lead articles
Threedimensional Euclidean nets from twodimensional hyperbolic tilings: kaleidoscopic examples
^{a}Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia
^{*}Correspondence email: stephen.hyde@anu.edu.au
We present a method for geometric construction of periodic threedimensional Euclidean nets by projecting twodimensional hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMSs). Our techniques extend the combinatorial tiling theory of Dress, Huson & DelgadoFriedrichs to enumerate simple reticulations of these TPMSs. We include a taxonomy of all networks arising from kaleidoscopic hyperbolic tilings with up to two distinct tile types (and their duals, with two distinct vertices), mapped to three related TPMSs, namely Schwarz's primitive (P) and diamond (D) surfaces, and Schoen's gyroid (G).
Keywords: threedimensional Euclidean nets; twodimensional hyperbolic tilings; triply periodic minimal surfaces; kaleidoscopic hyperbolic tilings.
1. Introduction
The science of networks has grown beyond the inherent mathematical interest of graph theory, thanks to the intimate connection between complex systems and networks. Those connections are now a prime focus of statistical physics, where the study of random networks has been catalysed by interesting results concerning `smallworld' and `scalefree' systems (Watts, 2003; Barabasi, 2002). Complementary to these endeavours is the study of `crystal nets': triply periodic embeddings of graphs in threedimensional Euclidean space. Interest in crystal nets stems from their fundamental relevance to condensed materials (Hyde et al., 2008) from atomic crystals (Wells, 1977; O'Keeffe & Hyde, 1996; Klee, 2004) to novel framework materials (Ockwig et al., 2005; Öhrström & Larsson, 2005), including carbon polymorphs (Strong et al., 2004), zeolites (Treacy et al., 1997; International Zeolite Association, 2008), related oxide (Zou et al., 2008) and aluminophosphate (AlPO) materials (Li et al., 2008), imidazolate (ZIF) frameworks (Banerjee et al., 2008) and metalcoordination polymeric materials (Blatov et al., 2004; Batten, 2001). The microdomains of some liquidcrystalline phases of soft materials, including amphiphile and assemblies and derivative mesoporous solids, are also characterized by crystal nets (Hyde & Schroeder, 2003).
This paper is concerned with the enumeration of crystal nets (or just `nets'), a topic that has gained momentum in recent years, helped by advances in tiling theory that we also exploit here. The technique we use is complementary to a more conventional approach that employs threedimensional tiling theory (DelgadoFriedrichs et al., 1999; Blatov et al., 2007), in that we build the nets largely within nonEuclidean (hyperbolic) twodimensional space, then project to threedimensional (Euclidean) space. Any systematic enumeration of crystal nets is constrained by the issue of combinatorial explosion; distinct approaches therefore explore different regions of the universe of crystal nets. Indeed, our results are largely unduplicated by threedimensional techniques (Hyde et al., 2006).
The underlying methodology of our approach is to decorate a surface with all allowed symmetric tilings of a given complexity, then use the topology of the tile boundaries and their embedding in the surface to define a net. To guide the reader through these various stages of net building, we adopt the convention that all tilings, composed of vertices, edges and faces, are denoted by uppercase names, while nets, composed of vertices and edges only, have lowercase names.
Our enumeration is constrained by considering a particular class of surfaces to reticulate. For various reasons that will be described in the paper, we choose triply periodic minimal surfaces (TPMSs), which give a natural bridge between twodimensional hyperbolic tilings and threedimensional Euclidean nets. A particular choice of TPMSs provides a filter with which to control the enumeration both in twodimensional hyperbolic and threedimensional Euclidean space. Here we use the simplest (cubic genus3) TPMS, namely Schwarz's primitive (P) and diamond (D) surfaces, and Schoen's gyroid (G).
Tilings of the TPMSs, called Etilings, may be lifted to their universal covering space, the twodimensional hyperbolic plane . Tilings there, that we call Utilings, give rise to twodimensional hyperbolic nets that we call hnets. Alternatively we can form a compact quotient space of the TPMS, where primitive lattice translations are made equivalent to the identity operation, and the resulting surface is the closed genus3 tritorus. Tilings on this finite surface define Otilings; subsequent embedding of the tritorus in threedimensional space leads to a specific threading of edges, forming the onet. Finally, the tilings of a TPMS can give rise to at least two different net types, depending on how much embedding information is used. A net embedded in the TPMS (following the edges of the Etiling) will share its threedimensional symmetry, and we label these surface reticulations enets. Ignoring the surface, and looking only at network topology, we can relax the net in threedimensional space () to give canonical, maximally symmetric forms called snets.
Here we explain in detail the process of generating the various tilings and nets. For ease of reference, we summarize these in Table 1 and display key steps of the enumeration in the flowchart of Fig. 1. A specific example is illustrated in Fig. 2.

