Toric Geometry of the Regular Convex Polyhedra

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.


Introduction
Among the five regular convex polyhedra, the regular tetrahedron and the cube are examples of simple rational convex polyopes. To these, the standard smooth toric geometry applies, both in the symplectic and complex category. From the symplectic viewpoint, in fact, the regular tetrahedron and the cube satisfy the hypotheses of the Delzant Theorem [10], and it is easily seen that they correspond, respectively, to S 7 /S 1 and S 2 × S 2 × S 2 . From the complex viewpoint, on the other hand, the toric variety associated to the regular tetrahedron is CP 3 while CP 1 × CP 1 × CP 1 corresponds to the cube (see for example [11,Section 1.5]); these can also be obtained as quotients by a complex version of the Delzant procedure described by Audin in [1,Chapter VI].
The regular octahedron is still rational, but it is no longer simple. The toric variety associated to the regular octahedron is however well known and is described, for example, by Fulton in [11,Section 1.5]; it can also be obtained as a complex quotient by applying the Cox construction [9, 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 [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.
In this article, we recall all of the above and we complete the picture, first of all, by associating to the regular dodecahedron a complex toric quasifold. We do so by applying a generalization, given by the authors in [4], of the procedure described by Audin in [1, Chapter VI] to the case of general simple convex polytopes. As in the smooth case, the symplectic and complex quotients can be identified [4,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 [5] and Battaglia [2,3] of toric quasifolds [14,4] and of the Cox construction [9] to arbitrary convex polytopes; this allows us to associate to the regular icosahedron, both in 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 [3,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 [2,Remark 6.6]); moreover, by [3,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 Section 1 we recall a few necessary facts on convex polytopes; in Section 2 we recall from [14,4] how to construct symplectic and complex toric quasifolds from simple convex polytopes; in Section 3 we recall from [2,3] the construction of the symplectic and complex toric spaces corresponding to arbitrary convex polytopes; finally, in Sections 4, 5, 6, and 7 we describe the symplectic and complex toric spaces corresponding to the five regular convex polyhedra.

Facts on convex polytopes
Consider a dimension n convex polytope ∆ ⊂ (R n ) * . Definition 1.1 (Simple polytope) ∆ is said to be simple if each of its vertices is contained in exactly n facets.
Assume now that ∆ has d facets. Then there exist elements X 1 , . . . , X d in R n and real numbers λ 1 , . . . , λ d such that Let us consider the open faces of ∆. They can be described as follows. For each such face F there exists a, possibly empty, subset I F ⊂ {1, . . . , d} such that A partial order on the set of all open faces of ∆ is defined by setting The polytope ∆ is the disjoint union of its open faces. Let r F = card(I F ); we have the following definitions: Remark 1.4 Let F be a p-dimensional singular face in (R n ) * , then p < n − 2. Therefore any polytope in (R 2 ) * is simple and the singular faces of a nonsimple polytope in (R 3 ) * are vertices.
We refer the reader to Ziegler's book [18] for additional basic facts on convex polytopes. We now go on to recall what is meant by quasilattice, and quasirational polytope.
Notice that Span Z {Y 1 , . . . , Y q } is a lattice if, and only if, it is generated by a basis of R n . Definition 1.6 (Quasirational polytope) Let Q be a quasilattice in R n . A convex polytope ∆ ⊂ (R n ) * is said to be quasirational with respect to the quasilattice Q if the vectors X 1 , . . . , X d in (1) can be chosen in Q.
Remark that each polytope in (R n ) * is quasirational with respect to the quasilattice Q that is generated by the elements X 1 , . . . , X d in (1). We note that if X 1 , . . . , X d can be chosen inside a lattice, then the polytope is rational.

