Hecke Groups, Dessins d'Enfants and the Archimedean Solids

Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts.


Introduction
The Platonic solids -the convex polyhedra with equivalent faces composed of congruent convex regular polygons -have been known and studied by mathematicians and philosophers alike for millennia. More broadly, another class of convex polyhedra which are also well-known to mathematicians are the Archimedean solids: the semiregular convex polyhedra composed of two or more types of regular polygons meeting in identical vertices, with no requirement that faces be equivalent. There are three categories of such Archimedean solids: (I) the Platonic solids; (II) two infinite series solutions -the prisms and anti-prisms; and (III) fourteen further exceptional cases.
In [1], a novel approach to the Archimedean solids was taken by interpreting the graphs of these solids as clean dessins d'enfants in the sense of Grothendieck. We recall that a dessin is a bipartite graph drawn on a Riemann surface and that clean is the criterion that all the nodes of one of the two possible colours have valency two [2,3].
The underlying Riemann surface we will choose throughout will be simply the sphere CP 1 onto which we can embed the solids; furthermore, we will draw them in a planar projection. Now, the planar graph of any polytope can be interpreted as a clean dessin by inserting a black node into every edge of the graph, and colouring every vertex white.
Given any such clean dessin, one can extract a remarkable array of group theoretic and algebraic properties. Indeed, the motivation of [1] was to investigate some of these properties for the interesting class of highly symmetric dessins which correspond to the Archimedean solids.
In a parallel vein, trivalent clean dessins can be naturally associated to conjugacy classes of subgroups of the modular group in a manner we will soon describe; this further extends to a correspondence with K3 surfaces [4,5] and to gauge theories [6].
This paper constitutes an codification of the above directions into a unified outlook.
While [1] focused on finding the Belyi maps corresponding to each of the dessins for the Archimedean solids, here we shall approach these dessins from the group-theoretic angle discussed in [4], by considering their connections to subgroups of the famous modular group, and a more general class of groups known as Hecke groups. Recall that the modular group Γ ≡ Γ (1) = PSL(2; Z) = SL (2; Z) / {±I} is the group of linear fractional transformations z → az+b cz+d , with a, b, c, d ∈ Z and ad − bc = 1. The presentation of Γ is x, y|x 2 = y 3 = I . A natural generalisation is to consider groups with presentation x, y|x 2 = y n = I : such groups, which we shall denote H n , are the Hecke groups. Clearly, H 3 is the modular group.
With this in mind, every clean dessin d'enfant can be associated to the conjugacy class of a subgroup of a certain Hecke group, or free product of Hecke groups, in the following way: black nodes are associated with elements of the cyclic group C 2 = x|x 2 = I , while n-valent white nodes are associated with elements of the cyclic group C n = y|y n = I . For a dessin with white nodes with multiple valencies, the different valency white nodes are associated with the different C n , so that a clean dessin with white nodes of a single valency is associated with a conjugacy class of subgroups of a certain Hecke group, while a clean dessin with white nodes of multiple valencies is associated with a conjugacy class of subgroup of a group that is isomorphic to the free product of the relevant Hecke groups.
This association made, each dessin can be interpreted as the Schreier coset graph of the corresponding Hecke group [18], which is the Cayley graph of the Hecke group quotiented by some finite index subgroup. From this coset graph, the permutation representation for the associated conjugacy class of subgroups can be extracted. In this paper, we shall use these data to compute the generators for a representative of the conjugacy class of Hecke subgroups corresponding to each of the exceptional Archimedean solids as explicit 2 × 2 matrices. Once we have obtained these results, we shall moreover investigate whether any of these subgroups are so-called congruence subgroups.
The structure of this paper is as follows. First, in section 2 we present some technical details regarding Hecke groups and dessins d'enfants. We show that it is possible to interpret each clean dessin as the Schreier coset graph for a conjugacy class of subgroups of a certain Hecke group, or free product of Hecke groups. We then show that the Belyi map associated to the particular clean dessin in question is the map from the generalisation of the modular curve associated to each conjugacy class of subgroups of a Hecke group in question to P 1 . In section 3, we find the permutations for the conjugacy classes of subgroups of Hecke groups corresponding to every Archimedean solid, and provide explicit generating sets of matrices for representatives of those conjugacy classes of subgroups in the exceptional cases. Finally, we remark on the congruence properties of these subgroups.

