e2πiθ2,e2πiθ3,e2πiθ3∣θ1,θ2,θ3∈R=S1×S1×S1⊂T6and, therefore, the symplectic toric manifold MΔ is given by (26)z_∈C6∣z12+z22=2,z32+z42=2,z52+z62=2N=S2×S2×S2,where the S2’s have all radius 2. The corresponding complex toric manifold XΔ, on the other hand, is given by (27)C2∖0×C2∖0×C2∖0C∗×C∗×C∗=CP1×CP1×CP1.This provides an elementary example of a general fact: the symplectic toric manifold depends on the polytope, while the complex toric manifold only depends on the fan that is normal to the polytope. For instance, let us consider the cube having vertices ±a,±a,±a, with a a positive real number, with the same vectors X1,…,X6 as above. Then the corresponding symplectic toric manifold is the product of three spheres of radius 2a. In conclusion, the symplectic structure varies, while the complex toric manifold remains the same.Figure 1
The regular tetrahedron.Figure 2
The cube.


## 6. Nonsimple and Rational: The Regular Octahedron

Let us consider the regular octahedronΔ that is dual to the cube of Section 5 (see Figure 3). Its vertices are given by ν1=e1,ν2=-e1,ν3=e2,ν4=-e2,ν5=e3,ν6=-e3. This polytope is rational with respect to the lattice L defined in Section 5.Figure 3
The regular octahedron.In fact(28)Δ=⋂j=18μ∈R3∗∣μ,Xj≥-1,where Xi=Yi, X4+i=-Yi, i=1,…,4. However Δ is not simple. More precisely, consider the planes (29)Hi=μ∈R3∗∣μ,Xj=-1,i=1,…,8, containing the eight facets of the octahedron. Notice that each such plane contains exactly three vertices, as shown in Table 1.Table 1
Plane
Vertices
H1
(
ν
2
,
ν
4
,
ν
6
)
H2
ν
2
,
ν
3
,
ν
5
H3
(
ν
1
,
ν
4
,
ν
5
)
H4
(
ν
1
,
ν
3
,
ν
6
)
H5
(
ν
1
,
ν
3
,
ν
5
)
H6
(
ν
1
,
ν
4
,
ν
6
)
H7
(
ν
2
,
ν
3
,
ν
6
)
H8
(
ν
2
,
ν
4
,
ν
5
)Furthermore, each vertex is given by the intersection of four planes:(30)ν1=H3∩H4∩H5∩H6,ν2=H1∩H2∩H7∩H8,ν3=H2∩H4∩H5∩H7,ν4=H1∩H3∩H6∩H8,ν5=H2∩H3∩H5∩H8,ν6=H1∩H4∩H6∩H7.Therefore, each vertex is a singular face of Δ. If we apply the generalized Delzant construction we find that N=exp⁡n, with n given by (31)θ1,θ2,θ3,θ4,θ1+s,θ2+s,θ3+s,θ4+s∣θ1,θ2,θ3,θ4,s∈R.The level set Ψ-10 for the moment mapping Ψ with respect to the induced N-action on C8 is therefore given by the compact subset of points z_ in C8 such that (32)z12+z52=2,z22+z62=2,z32+z72=2,z42+z82=2,z32+z42=z52+z62.Notice that the N-action has isotropies of dimension 1 at the T8-orbits corresponding to the 6 (singular) vertices. For example, consider the first vertex: ν1=H3∩H4∩H5∩H6. The corresponding orbit in Ψ-10 is (33)z_∈C8∣z12=2,z22=2,z72=2,z82=2,with isotropy 1,1,e2πit,e2πit,1,1,e2πit,e2πit∣t∈R. Away from the orbits corresponding to the vertices, the level set is smooth. By Theorem 12, the orbit space (34)MΔ=Ψ-10Nis an elementary example of symplectic toric space stratified by manifolds. The six orbits in Ψ-10 corresponding to the vertices yield six singular points in the quotient MΔ. Their complement is the regular stratum of MΔ, which is a smooth symplectic manifold.From the complex viewpoint, by (14), the set CΔ8 is the union of 6 open sets, each of which corresponds to a vertex. The Cox quotient XΔ=CΔ8//NC is an