In order to illustrate hyperbolic tilings, we require a model of the hyperbolic plane, that – like the surface of a sphere – cannot be readily imaged on the page without distortions. The hyperbolic plane is illustrated in this paper using the Poincaré disc model (Stillwell, 1989). The vast area of hyperbolic space is compressed into a unit disc at the expense of significant foreshortening of distances, that are shrunk more and more towards the perimeter of the disc. This model is attractive in that it is conformal, with no distortion of angles induced by the map from hyperbolic space to the disc; however, hyperbolic lines are imaged as circular arcs that intersect the disc boundary at 90°. The remainder of this introductory section provides an outline of the paper.
The observation that underlies our enumeration scheme is that the intrinsic geometry and symmetry of TPMSs are related to discrete groups of twodimensional hyperbolic isometries. It is therefore possible to wrap the hyperbolic plane onto a TPMS in almost the same way as the Euclidean plane wraps onto a cylinder. This wrapping is formally defined by a covering map, described in detail for the primitive (P), diamond (D) and gyroid (G) surfaces in §2. The determination of suitable covering maps is an essential component of the enumeration process. Further, among the infinite variety of twodimensional hyperbolic tilings, we choose those examples whose symmetries in are commensurate with both the local and translational symmetries of the TPMS. Those allowed symmetries were presented in an earlier paper (Robins et al., 2004a). In the current paper we focus on a subset of these: the kaleidoscopic groups which are generated by reflections only.
The enumeration of suitable hyperbolic tilings involves a straightforward application of combinatorial tiling theory as developed by Dress, Huson & DelgadoFriedrichs (Dress, 1987; Dress & Huson, 1987; DelgadoFriedrichs, 1994). We give an overview of their techniques in §3; technical definitions are summarized in Appendix B.
In §4 we describe how to embed and unfold these hyperbolic tilings in so they are compatible with the symmetries of the covering maps. This process involves a number of steps that extend Delaney–Dress theory to account for equivalent tilings on topologically complex TPMSs. This is the technical core of the current paper and involves both combinatorial and grouptheoretic algorithms. We then discuss, in §5.1, how these hyperbolic tilings project onto a TPMS, and present the spacegroup symmetries of the resulting surface reticulations.
In §6 we discard the tiling information and examine the topological structure of the nets defined by edges and vertices.
To further clarify our techniques, we illustrate the procedure from start to finish by a fully worked example in §7.
The output of the enumeration process for kaleidoscopic tilings is summarized in §§6 and 9, with emphasis on the variety of e and snets that result. These results are far too extensive to capture in any detail in a publication. Accordingly, we have built the online Epinet database (Ramsden et al., 2005), which has detailed searchable catalogues of data and images linking tilings and nets.
In the future, this project will grow to explore nonkaleidoscopic tilings, in directions outlined in §8. Accordingly, we have written this paper to provide a detailed foundation on which an evolving suite of network data will be built. Although foundations invariably make dull reading, they are needed to explicate a richer and more interesting The reader is therefore urged to read this paper in conjunction with the Epinet database, accessible at https://epinet.anu.edu.au .
2. Hyperbolic symmetries and the covering maps
Note. Throughout this paper we refer to twodimensional symmetry groups by their orbifold symbol, a notation introduced by Conway (1992) and described in Appendix A.
The P, D and G surfaces (illustrated in Figs. 3 and 4) each have intrinsic surface symmetry related to the group of hyperbolic reflections. This group is generated by three reflections, R_{1}, R_{2} and R_{3}, whose mirror lines bound a triangle in with corner angles of π/4, π/6 and π/2. This geometry induces a set of relations for the group,
and because the operations are reflections we also have R_{1}^{2} = R_{2}^{2} = R_{3}^{2} = I (the identity). Sadoc & Charvolin (1989) identified translational unit cells of the oriented P, D and G surfaces that pull back to the same dodecagon in the hyperbolic plane. These primitive rhombohedral unit cells and the corresponding dodecagon are shown in Figs. 3 and 5. Euclidean translation vectors for each surface pull back to the hyperbolic group generated by the six translations that pair opposite edges of the dodecagon. These translations were originally given in Sadoc & Charvolin (1989) and are rewritten here in terms of the reflections:
They satisfy the following identity:
The T generated by the t_{i} and translations is isomorphic to the fundamental group of a genus3 torus and therefore has orbifold symbol .