Dessins d'Enfants and Hecke Groups
In this section, we first recall some essential details regarding both Hecke groups and clean dessins d'enfants. We begin by considering the modular group Γ ∼ = H 3 , as although this is isomorphic to only one particular Hecke group, it is by far the most wellstudied, and we shall draw upon the presented results at many points in the ensuing discussion. Subsequently, we discuss Hecke groups more generally, before moving on to consider clean dessins and their associated Belyi maps.
With these well-known results in hand, we then describe how every such clean dessin is isomorphic to the Schreier coset graph for a certain conjugacy class of subgroups of a Hecke group, or free product of Hecke groups (henceforth we may refer to such conjugacy classes as classes of Hecke subgroups; where there is no ambiguity we may use C to denote such classes). We then take a short digression in order to resolve some tangential questions regarding the construction of these Schreier coset graphs which have yet to be presented explicitly in the literature. Next, we describe the connection between the Belyi maps associated to the clean dessins in question and the algebraic curves corresponding to precisely the same classes of Hecke subgroups as arose in the discussion of coset graphs.

The Modular Group
To begin our discussion, we recall some essential details regarding the modular group Γ. This is the group of linear fractional transformations Z z → az+b cz+d , with a, b, c, d ∈ Z and ad − bc = 1. It is generated by the transformations T and S defined by: The presentation of Γ is S, T |S 2 = (ST ) 3 = I , and we will later discuss the presentations of certain modular subgroups. The 2 × 2 matrices for S and T are as follows: Letting x = S and y = ST denote the elements of order 2 and 3 respectively, we see that Γ is the free product of the cyclic groups C 2 = x|x 2 = I and C 3 = y|y 3 = I . It follows that 2 × 2 matrices for x and y are: With these basic details in hand, we should consider some notable subgroups of Γ.

Congruence Modular Subgroups
The most important subgroups of Γ are the so-called congruence subgroups, defined by having the the entries in the generating matrices S and T obeying some modular arithmetic. Some conjugacy classes of congruence subgroups of particular note are the following: • Principal congruence subgroups: • Congruence subgroups of level m: subgroups of Γ containing Γ (m) but not any Γ (n) for n < m; • Unipotent matrices: • Upper triangular matrices: for certain choices of m, d, , χ.
We note here that: In section 3 of this paper, we shall remark on the connections between some specific conjugacy classes of congruence modular subgroups and the Archimedean solids.

Hecke Groups
We can now extend our discussion of the modular group Γ ∼ = H 3 to the more general Hecke groups H n . The Hecke group H n has presentation x, y|x 2 = y n = I , and is thus the free product of cyclic groups C 2 = x|x 2 = I and C n = y|y n = I . H n is generated by transformations T and S now defined by: where λ n is some real number to be determined. The 2 × 2 matrices for these S and T are: Letting x = S and y = ST as in our discussion of Γ, we see that 2 × 2 matrices for x and y are: For H n , we clearly have (ST ) n = I, thereby constraining λ n for a given n [12].
Diagonalizing y to compute y n explicitly places a constraint which allows for a solution of λ n , we find that the following general expression, as well as some important values for small n: λ n = 2 + 2 cos 2π n , n 3 4 5 6 In particular, λ n are algebraic numbers.

Congruence Subgroups of Hecke Groups
By way of extension of the above discussion of subgroups of the modular group Γ, it is useful to consider congruence subgroups of Hecke groups. As stated in [11], the Hecke groups are discrete subgroups of PSL (2, R); in fact, the matrix entries are in Z [λ n ], the extension of the ring of integers by the algebraic number λ n . Note that: However, unlike the special case of the modular group, this inclusion is strict. With this point in mind, we can define the congruence subgroups of Hecke groups in the following way [12]. Let I be an ideal of Z [λ n ]. We then define: By analogy, we also define: As noted in [12], we clearly have: By analogy with our discussion of the modular group, we define congruence subgroups of level m of the Hecke group H n as subgroups of H n containing H n (m) but not any H n (p) for p < m [11]. With these details regarding Hecke groups and their subgroups in hand, we can now consider their connections to clean dessins d'enfants.