elementary example of complex toric space stratified by manifolds. The quotients MΔ and XΔ can be identified according to Theorem 15.Let us briefly describeMΔ as a symplectic toric stratified space. By Theorem 12, the action of the torus D3=R3/L on MΔ is effective and continuous. Moreover, the image of the continuous mapping Φ=π∗-1∘J is exactly Δ and each singular point is sent by Φ to the corresponding vertex. The action of the torus D3=R3/L on the regular stratum is smooth and Hamiltonian, and the mapping Φ is the moment mapping with respect to the D3-action. By Theorem 15, the regular stratum is a 6-dimensional Kähler manifold [9, Remark  6.6].We now describe the local structure of the regular stratum and of the stratification around the singular points. Consider, for example, the vertexν1. We recall that the singular point m1∈MΔ corresponding to ν1 is given by (35)z_∈C8∣z12=2,z22=2,z72=2,z82=2N.We compute a symplectic chart for the regular stratum in a neighborhood of m1. Consider the open subset U given by (36)u3,u4,u5∈C3∖0∣u52<u32+u42<2+u52,ui2<2,i=3,4,5and the mapping (37)τ:U⟶z_∈Ψ-10∣zi≠0,i=1,2,6,7,8,z3,z4,z5≠0,0,0u3,u4,u5⟼τ1u_,τ2u_,u3,u4,u5,τ6u_,τ7u_,τ8u_,where (38)τ1u_=2-u52,τ2u_=2-u32-u42+u52,τ6u_=u32+u42-u52,τ7u_=2-u32,τ8u_=2-u42.The mapping τ is a homeomorphism (see Proof of [9, Theorem  5.3]). The pair U,ω0,τ, where ω0 is the standard symplectic structure of C3, defines a symplectic chart for the regular part in a neighborhood of m1. To find other charts, repeat the procedure by replacing the indices {3,4,5} with any triple of indices i,j,k contained in one of the six sets(39)3,4,5,6,1,2,7,8,2,4,5,7,1,3,6,8,2,3,5,8,1,4,6,7.We recall that, according to (30), each of these sets corresponds to a different vertex. We need at least 12 charts to obtain a symplectic atlas for the regular stratum.The regular part can also be seen as a complex manifold. Consider the open subset(40)V=C3∖0and the mapping (41)τC:V⟶z_∈CΔ8∣zi≠0,i=1,2,6,7,8,z3,z4,z5≠0,0,0u3,u4,u5⟼1,1,u3,u4,u5,1,1,1.The mapping τC is a homeomorphism. The pair V,τC defines a complex chart for the regular part in a neighborhood of the singular point x1=χm1. Again, a complex atlas can be obtained by considering all the triples of indices i,j,k contained in one of the six sets (39). Similarly to [7, Example  3.8], one can compute the local expression of the mapping χ as a diffeomorphism from U to V and find the local expression for the Kähler form on V.Let us now give an example of the existence of nonclosedA-orbits for nonsimple polytopes. Consider a point in CΔ8 that does not lie in any CF8; for instance, (42)z_=z1:z2:z3:0:0:0:z7:z8,z1,z2,z3,z7,z8≠0.Take the element at=1,1,e-2πt,e-2πt,e2πt,e2πt,1,1∈A. When t tends to +∞, at·z_ tends to z1:z2:0:0:0:0:z7:z8, which therefore lies in the closure of the orbit A·z_. Notice that, by Theorem 13, the orbit through z1:z2:0:0:0:0:z7:z8 is the only closed orbit contained in A·z_¯.Let us now show that the quotientMΔ≃XΔ is a stratified space in the sense of [14] with isolated singularities. More precisely we show that, in this case, a