The simple case
Let ∆ ⊂ (R n ) * be an n-dimensional simple convex polytope. We are now ready to recall from [14] and [4] the construction of the symplectic and complex toric quasifolds associated to ∆. For the definition and main properties of symplectic and complex quasifolds we refer the reader to [14,6] and [4]. 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 2.1 (Quasitorus) Let Q be a quasilattice in R 3 . We call quasitorus of dimension 3 the group and quasifold Notice that, if the quasilattice is a lattice, we obtain the classical notion of torus. The quasilattice Q also acts naturally on C 3 : Therefore, in the complex category we have Definition 2.2 (Complex quasitorus) Let Q be a quasilattice in R 3 . We call complex quasitorus of dimension 3 the group and complex quasifold D 3 C = C 3 /Q. In analogy with the smooth case, we will say that D 3 C is the complexification of D 3 . Assume now that our polytope ∆ is quasirational with respect to a quasilattice Q and write for some elements X 1 , . . . , X d ∈ Q and some real numbers λ 1 , . . . , λ d ; again, d here is the number of facets of ∆. Let {e 1 , . . . , e d } denote the standard basis of R d and C d . Consider the surjective linear mapping and its complexification Consider the quasitori D 3 = R 3 /Q and D 3 C = C 3 /Q. The mappings π and π C each induce group epimorphisms We define N to be the kernel of the mapping Π and N C to be the kernel of the mapping Π C . Notice that neither N nor N C are honest tori unless Q is a honest lattice. The Lie algebras of N and N C are, respectively, n = ker π and n C = ker π C . The mappings Π and Π C induce isomorphisms Let us begin with the symplectic construction. Consider the space C d , endowed with the symplectic form ω 0 = 1 2πi d j=1 dz j ∧ dz j and the action of the torus T d = R d /Z d : This action is effective and Hamiltonian, with moment mapping given by Choose now λ = d j=1 λ j e * j , with λ 1 , . . . , λ d as in (3). Denote by i the Lie algebra inclusion n → R d and notice that Ψ = i * • J is a moment mapping for the induced action of N on C d . Consider now the orbit space M ∆ = Ψ −1 (0)/N . Then we have, from [14, Theorem 3.3]: Theorem 2.3 (Generalized Delzant construction) Let Q be a quasilattice in R 3 and let ∆ ⊂ (R 3 ) * be a 3-dimensional simple convex polytope that is quasirational with respect to Q. Assume that d is the number of facets of ∆ and consider vectors X 1 , . . . , X d in Q that satisfy (3). For each (∆, Q, {X 1 , . . . , X d }), the orbit space M ∆ is a compact, connected 6-dimensional symplectic quasifold endowed with an effective Hamiltonian action of the quasitorus We say that the quasifold M ∆ with the effective Hamiltonian action of D 3 is the symplectic toric quasifold associated to (∆, Q, {X 1 , . . . , X d }).
Let us now pass to the complex construction. Following the notation of the previous section, consider, for any open face F of ∆, the T d C -orbit Consider the open subset of C d given by Notice that where µ ranges over all the vertices of the polytope ∆. Moreover, since the polytope is simple, we have that In fact, in this case, Consider the space of orbits X ∆ = C d ∆ /N C . We then have, from [4, Theorem 2.2]: Theorem 2.4 Let Q be a quasilattice in R 3 and let ∆ ⊂ (R 3 ) * be a 3-dimensional simple convex polytope that is quasirational with respect to Q. Assume that d is the number of facets of ∆ and consider vectors X 1 , . . . , X d in Q that satisfy (3). For each (∆, Q, {X 1 , . . . , X d }), the corresponding quotient X ∆ is a complex quasifold of dimension 3, endowed with a holomorphic action of the complex quasitorus D 3 We say that the quasifold X ∆ with the holomorphic action of D 3 C is the complex toric quasifold associated to (∆, Q, {X 1 , . . . , X d }).
Finally, we conclude this section by recalling that the natural embedding