Dessins d'Enfants and Belyi Maps
A dessin d'enfant in the sense of Grothendieck is an ordered pair X, D , where X is an oriented compact topological surface and D ⊂ X is a finite graph satisfying the following conditions [3]: 1. D is connected.
2. D is bipartite, i.e. consists of only black and white nodes, such that vertices connected by an edge have different colours.
3. X \ D is the union of finitely many topological discs, which we call the faces.
We can interpret any polytope as a dessin by inserting a black node into every edge, and colouring all vertices white. This process of inserting into each edge a bivalent node of a certain colour is standard in the study of dessins d'enfants and gives rise to socalled clean dessins, i.e., those for which all the nodes of one of the two possible colours have valency two. An example of this procedure for the cube is shown in Figure 1.

Now recall that there is a one-to-one correspondence between dessins d'enfants and
Belyi maps [3]. A Belyi map is a holomorphic map to P 1 ramified at only {0, 1, ∞}, i.e. for which the only pointsx where d dx β (x) |x = 0 are such that β (x) ∈ {0, 1, ∞}. We can associate a Belyi map β (x) to a dessin via its ramification indices: the order of vanishing of the Taylor series for β (x) atx is the ramification index r β(x)∈{0,1,∞} (i) at that ith ramification point [4,6]. To draw the dessin from the map, we mark one white node for the ith pre-image of 0, with r 0 (i) edges emanating therefrom; similarly, we mark one black node for the jth pre-image of 1, with r 1 (j) edges. We connect the nodes with the edges, joining only black with white, such that each face is a polygon   with 2r ∞ (k) sides [4]. The converse direction (from dessins to Belyi maps) is detailed in e.g. [5].

Schreier Coset Graphs
As already discussed, the Hecke group H n has presentation x, y|x 2 = y n = I , and is thus the free product of cyclic groups C 2 = x|x 2 = I and C n = y|y n = I . Given the free product structure of H n , we see that its Cayley graph is an infinite free n-valent tree, but with each node replaced by an oriented n-gon. Now, for any finite index subgroup G of H n , we can quotient the Cayley graph to arrive at a finite graph by associating nodes to right cosets and edges between cosets which are related by action of a group element. In other words, this graph encodes the permutation representation of H n acting on the right cosets of G. This is called a Schreier coset graph, sometimes also referred to in the literature as a Schreier-Cayley coset graph, or simply a coset graph.
There is a direct connection between the Schreier coset graphs and the dessins d'enfants for each class of Hecke subgroups [18]: The dessins d'enfants for a certain conjugacy class of subgroups of a Hecke group H n can be constructed from the Schreier coset graphs by replacing each positively oriented n-gon with a white node, and inserting a black node into every edge.
Conversely, the Schreier coset graphs can be constructed from the dessins by replacing each white node with a positively oriented n-gon, and removing the black node from every edge.
Let σ 0 and σ 1 denote the permutations induced by the respective actions of x and y on the cosets of each subgroup. We can find a third permutation σ ∞ by imposing the following condition, thereby constructing a permutation triple: (2.14) The permutations σ 0 , σ 1 and σ ∞ give the permutation representations of H n on the right cosets of each subgroup in question. As elements of the symmetric group, σ 0 and σ 1 can be easily computed from the Schreier coset graphs by following the procedure elaborated in [17], i.e., by noting that the doubly directed edges represent an element x of order 2, while the positively oriented triangles represent an element y of order n. Since the graphs are connected, the group generated by x and y is transitive on the vertices. Clearly, σ 0 and σ 1 tell us which vertex of the coset graph is sent to which, i.e. which coset of the modular subgroup in question is sent to which by the action of H n on the right cosets of this subgroup.
Consider now a clean dessin d'enfant with white nodes with m different valencies (in the above, we assumed that all the white nodes of the dessin had the same valency).
It is still possible to interpret this dessin as a Schreier coset graph, though in this case the coset graph will correspond to a conjugacy class of subgroups of a group isomorphic to the free product of cyclic groups C 2 * C k 1 * . . . * C km , where the k i denote the different valencies of the white nodes of the dessin. Since, to recall, H n ∼ = C 2 * C n , we see that dessins with multiple white node valencies correspond to conjugacy classes of subgroups of free products of Hecke groups.