neighborhood of each singular point mj, with j=1,…,6, can be identified with a cone over a manifold, called the link of m. We consider the first vertex; the same argument applies to the others. Consider the cone stemming from ν1(43)C=⋂j=36μ∈R3∗∣μ,Xj≥-1.The complex and symplectic toric spaces corresponding to this cone are XC=C4//NCC and(44)MC=z_∈C4∣z32+z42=z52+z62NC, where (45)NC=exp⁡s,s,-s,-s∣s∈R.We now see explicitly that MC and XC are cones (diffeomorphic by the proof of [10, Theorem  3.3]) which are local models for our toric space MΔ≃XΔ near the singular point m1=χ-1x1. This will prove that MΔ≃XΔ is a stratified space. Following the recipes given in [10, Sections  3.2, 3.4], the link for m1=χ-1x1 can be found by cutting the cone C with the yz-plane; this gives the square ΔL of vertices 0,±1,±1. The 5-dimensionalreal linkL is then given by (46)z_∈C4∣z32+z42=z52+z62=2NC≃S3×S3S1,where the spheres S3 have both radius 2 and S1 acts with weights +1 on the first two coordinates and with weights -1 on the last two. From the complex point of view, we have(47)LC=C2∖0×C2∖0NCCexp⁡is,s,s,s∣s∈R.Notice that the inclusion(48)z_∈C4∣z32+z42=z52+z62=2↪C4induces a diffeomorphism L≃LC. By applying the Delzant procedure to the square ΔL, with respect to the normal vectors ±e2,±e3, one finds NΔL=exp⁡s,s,t,t∣s,t∈R. Thus the symplectic toric manifold MΔL is given by (49)z_∈C4∣z32+z42=z52+z62=2NΔL≃S2×S2,where the spheres S2 also have radius 2. We follow the terminology of [15] and refer to this manifold as thesymplectic link. Correspondingly,the complex link in the sense of [14, p. 15] is given by (50)XΔL=C2∖0×C2∖0NCΔL≃CP1×CP1.Notice that (51)C4//NCC∖0=C2∖0×C2∖0NCC;this shows that the cone XC is a complex cone over the complex link. Finally, remark that our space MC≃XC, without the singular point that corresponds to the cone apex, projects first onto the real link L≃LC and then onto the symplectic and complex links as follows:(52)On the left hand side,p1z_=rz1:rz2:rz3:rz4, with r=1/2(|z3|2+z4|2-1/2, p2 is the natural projection, and p is the composite. On the right hand side the mappings are the natural projections.The fiber of the projectionsp1 and pC1 is R>0, the fiber of the projections p2 and pC2 is (53)NΔLNC≃exp⁡t,t,t,t∣t∈R≃S1,and the fiber of the projections p and pC is (54)NCΔLNCC≃exp⁡u,u,u,u∣u∈C≃R>0×S1≃C∗.Therefore, XC≃MC is a real cone over LC≃L and a complex cone over the symplectic/complex link. Finally, notice that L/S1 gives the symplectic link.We now show thatMC and XC are local models for MΔ and XΔ, respectively. Consider, in the cone(55)z_∈C4∖0∣z32+z42=z52+z62/NC,the neighborhood C1 of the apex given by the set of points such that zi2<2,i=3,4,5,6. The continuous mapping (56)C1⟶z_∈Ψ-10∣zi≠0,i=1,2,7,8z3:z4:z5:z6⟼2-z32:2-z42:z3:z4:z5:z6:2-z72:2-z82is a symplectomorphism on the regular part and sends singular point to singular point. On the other hand, the continuous mapping (57)C4//NCC⟶z_∈C8∣zi≠0,i=1,2,7,8//NC⊂CΔ8//NCz3:z4:z5:z6⟼1:1:z3:z4:z5:z6:1:1is a biholomorphism on the regular part and also sends singular point to singular point.