The nonsimple case
Consider now a nonsimple convex polytope ∆ ⊂ (R 3 ) * and assume that ∆ 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 N C and the quasitori D 3 and D 3 C , isomorphic to T d /N and T d C /N C respectively. However, the symplectic construction produces spaces that are stratified by symplectic quasifolds, while the complex construction yields spaces that are stratified by complex quasifolds. For the exact definitions of these notions we refer the reader to [2, Section 2] and [3, Definition 1.5]. We remark that in the smooth case these definitions yield the classical definition of Goresky and MacPherson [12]. Many of the important features of these stratified structures will be clarified when addressing the relevant examples (see Sections 5 and 7).
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 set Ψ −1 (0) that have isotropy groups of positive dimension; therefore Ψ −1 (0) is no longer a smooth manifold. From Proposition 3.3 and Theorems 5.3, 5.10, 5.11, 6.4 in [2] we have Theorem 3.1 (Generalized Delzant construction: nonsimple case) Let Q be a quasilattice in R 3 and let ∆ ⊂ (R 3 ) * be a 3-dimensional convex polytope that is quasirational with respect to Q. Assume that d is the number of facets of ∆ and consider vectors X 1 , . . . , X d in Q that satisfy (3). For each (∆, Q, {X 1 , . . . , X d }), the quotient M ∆ is a compact, connected 6-dimensional space stratified by symplectic quasifolds, endowed with an effective continuos action of the quasitorus D 3 = R 3 /Q. Moreover, there exists a continuos mapping Φ : M → (R 3 ) * such that Φ(M ∆ ) = ∆. Finally, the restriction of the D 3 -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-Lerman in [17]; in particular, the strata are either manifolds or orbifolds [2, Remark 6.6]. We remark that the nonsimple rational case was addressed also by Burns-Guillemin-Lerman in [7,8]; 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 subset C d ∆ as defined in (4) but, while in the simple case the orbits of exp(in) on C d ∆ were closed, here there are nonclosed orbits. We recall first that N C = N A, where A = exp(in); this actually happens also in the simple case. Then, from [3, Theorem 2.1], we have Then the A-orbit through z, A · z, is closed if, and only if, there exists a face F such that z is in C d F . 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 points z and w in C d ∆ are equivalent with respect to the action of the group N C if, and only if, Let Q be a quasilattice in R 3 and let ∆ ⊂ (R 3 ) * be a 3-dimensional convex polytope that is quasirational with respect to Q. Assume that d is the number of facets of ∆ and consider vectors X 1 , . . . , X d in Q that satisfy (3). For each (∆, Q, {X 1 , . . . , X d }), the corresponding quotient X ∆ is endowed with a stratification by complex quasifolds of dimension 3. The complex quasitorus D 3 C acts continuously on X ∆ , with a dense open orbit. Moreover, the restriction of the D 3 C -action to each stratum is holomorphic.
We remark that, when Q is a lattice and the vectors X 1 , . . . , X d are primitive in Q, the quotient X ∆ coincides with the Cox presentation [9] of the classical toric variety that corresponds to the fan normal to the polytope ∆. As for classical toric varieties, there is a one-to-one correspondence between p-dimensional orbits of the quasitorus D 3 C 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 embedding Ψ −1 (0) → C d ∆ induces an identification between symplectic and complex quotients. From [3, Theorem 3.3] we have The mapping χ ∆ : M ∆ −→ X ∆ is a homeomorphism which is equivariant with respect to the actions of D 3 and D 3 C , respectively. Moreover, the restriction of χ ∆ 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 that M ∆ X ∆ 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, whilst 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 [2, Section 2].

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 Section 2, which also applies to the smooth case. Let us begin with the regular tetrahedron ∆ (see Figure 1) having vertices Consider the sublattice L of Z 3 that is generated by the corresponding four vectors Notice that Y 1 + Y 2 + Y 3 + Y 4 = 0, therefore any three of these four vectors form a basis of L. Moreover, Thus the regular tetrahedron ∆ satisfies the hypotheses of Delzant's Theorem [10] with respect to L. From the symplectic viewpoint, it is readily verified here that N = { (e 2πiθ , e 2πiθ , e 2πiθ , e 2πiθ ) | θ ∈ R } = S 1 and therefore that where S 7 denotes the 7-sphere of radius 2. From the complex viewpoint, it is easy to see that C 4 ∆ = C 4 {0}, that N C = C * and therefore that Consider now the cube ∆ having vertices (±1, ±1, ±1) (see Figure 2). Notice that We can again apply the Delzant procedure, this time relatively to the lattice Z 3 , and we get the 3-dimensional group and, therefore, the symplectic toric manifold where the S 2 's have all radius √ 2. The corresponding complex toric manifold, on the other hand, is given by 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 X 1 , . . . , X 6 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.
The level set Ψ −1 (0) for the moment mapping Ψ with respect to the induced N -action on C 8 is therefore given by the compact subset of points z in C 8 such that Notice that the N -action has isotropies of dimension 1 at the T 8 -orbits corresponding to the 6 (singular) vertices. For example, consider the first vertex: with isotropy {(1, 1, e 2πit , e 2πit , 1, 1, e 2πit , e 2πit ) | t ∈ R}. Away from the orbits corresponding to the vertices, the level set is smooth. By Theorem 3.1, the orbit space is an elementary example of symplectic toric space stratified by manifolds. The six orbits in Ψ −1 (0) corresponding to the vertices yield six singular points in the quotient M . Their complement is the regular stratum of M , which is a smooth symplectic manifold.
From the complex viewpoint, by (5), the set C 8 ∆ is the union of 6 open sets, each of which corresponds to a vertex. The Cox quotient X ∆ = C 8 ∆ //N C is an elementary example of complex toric space stratified by manifolds. The quotients M ∆ and X ∆ can be identified according to Theorem 3.4.
Let us briefly describe M ∆ as a symplectic toric stratified space. By Theorem 3.1, the action of the torus D 3 = R 3 /L on M ∆ is effective and continuous. Moreover, the image of the continuous mapping Φ = (π * ) −1 • J is exactly ∆ and each singular point is sent by Φ to the corresponding vertex. The action of the torus D 3 = R 3 /L on the regular stratum is smooth and Hamiltonian, and the mapping Φ is the moment mapping with respect to the D 3 -action. By Theorem 3.4, the regular stratum is a 6-dimensional Kähler manifold [2, 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 vertex ν 1 . We recall that the singular point m 1 ∈ M ∆ corresponding to ν 1 is given by The mapping τ is a homeomorphism (see We recall that, according to (7), each of these sets correspond 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 and the mapping The mapping τ C is a homeomorphism. The pair (V, τ C ) defines a complex chart for the regular part in a neighborhood of the singular point x 1 = χ(m 1 ). Again, a complex atlas can be obtained by considering all the triples of indices {i, j, k} contained in one of the six sets (8). Analogously to [4,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 show that the quotient M ∆ X ∆ is a stratified space in the sense of [12] with isolated singularities. More precisely we show that, in this case, a neighborhood of each singular point m j , 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 ν 1 The symplectic and complex toric spaces corresponding to this cone are We now see explicitly that M C and X C are cones (diffeomorphic by the proof of [3, Theorem 3.3]) which are local models for our toric space M ∆ X ∆ near the singular point m 1 = χ −1 (x 1 ). This will prove that M ∆ X ∆ is a stratified space. Following the recipes given in [3,Sections 3.2,3.4], the link for m 1 = χ −1 (x 1 ) can be found by cutting the cone C with the yz-plane; this gives the square ∆ L of vertices (0, ±1, ±1). The 5-dimensional real link is then given by where the spheres S 3 have both radius √ 2 and S 1 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 Notice that the inclusion {z ∈ C 4 | |z 3 | 2 + |z 4 | 2 = |z 5 | 2 + |z 6 | 2 = 2} → C 4 induces a diffeomorphism L L C . By applying the Delzant procedure to the square ∆ L , with respect to the the normal vectors ±e 2 , ±e 3 , one finds N (∆ L ) = exp{(s, s, t, t) | s, t ∈ R}. Thus the symplectic toric manifold M ∆ L is given by where the spheres S 2 also have radius √ 2. We follow the terminology of [17] and refer to this manifold as the symplectic link. Correspondingly, the complex link in the sense of [12, p.15], is given by