Polygon Orientation in Schreier Coset Graphs
As detailed above, Schreier coset graphs are built from simple edges (x) and n-gons (y) which are assumed to be positively oriented. Reversal of n-gon orientation corresponds to applying the automorphism of H n that inverts each of the two generators x and y.
At this point though, it is worth answering an intriguing tangential question: to what conjugacy classes of subgroups do the coset graphs correspond when arbitrary n-gons are reversed in orientation?
To answer this question, first consider the Schreier coset graphs for the conjugacy classes of congruence subgroups of the modular group Γ 0 (8) and Γ 1 (5), as shown in Figure 2 (these coset graphs were originally presented in [13]; the associated dessins d'enfants are drawn in [4]). Blue lines correspond to permutations σ 0 generated by x, while red triangles correspond to permutations σ 1 generated by y, and we assume initially that all such triangles are positively oriented. As stated in [17], these two coset graphs are not isomorphic, as they have different circuits.
What happens if we swap the orientation of an arbitrary triangle in one of these coset graphs? For example, what happens if we choose the third-from-left triangle in the coset graph for Γ 0 (8) to be negatively oriented rather than positively oriented?
Such a modified coset graph is drawn in Figure 3.
To answer this question, we read the relevant permutations off of the original and altered coset graph. In the original case, the corresponding conjugacy class of modular subgroups Γ 0 (8) is generated by the permutations: (3,4) , (6,9) , (5, 7) , (8, 10) , (11,12) σ 1 : (1, 2, 3) , (4, 5, 6) , (7,8,9) , (10, 11, 12) (2. 15) In the second, altered case, the corresponding conjugacy class of modular subgroups is that generated by the permutations: (6,9) , (5, 7) , (8, 10) , (11,12) σ 1 : (1, 2, 3) , (4, 5, 6) , (7,9,8) , (10, 11, 12) (2.16) Note that the triple (7,8,9) has changed to (7,9,8): this is the only change. Now, at this point we can use the GAP [16] algorithm presented in section 3.2 of [4] to compute a generating set of matrices for a representative of the conjugacy class of subgroups corresponding to our original and modified coset graphs, taking the above permutations as input. Of course, we would need to modify the algorithm in order to study classes of Hecke subgroups more generally; we present this modified code explicitly in the Appendix. In the original case, such a generating set is: From the definition of Γ 0 (m), we see that these are generators for a representative of the conjugacy class of congruence subgroups Γ 0 (8), as expected. In the modified case, a generating set is given by: Brief reflection on these matrices suffices to make apparent that these are generators of a representative of the conjugacy class of congruence subgroups Γ 1 (5). By reversing the orientation of the chosen triangle in the coset graph for Γ 0 (8), we have obtained the permutations for Γ 1 (5). Hence, this modified coset graph no longer corresponds to Γ 0 (8), but rather to Γ 1 (5). The reasons for this are easily stated in general terms: swapping n-gon orientations in Schreier coset graphs changes the permutations defining the associated conjugacy class of subgroups of the relevant Hecke group.
These permutations need not define the same conjugacy class of subgroups as the original permutations. In fact, in general they will not, and will instead define the subgroup whose Schreier coset graph with all positively oriented n-gons has the same cycles as the modified coset graph, with arbitrary n-gon orientations. This final point is nicely viewed in geometrical terms for our specific example: by choosing the thirdfrom-left triangle in the coset graph for Γ 0 (8) to be negatively oriented, we obtain a coset graph which has the same cycles as that for Γ 1 (5) when all the triangles are chosen to be positively oriented.

Algebraic Curves
Let p be prime and let Γ (p) ⊆ Γ (1) be the subgroup of matrices congruent to (plus or minus) the identity modulo p. Then Γ (p) acts on the completed upper half-plane H * , and the quotient X (p) = Γ (p) \ H * can be given the structure of a Riemann surface, known as a modular curve. Defining the subgroup G = Γ (1) /Γ (p) ⊆ Aut (X (p)), the j-map j : X (p) → X (p) /G ∼ = P 1 is a Galois cover ramified (with suitable normalisation) at {0, 1, ∞}. j is a Belyi map from X (p) to P 1 associated to a dessin drawn on the Riemann surface X (p).
Generalising now from Γ ∼ = H 3 to all Hecke groups H n , we first note that, as detailed in [3,11], the quotient space H/H n is a sphere with one puncture and two elliptic fixed points of order 2 and n. Therefore, all Hecke groups H n can be considered as triangle groups, and the Hecke surface H/H n is a Riemann surface [3,11]. Clearly, we are interested in the class of algebraic curves defined in analogy to the modular curve above, with the property that there exists a Hecke subgroup G ⊆ Aut (X) such that the map X → X/G ∼ = P 1 is a Galois cover ramified at {0, 1, ∞}.
With this in mind, let X be a compact Riemann surface which admits a regular tessellation by hyperbolic n-gons. Such a surface admits reconstruction from a torsionfree subgroup G of finite index in a Hecke group H n in the following way. The compact surface X = X G obtained from the group G has finitely many cusps added to the surface H/G, one for each G-orbit of boundary points at which the stabiliser is non-trivial.
The holomorphic projection π : H/G → H/H n ∼ = C induced from the identity map on the universal covering extends to a Belyi map β : X → P 1 from the compactification, with each cusp of X mapped by β to the single cusp {∞} of H n , which compactifies [9] the plane H/H n to C ∪ ∞ = P 1 .
The connection to dessins is then as follows: choose any particular clean dessin d'enfant. We have already described how this dessin can be interpreted as a Schreier coset graph for a particular Hecke group H n (or free product of Hecke groups). In addition though, the above results demonstrate that the Belyi rational map associated to the dessin is precisely the map X → P 1 , with each cusp of X mapped by β to the single cusp {∞} of H n .