## 7. Simple and Nonrational: The Regular Dodecahedron

Letϕ=1+5/2 be thegolden ratio and remark that it satisfies the equation ϕ=1+1/ϕ. Let Δ be the regular dodecahedron having vertices (58)±1,±1,±1,0,±ϕ,±1ϕ,±1ϕ,0,±ϕ,±ϕ,±1ϕ,0(see Figure 4). The polytope Δ is simple but not rational. However, it is quasirational with respect to the quasilattice P, known in physics as thesimple icosahedral lattice [18], that is generated by the following vectors in R3: (59)Y1=1ϕ2,1ϕ,0,Y2=0,1ϕ2,1ϕ,Y3=1ϕ,0,1ϕ2,Y4=-1ϕ2,1ϕ,0,Y5=0,-1ϕ2,1ϕ,Y6=1ϕ,0,-1ϕ2.In fact, a straightforward computation shows that (60)Δ=⋂j=112μ∈R3∗∣μ,Xj≥-1,where Xi=Yi, X6+i=-Yi, i=1,…,6. Notice that the vectors X1,…,X12 point to the twelve vertices of the regular icosahedron that is dual to Δ (see Figures 5 and 6).Figure 4
The regular dodecahedron.Figure 5
The vectorsX1,…,X12.Figure 6
The dual icosahedron.We recall from [5] that if we apply the generalized Delzant construction to Δ with respect to the quasilattice P and the vectors X1,…,X12, we get N=exp⁡(n), n being the 9-dimensional subspace of R12 that is spanned by the vectors (61)e1+e7,e2+e8,e3+e9,e4+e10,e5+e11,e6+e12,e1+e2-ϕe3+e4,e2+e3-ϕe1+e5,e1+e3-ϕe2+e6.The level set Ψ-10 for the moment mapping Ψ with respect to the induced N-action on C12 is therefore given by the compact subset of points z_ in C12 such that (62)z12+z72=2,z22+z82=2,z32+z92=2,z42+z102=2,z52+z112=2,z62+z122=2,z12+z22-ϕz32+z42=-2ϕ,z22+z32-ϕz12+z52=-2ϕ,z12+z32-ϕz22+z62=-2ϕ.The quotient MΔ=Ψ-10/N is a symplectic quasifold; it has an atlas made of 20 charts, each corresponding to a different fixed point of the D3-action; we refer the reader to [5] 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 vectorsXjj=1,…,12 above, is given by the quotient (63)XΔ=CΔ12NC,where CΔ12 is the open subset of C12 given by the union of the 20 open subsets defined in (14). The symplectic and complex quotients can be identified by Theorem 11.Let us describe a chart for the complex toric quasifoldXΔ around the fixed point corresponding to the vertex -1,-1,-1. Let V~=C3 and consider the following slice of CΔ12 that is transversal to the NC-orbits (64)τ~C:V~⟶w_∈CΔ12∣wi≠0,i=4,…,12z1,z2,z3⟼z1,z2,z3,1,…,1.The mapping τ~C induces a homeomorphism (65)τC:V~Γ⟶Uz1,z2,z3⟼τ~Cz1,z2,z3,where the open subset V of XΔ is the quotient (66)w_∈CΔ12∣wi≠0,i=4,…,12NCand the discrete group Γ is given by (67)e2πiϕh+l,e2πiϕh+k,e2πiϕk+l∈T3∣h,k,l∈Z.The triple (V,τ,V~/Γ) defines a complex chart for XΔ. The others can be described similarly.


## 8. Nonsimple and Nonrational: The Regular Icosahedron

LetΔ be the regular icosahedron that is dual to the dodecahedron in Section 7 (see Figure 7). Its vertices are given by (68)ν1=1ϕ2,1ϕ,0,ν2=0,1ϕ2,1ϕ,ν3=1ϕ,0,1ϕ2,ν4=-1ϕ2,1ϕ,0,ν5=0,-1ϕ2,1ϕ,ν6=1ϕ,0,-1ϕ2,ν7=-ν1,ν8=-ν2,ν9=-ν3,ν10=-ν4,ν11=-ν5,ν12=-ν6.It is not a rational polytope. However it is quasirational with respect to the quasilattice B, known in physics as thebody-centered lattice [18], that is generated by the six vectors (69)V1=ϕ,1ϕ,0,V2=0,ϕ,1ϕ,V3=1ϕ,0,ϕ,V4=-ϕ,1ϕ,0,V5=0,-ϕ,1ϕ,V6=1ϕ,0,-ϕ.Consider in fact the four additional vectors in B given by (70)V7=-V4-V5-V6=1,1,1,V8=-V1+V3-V5=-1,1,1,V9=+V1-V2-V6=1,-1,1,V10=+V2-V3-V4=1,1,-1.Then (71)Δ=⋂j=120μ∈R3∗∣μ,Xj≥-1,where Xi=Vi, X10+i=-Vi,i=1,…,10. Notice that the vectors X1,…,X20 point to the vertices of the dual regular dodecahedron of Section 7 (see Figures 8 and 9).Figure 7