Notice that
this shows that the cone X C is a complex cone over the complex link. Finally, remark that our space M C X C , without the singular point that corresponds to the cone apex, projects first onto the real link L L C and then onto the symplectic and complex links as follows: On the left hand side, p 1 ([z]) = [rz 1 : rz 2 : rz 3 : rz 4 ], with r = 1 2 (|z 3 | 2 + |z 4 | 2 ) −1/2 , p 2 is the natural projection and p is the composite. On the right hand side the mappings are the natural projections. The fiber of the projections p 1 and p 1 C is R >0 , the fiber of the projections p 2 and p 2 and the fiber of the projections p and p C is Therefore, X C M C is a real cone over L C L and a complex cone over the symplectic/complex link. Finally, notice that L/S 1 gives the symplectic link. We now show that M C and X C are local models for M ∆ and X ∆ respectively. Consider, in the cone {z ∈ C 4 {0} | |z 3 | 2 +|z 4 | 2 = |z 5 | 2 +|z 6 | 2 }/N (C), the neighborhood C 1 of the apex given by the set of points such that |z i | 2 < 2, i = 3, 4, 5, 6. The continuos mapping is a symplectomorphism on the regular part and sends singular point to singular point. On the other hand, the continuos mapping be the golden ratio and remark that it satisfies the equation φ = 1 + 1 φ . Let ∆ be the regular dodecahedron having vertices Figure 4). The polytope ∆ is simple but not rational. However, it is quasirational with respect to the quasilattice P , known in physics as the simple icosahedral lattice [16], that is generated by the following vectors in R 3 : In fact, a straightforward computation shows that: . . , 6. Notice that the vectors X 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 Figure 5: The vectors X 1 , . . . , X 12 Figure 6: The dual icosahedron respect to the quasilattice P and the vectors X 1 , . . . , X 12 , we get N = exp(n), n being the 9-dimensional subspace of R 12 that is spanned by the vectors e 1 + e 7 e 2 + e 8 e 3 + e 9 e 4 + e 10 e 5 + e 11 e 6 + e 12 e 1 + e 2 − φ(e 3 + e 4 ) e 2 + e 3 − φ(e 1 + e 5 ) e 1 + e 3 − φ(e 2 + e 6 ).
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. From the complex viewpoint, the complex toric quasifold corresponding to the dodecahedron, with the choice of the same vectors X j (j = 1, . . . , 12) above is given by the quotient ∆ is the open subset of C 12 given by the union of the 20 open subsets defined in (5). The symplectic and complex quotients can be identified by Theorem 2.5.
Let us describe a chart for the complex toric quasifold X ∆ around the fixed point corresponding to the vertex (−1, −1, −1). LetṼ = C 3 and consider the following slice of C 12 ∆ that is transversal to the N C -orbits The mappingτ C induces a homeomorphism where the open subset V of X ∆ is the quotient {w ∈ C 12 ∆ | w i = 0, i = 4, . . . , 12}/N C and the discrete group Γ is given by Γ = (e 2πiφ(h+l) , e 2πiφ(h+k) , e 2πiφ(k+l) ) ∈ T 3 | h, k, l ∈ Z .
The triple (V, τ,Ṽ /Γ) defines a complex chart for X ∆ . The others can be described similarly.