Archimedean Solids
With the results of the previous section in hand, we can proceed to apply the theory to the so-called Archimedean solids. In this section, we first give a technical definition of these geometrical entities. We then identify the Hecke group for which each of the Archimedean solids has a corresponding conjugacy class of subgroups, before giving the permutation representations σ 0 and σ 1 for each corresponding class. We then (in every case where possible) find a representative of these classes of subgroups associated to each Archimedean solid by specifying a generating set for that representative, following the procedure in [4]. Finally, we follow up some residual questions from [4], investigating whether the trivalent Archimedean solids, when interpreted as dessins d'enfants, correspond in general to conjugacy classes of congruence subgroups of the modular group Γ.

Platonic and Archimedean Solids
The Platonic solids are well-known to us, and are the regular, convex polyhedra. They are the tetrahedron, cube, octahedron, dodecahedron and icosahedron. In order to introduce the wider class of so-called Archimedean solids, consider planar graphs without loops, and with vertices of degree k > 2. Following [1], let us call the list of numbers We emphasize a subtlety here. As stated in the Introduction, one informal way of defining the Archimedean solids is as the semi-regular convex polyhedra composed of two or more types of regular polygons meeting in identical vertices, with no requirement that faces be equivalent. Identical vertices is usually taken to mean that for any two vertices, there must be an isometry of the entire solid that takes one vertex to the other. Sometimes, however, it is only required that the faces that meet at one vertex are related isometrically to the faces that meet at the other. On the former definition, the so-called pseudorhombicuboctahedron, otherwise known as the elongated square gyrobicupola, is not considered an Archimedean solid; on the latter it is. The formal definition of the Archimedean solids above corresponds to the second, broader definition of identical vertices; following [1], it is this definition which we shall use. For a more extended discussion of this point, the reader is referred to [19].
It turns out the the solids which satisfy the above Archimedean condition are:  We record the number of vertices V , faces F and edges E for each. The vertex type is the number of edges which the adjacent faces to the vertex, counter-clockwisely, have; the fact that that all vertices have the same vertex type is the definition of Archimedean. "Sym" is the symmetry group of the solid, given in standard Schönflies notation. We also indicate whether the vertices are all trivalent, in which case the solid yields a planar Schreier coset graph for a conjugacy class of subgroups of the modular group; if not, it can be accommodated by a more general Hecke group H n .
All these solids are listed in Table 1, and are drawn in Figures 4 to 6 (for the n-prisms and n-antiprisms, we give examples for small n). All these images were constructed using Mathematica Version 8.0 [15]. For completeness, we also give the symmetry groups of each of these solids in the column labelled "Sym" in this table. These are given in standard so-called Schönflies notation. We recognize the standard tetrahedral, octahedral and icosahedral groups    Figure 6: Type III Archimedean solids: the 14 remaining exceptional cases. Note that in some references, the Pseudorhombicuboctahedron is not included because there is no isometry of the entire solid which takes each vertex to another.
rial to that work. By the correspondences detailed in the previous section, we should be able to associate a class of Hecke subgroups to every Archimedean solid. This task shall be undertaken in the following subsection.