The regular icosahedron.Figure 8
The normal vectorsX1,…,X20.Figure 9
The dual dodecahedron.The regular icosahedron is not a simple polytope. More precisely, consider the planes(72)Hi=μ∈R3∗∣μ,Xj=-1,i=1,…,20,supporting the facets of Δ. Notice that each such plane contains exactly three vertices as shown in Table 2.Table 2
Plane
Vertices
H1
(
ν
7
,
ν
9
,
ν
12
)
H2
(
ν
7
,
ν
8
,
ν
10
)
H3
(
ν
8
,
ν
9
,
ν
11
)
H4
(
ν
3
,
ν
6
,
ν
10
)
H5
(
ν
1
,
ν
4
,
ν
11
)
H6
(
ν
2
,
ν
5
,
ν
12
)
H7
(
ν
7
,
ν
8
,
ν
9
)
H8
(
ν
6
,
ν
8
,
ν
10
)
H9
(
ν
4
,
ν
9
,
ν
11
)
H10
(
ν
5
,
ν
7
,
ν
12
)
H11
(
ν
1
,
ν
3
,
ν
6
)
H12
(
ν
1
,
ν
2
,
ν
4
)
H13
(
ν
2
,
ν
3
,
ν
5
)
H14
(
ν
4
,
ν
9
,
ν
12
)
H15
(
ν
5
,
ν
7
,
ν
10
)
H16
(
ν
6
,
ν
8
,
ν
11
)
H17
(
ν
1
,
ν
2
,
ν
3
)
H18
(
ν
2
,
ν
4
,
ν
12
)
H19
(
ν
3
,
ν
5
,
ν
10
)
H20
(
ν
1
,
ν
6
,
ν
11
)Furthermore, each vertex is given by the intersection of five planes:(73)ν1=H5∩H11∩H12∩H17∩H20,ν2=H6∩H12∩H13∩H17∩H18,ν3=H4∩H11∩H13∩H17∩H19,ν4=H5∩H9∩H12∩H14∩H18,ν5=H6∩H10∩H13∩H15∩H19,ν6=H4∩H8∩H11∩H16∩H20,ν7=H1∩H2∩H7∩H10∩H15,ν8=H2∩H3∩H7∩H8∩H16,ν9=H1∩H3∩H7∩H9∩H14,ν10=H2∩H4∩H8∩H15∩H19,ν11=H3∩H5∩H9∩H16∩H20,ν12=H1∩H6∩H10∩H14∩H18.Let us now apply the generalized Delzant construction. It is easy to see that the relations (74)V1=-12ϕ+1ϕV4+ϕV5+1ϕV6,V2=-121ϕV4+ϕ+1ϕV5+ϕV6,V3=-12ϕV4+1ϕV5+ϕ+1ϕV6,V8=121ϕV4-V5-ϕV6,V9=12-ϕV4+1ϕV5-V6,V10=12-V4-ϕV5+1ϕV6imply that the kernel of π, n, is the 17-dimensional subspace of R20 that is spanned by the vectors (75)ei+e10+i,i=1,…,10,e1+12ϕ+1ϕe4+ϕe5+1ϕe6,e2+121ϕe4+ϕ+1ϕe5+ϕe6,e3+12ϕe4+1ϕe5+ϕ+1ϕe6,e7+e4+e5+e6,e8+12-1ϕe4+e5+ϕe6,e9+12ϕe4-1ϕe5+e6,e10+12e4+ϕe5-1ϕe6.Since the vectors Xi,i=1,…,20, generate the quasilattice B, the group N is connected and given by the group exp⁡n. The level set Ψ-10 for the moment mapping Ψ with respect to the induced N-action on C20 is given by the compact subset of points z_ in C20 such that (76)zi2+z10+i2=2i=1,…,10,z12+12ϕ+1ϕz42+ϕz52+1ϕz62=2ϕ,z22+121ϕz42+ϕ+1ϕz52+ϕz62=2ϕ,z32+12ϕz42+1ϕz52+ϕ+1ϕz62=2ϕ,z72+z42+z52+z62=4,z82+12-1ϕz42+z52+ϕz62=2,z92+12ϕz42-1ϕz52+z62=2,z102+12z42+ϕz52-1ϕz62=2.By Theorem 12, the quotient MΔ=Ψ-10/N is stratified by symplectic quasifolds.From the complex viewpoint, consider the open subsetCΔ20 of C20 obtained by (14) using (73). By Theorem 14, the quotient XΔ=CΔ20//NC is stratified by complex quasifolds.By Theorem15, MΔ and XΔ can be identified. Their global description is similar to the one given for the symplectic and complex toric spaces corresponding to the octahedron, except that here D3=R3/B is a quasitorus and the regular stratum is a quasifold.We describe a chart for