Nonsimple and nonrational: the regular icosahedron
Let ∆ be the regular icosahedron that is dual to the dodecahedron in Section 6 (see Figure 7). Its vertices are given by Figure 7: The regular icosahedron It is not a rational polytope. However it is quasirational with respect to the quasilattice B, known in physics as the body-centered lattice [16], that is generated by the six vectors Consider in fact the four additional vectors in B given by . . , 10. Notice that the vectors X 1 , . . . , X 20 point to the vertices of the dual regular dodecahedron of Section 6 (see Figures 8 and 9). The regular icosahedron is not a simple polytope. More precisely, consider the planes Figure 8: The normal vectors X 1 , . . . , X 20 Figure 9: The dual dodecahedron and that each vertex is given by the intersection of five planes: Let us now apply the generalized Delzant construction. It is easy to see that the following relations imply that the kernel of π, n, is the 17-dimensional subspace of R 20 that is spanned by the vectors e i + e 10+i , i = 1, . . . , 10 ]. Since the vectors X i , i = 1, . . . , 20, generate the quasilattice B, the group N is connected and given by the group exp(n). The level set Ψ −1 (0) for the moment mapping Ψ with respect to the induced N -action on C 20 is given by the compact subset of points z in C 20 such that By Theorem 3.1, the quotient M ∆ = Ψ −1 (0)/N is stratified by symplectic quasifolds. From the complex viewpoint, consider the open subset C 20 ∆ of C 20 obtained by (5) using (10). By Theorem 3.3, the quotient X ∆ = C 20 ∆ //N C is stratified by complex quasifolds.
By Theorem 3.4, M ∆ and X ∆ 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 D 3 = R 3 /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 vertex ν 4  Similarly to what happens for the octahedron, the quotient X ∆ around the singular point corresponding to ν 4 is a cone. We will also have a diagram similar to (9). In this case n(C) = {(−φs − t, φ(s + t), s, −s − φt, t) | s, t ∈ R} and N (C) = exp{(−φs − t, φ(s + t + 2l), s, −s − φt, t) | s, t ∈ R, l ∈ Z}.
This gives ∆ L = i=5,9,12,14,18 The above vectors are obtained from X 5 , X 9 , X 12 , X 14 , X 18 by rotating with R t and then projecting onto the xz-plane. They generate the pentagonal quasilattice B L given by rotating and projecting the icosahedral quasilattice B onto the xz-plane. In fact, it is easy to check that X 5 , X 9 , X 12 , X 14 , X 18 , X ν 4 generate B and, moreover, that R t (X ν 4 ) is parallel to (0, 1, 0). Now we apply the generalized Delzant procedure to ∆ L with respect to the quasilattice B L and the vectors Y i above. Again here N (∆ L ) is connected and is given by exp({(−φr − φs − t, r + φs + φt, s, −φr − s − φt, t) | r, s, t ∈ R}). The level set Ψ −1 (0) for the moment mapping Ψ with respect to the induced N (∆ L )-action on C 5 is therefore the compact submanifold of C 5 given by the points z in C 5 such that We thus obtain the 6-dimensional symplectic quasifold M ∆ L = Ψ −1 (0)/N (∆ L ). From the complex viewpoint, the toric quasifold X ∆ L is given by the quotient C 5 ∆ L /N C (∆ L ), where the open subset C 5 ∆ L can be easily determined from Figure 10. We know that N (C) is a subgroup of N (∆ L ); in fact N (C) = exp(−φr − φs − t, r + φs + φt, s, −φr − s − φt, t) ∈ N (∆ L ) | r = − 2 φ l, with l ∈ Z .

Conflict of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.