Hecke Subgroups for the Archimedean Solids
Can the Archimedean solids all be associated with conjugacy classes of subgroups of the modular group, and are those subgroups congruence? The answer to the first question is straightforwardly no: only the trivalent solids are such that, interpreted as dessins, they are subgroups of the modular group, which is isomorphic to the free product of cyclic groups C 2 and C 3 , as detailed above. Hence, only those Archimedean solids labelled 'Yes' in the column 'Trivalent?' of Table 1 can be associated with subgroups of the modular group. The rest can be associated with conjugacy classes of subgroups of other Hecke groups, however. For example, the cuboctahedron is 4-valent [1], and can thus be associated with a conjugacy class of subgroups of the Hecke group H 4 ∼ = C 2 * C 4 .
The Hecke group related to each of the Archimedean solids in this way is presented in the final column of Table 1.
In the remainder of this subsection, we give the permutations σ 0 and σ 1 for the conjugacy class of subgroups of the relevant Hecke group for each Archimedean solid.
We also find explicitly a representative of the conjugacy class of subgroups of the relevant Hecke group to which each of the Archimedean solids corresponds (bar the two infinite series cases). The GAP [16] algorithm to achieve this task is presented in the Appendix of this paper. Since the Platonic solids have already been partly studied in e.g. [4], we divide the task into three sections, based upon the three categories of Archimedean solid presented in the previous subsection.

Platonic Solids
Interpreted as clean dessins d'enfants, the tetrahedron, cube, and dodecahedron correspond to the conjugacy classes of principal congruence subgroups Γ (3), Γ (4) and Γ (5), respectively. This can be verified explicitly from the dessins d'enfants presented in [4], where we can also see that the index of the associated class of modular subgroups is twice the number of edges of the Platonic solid in question. For convenience, these results are tabulated in Table 2. Now, reflection on Table 1 suffices to show that the remaining two Platonic solids -the octahedron and the icosahedron -will not correspond to subgroups of Γ, but rather to H 4 and H 5 , respectively. What are these subgroups?  We can answer this question by finding a generating set for a representative of the corresponding class in each case.
Octahedron: Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the octahedron are:

(3.2)
Using the procedure detailed in the Appendix, generators for a representative of the conjugacy class of subgroups of H 4 for the octahedron are: Comparison with our results concerning congruence subgroups of Hecke groups in section 2.2.1 above suffices to show that the octahedron corresponds to the principal congruence subgroup H 4 (3). We remark here that, as stated in [12], the matrices of H 4 are of the following two types: The elements of the first type form a subgroup of index 2 in H 4 ; all the subgroups of H 4 corresponding to the Archimedean solids will turn out to be subgroups of this subgroup.
Icosahedron: The elements of H 5 need not fall into such simple categories as those for H 4 as given in equation (3.4) above. The consequence is that the elements of the matrices of the respesentiative generating set for the icosahedron, snub cube and snub dodecahedron (to be discussed below), which correspond to conjugacy classes of subgroups of H 5 , are too complicated to present on paper, though they are given in the document which accompanies this paper. For this reason, in these three cases we resort in this paper proper to providing just the permutations σ 0 and σ 1 .
Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the icosahedron are:

Prisms and Antiprisms
The Prisms: Call the n-gon faced prism the n-prism. The n-prism has the following permutations σ 0 and σ 1 for x and y, respectively: (3i + 1, 3i + 6) (3i + 2, 3n + 3i + 1) (3n + 3i + 2, 3n + 3i + 6) This renders it a trivial matter to find a generating set for a representative of the relevant conjugacy class of modular subgroups for any prism, and moreover to check whether any n-prism corresponds to a conjugacy class of congruence subgroups of Γ (a topic to which we shall return in the following subsection).
Antiprisms: Call the n-gon faced anti-prism the n-antiprism. The n-antiprism has the following permutations σ 0 and σ 1 for x and y, respectively: Again, this renders it straightforward to find a generating set for a representative of the relevant conjugacy class of subgroups for any antiprism.

Exceptional Archimedean Solids
In addition to the Platonic solids and the prisms and antiprisms, there remain fourteen exceptional Archimedean solids, as given in Table 1. Here we give the permutations and compute a generating set for a representative of the corresponding class of Hecke subgroups.
Truncated tetrahedron: Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the truncated tetrahedron are:       Generators for a representative of the conjugacy class of subgroups of Γ for the truncated icosahedron are:

Congruence Archimedean Solids
We have seen that, interpreted as clean dessins d'enfants, the tetrahedron, cube, and dodecahedron correspond to the conjugacy classes of principal congruence subgroups Γ (3), Γ (4) and Γ (5), respectively [4]. In addition to these three arising as the dessins corresponding to certain congruence subgroups of the modular group, the dessins for the 33 conjugacy classes of genus zero, torsion-free congruence subgroups (discussed in detail in [4]) reveal that other Archimedean solids also correspond to certain conjugacy classes of congruence subgroups of the modular group. Specifically, the truncated tetrahedron is associated with the modular subgroup Γ 0 (2) ∩ Γ (3), while the 8-prism is associated with the subgroup Γ (8; 2, 1, 2), as shown in Figure 3.3.
Given these results, the question arises as to whether the other trivalent Archimedean solids can also be associated with congruence subgroups of the modular group. In order to answer this question, we use the Sage [14] command is congruence() to determine whether that conjugacy class of modular subgroups, defined by permutations σ 0 and σ 1 , is congruence. Below we give two examples of this procedure: one for the truncated tetrahedron (which, for the reasons explained above, we expect to be identified as congruence), and one for the 3-prism (the congruence properties of which are unknown).

Example 1: The Truncated Tetrahedron
The clean dessin d'enfant of the truncated tetrahedron is drawn above. As discussed, this can be interpreted as a Schreier coset graph by replacing every white node with a positively oriented triangle. We can then extract the permutations σ 0 and σ 1 for x and y. These permutations are given in the previous subsection. With these in hand, we can then proceed to check whether the modular subgroup associated to the truncated tetrahedron is congruence by implementing the following code in Sage: sage: G = ArithmeticSubgroup Permutation( Hence we see that the 3-prism is an Archimedean solid whose corresponding modular subgroup is congruence. This is to be expected: we already know that the truncated tetrahedron corresponds to the congruence subgroup Γ 0 (2) ∩ Γ (3) [4].
Our result for the 3-prism above demonstrates that the prisms in general do not correspond to conjugacy classes of congruence subgroups of the modular group, when interpreted as clean dessins d'enfants. However, we know that some prisms do correspond to conjugacy classes of congruence subgroups: we have seen that the 8-prism corresponds to the conjugacy class of subgroups Γ (8; 2, 1, 2). We hence conclude that the prisms do not in general correspond to conjugacy classes of congruence subgroups of the modular group, though there are exceptions.
Repeating the working presented in the examples above for the remaining trivalent, exceptional Archimedean solids gives the following result:

PROPOSITION 2
The truncated tetrahedron is the only trivalent, exceptional Archimedean solid which corresponds to a conjugacy class of congruence subgroups of the modular group.

Conclusions
In this paper, we have elaborated the connections between clean dessins d'enfants and conjugacy classes of subgroups of Hecke groups, or free products of Hecke groups. We are not the first to notice this connection: though framed in terms of maps rather than dessins, the link between Hecke groups and finite graphs was discussed in depth in [12]. This current work extends the connections between dessins d'enfants, group theory, and algebraic curves -which has previously focused largely on the modular group Γ ∼ = H 3 -to a much wider class of groups.
When it comes to applying this theory, we have first shown how, through interpretation as clean dessins d'enfants, each of the Archimedean solids can be associated with a conjugacy class of subgroups of Hecke groups. We have explicitly computed the matrix generators of these corresponding Hecke subgroups. Moreover, we have presented a number of results regarding the congruence properties of these subgroups. This work is useful not only as an illustration of the connections between Hecke groups and clean dessins, but also because it develops one more area of study of the Archimedean solids, and extends the work in this area already carried out in [1].
Possible developments of this work abound. For example, one might be interested in establishing whether the conjugacy classes of subgroups of Hecke groups for the Archimedean solids which correspond to subgroups of H 4 and H 5 are congruence. No general algorithm for checking the congruence properties of a Hecke group currently exists. There is, however, an algorithm for determining whether a subgroup of Γ is congruence [10], which we have employed (via implementation in Sage [14]) in this paper. To extend this algorithm to general Hecke groups is clearly an interesting task for future pursuit.
From an algebro-geometrical point of view, the subgroups of the modular group can be quotiented, upon adjoining cusps, to give modular curves which are Riemann surfaces of some genus. The tetrahedron, cube and dodecahedron correspond to Γ(n) for n = 3, 4, 5, which are all genus 0, torsion free subgroups of Γ. It would be interesting to explore the geometry of all the Riemann surfaces obtainable from the list of Archimedean solids. Indeed, physical interpretations of such dessins in relation to the modular curves have been studied in [6,7], to further investigate this direction is clearly of interest.