the regular part of the complex quotient. We choose the vertexν4 and the triple 5,9,14. A local model for the following open subset of the regular part (77)z_∈CΔ20∣zi≠0,i≠5,9,14,z5,z9,z14≠0,0,0NCis given by C3∖0/Γ, where (78)Γ=exp⁡-ϕh,ϕh+k+2l,-ϕk∣h,k,l∈Z.Similarly to what happens for the octahedron, the quotient XΔ around the singular point corresponding to ν4 is a cone. We will also have a diagram similar to (52). In this case (79)nC=-ϕs-t,ϕs+t,s,-s-ϕt,t∣s,t∈R,while(80)NC=exp⁡-ϕs-t,ϕs+t+2l,s,-s-ϕt,t∣s,t∈R,l∈Z.The components of the moment mapping Ψ: C5→nC∗ with respect to the basis of nC∗ dual to -ϕ,ϕ,1,-1,0,-1,ϕ,0,-ϕ,1 are(81)-ϕz12+ϕz22+z32-z42,-z12+ϕz22-z42+z52.As in the case of the octahedron, we can find the symplectic and complex toric spaces corresponding to the cone C; these are regular symplectic and complex quasifolds away from the point corresponding to the cone apex. We now describe the local structure of the stratification. Consider the vector (82)Xν4=2-1,ϕ,0=-22+ϕX5+X9+X12+X14+X18=-V1+V2+V3+V4-V5+V6∈B.The plane Hν4=μ∈R3∗∣μ,Xν4=2/ϕ cuts the icosahedron in the regular pentagon ΔL whose vertices are given by ν1,ν2,ν9,ν11,ν12 (see Figure 10). Let ξν4=(-1/ϕ(2+ϕ),1/2+ϕ,0) be a point in the cutting plane Hν4. The rigid motion R+ξν4:R3∗→R3∗, where R is the rotation (83)ϕ2+ϕ-12+ϕ012+ϕϕ2+ϕ0001,maps the xz-plane into Hν4. We obtain that ΔL equals(84)⋂i=5,9,12,14,18μ=μ1,0,μ3∈R3∗∣Rμ+ξν4,Xi≥-1.This gives (85)ΔL=⋂i=5,9,12,14,18μ1,μ3∈R2∗∣μ1,μ3,Yi≥-22+ϕ,with (86)Y5=-ϕ2+ϕ,1ϕ,Y9=1ϕ2+ϕ,1,Y12=-ϕ2+ϕ,-1ϕ,Y14=22+ϕ,0,Y18=1ϕ2+ϕ,-1.The above vectors are obtained from X5,X9,X12,X14,X18 by rotating with Rt, the transpose of R, and then projecting onto the xz-plane. They generate the pentagonal quasilattice BL given by rotating and projecting the icosahedral quasilattice B onto the xz-plane. In fact, it is easy to check that X5,X9,X12,X14,X18,Xν4 generate B and, moreover, that RtXν4 is parallel to 0,1,0. Now we apply the generalized Delzant procedure to ΔL with respect to the quasilattice BL and the vectors Yi above. Again here NΔL is connected and is given by (87)exp⁡-ϕr-ϕs-t,r+ϕs+ϕt,s,-ϕr-s-ϕt,t∣r,s,t∈R.The level set Ψ-10 for the moment mapping Ψ with respect to the induced NΔL-action on C5 is therefore the compact submanifold of C5 given by the points z_ in C5 such that (88)ϕz12-z22+ϕz42=2ϕ,-ϕz12+ϕz22+z32-z42=0,-z12+ϕz22-ϕz42+z52=0.We thus obtain the 6-dimensional symplectic quasifold MΔL=Ψ-10/NΔL. From the complex viewpoint, the toric quasifold XΔL is given by the quotient CΔL5/NCΔL, where the open subset CΔL5 can be easily determined from Figure 10. We know that NC is a subgroup of NΔL; in fact it is given by(89)exp⁡-ϕr-ϕs-t,r+ϕs+ϕt,s,-ϕr-s-ϕt,t∈NΔL∣r=-2ϕl,withl∈Z.Interestingly, the quotient NΔL/NC is the quasitorus R/2ϕZ. Thus, in this case, the fibers of the projections p and pC of diagram (52) are given by (90)NCΔLNCC≃R>0×R2ϕZ.Figure 10
The pentagonΔL; its vertices and facets are suitably labeled.



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