A covering map, , from onto a surface in , is a continuous function such that any small disc on the surface pulls back to a countably infinite number of isomorphic copies that form a periodic pattern in the hyperbolic plane. The covering maps we construct have the additional properties of being conformal and compatible with the symmetries of the surface. This means that given a Euclidean symmetry of the surface , there is a corresponding symmetry of the hyperbolic plane, , such that . Thus, the covering map defines a relationship (a group homomorphism) between the hyperbolic symmetry group and the Euclidean
of the surface.In the case of the P surface, the hyperbolic reflections R_{1} and R_{2} map to Euclidean reflections in (110) and (100) mirror planes, respectively, whilst R_{3} maps to a twofold axis of rotation in the [110] directions, lying in the surface (and therefore swaps the sides of the surface). In the D surface R_{1} and R_{2} map to twofold rotation axes in the [110] and [100] directions, in the surface, and R_{3} maps to a reflection in the (110) plane. The Euclidean symmetries of the G surface are related to the rotational 246 of . While the hyperbolic reflections are intrinsic local symmetries of the G surface, they do not correspond to Euclidean isometries of the entire surface. Instead, the rotation R_{1}R_{2} maps to a twofold axis in the [110] direction, perpendicular to the G surface, R_{1}R_{3} maps to a inversion centre at Wyckoff sites 16a of the in the [111] direction, and R_{2}R_{3} maps to a inversion at Wyckoff sites 24d in the [001] direction. These symmetries are indicated in the surfaces in Fig. 4. Further details can be found elsewhere (Robins et al., 2005; Molnar, 2002).
The translation T, introduced above, maps onto the Euclidean translation lattice of each oriented surface We can summarize the action of a covering map by describing how the six generators of T map to three independent Euclidean translations a, b, c. For the P, D and G surfaces we choose the mappings given in Table 2 (other choices are possible and give identical results).

A tiling of the hyperbolic plane will map to a discrete reticulation on a surface only when the symmetries of the tiling are commensurate with the chosen covering map. This means that the tiling should have at least the symmetry of the fundamental group of the surface [i.e., the firstorder homotopy group ]. Since the P, D and G surfaces have unbounded genus, their fundamental groups have infinitely many generators. These groups are isomorphic, via the covering maps, to distinct subgroups of T. A tiling with only the symmetry of , say, will map to an aperiodic decoration of the P surface, and will be incompatible with the D and G surfaces. In this paper we study hyperbolic tilings with the full T symmetry, because these tilings are compatible with all three surfaces, and generate triply periodic patterns in . In principle, we can use the T as the starting point for tiling enumeration, but for convenience we enumerate tilings within symmetry groups that lie between and T; we established earlier that 131 distinct subgroups are suitable (including and T) (Robins et al., 2004a).
Among the 131 subgroups of , we have enumerated so far those tilings whose symmetries are generated by reflections only, namely the kaleidoscopic subgroups. There are 14 such subgroups, specified by the generators given in Table 3. The relationships are represented in Fig. 6. The kaleidoscopic subgroups are examples of a much broader class of groups known to mathematicians as Coxeter groups. To conform with a forthcoming classification scheme for families of twodimensional orbifolds, we refer to the orbifolds of kaleidoscopic groups as Coxeter orbifolds. Among the 14 kaleidoscopic subgroups there are only 11 different Coxeter orbifolds. The results presented in this paper are derived from these kaleidoscopic subgroups, but the concepts and techniques apply (with a little modification) to subgroups with other types of symmetry.
 Figure 6 and ellipsoidal nodes denote a of subgroups. The numbers in the righthand column are the index in . 
In the following §3 we give an overview of the relevant algorithms from combinatorial tiling theory. We then discuss in §4 how to take these abstract representations and generate tilings of that are compatible with the surfacecovering maps.
3. Enumerating Delaney–Dress symbols of hyperbolic tilings
The study of tilings has a long history (Grünbaum & Shephard, 1987), but it was only in the 1980s that a finite symbol was found to uniquely encode both the topology and symmetry of an infinite periodic tiling. This result is due to Andreas Dress (Dress, 1987) using earlier work of Matthew Delaney, so we refer to this description of a tiling as a Delaney–Dress symbol. Two students of Dress, Daniel Huson and Olaf DelgadoFriedrichs, have extended his mathematical formalism and developed efficient algorithms for enumerating tilings of the sphere, Euclidean and hyperbolic planes, and threedimensional Euclidean space (Huson, 1993; DelgadoFriedrichs, 2003; DelgadoFriedrichs & Huson, 1999). This body of work is usually referred to as combinatorial tiling theory.
Both the symmetry and topology of a twodimensional tiling are efficiently encoded by a twodimensional Delaney–Dress symbol, described in detail in Appendix B. In effect, these symbols represent a triangulation of an orbifold. Thus, the enumeration of twodimensional tilings amounts to the systematic enumeration of all possible triangulations (with certain properties) of a given orbifold.
Delaney–Dress (D) symbols allow exhaustive enumeration of tilings within a symmetry group up to a desired level of complexity. The complexity is initially quantified by the number of distinct symmetry classes of tile, also called the transitivity class. Thus, a tiling in which every tile is related to every other tile by a symmetry of the entire tiling is tile1transitive. A tiling with k symmetrically distinct tiles is tilektransitive. The transitivity class of a tiling is incremented by splitting all symmetric copies of a chosen tile. The split operations we allow involve adding a single edge to the interior of a tile – either between a vertex and a nonincident edge, between two different edges or between two nonincident vertices.
The second distinction in complexity concerns the internal symmetry of a tile. A tile is fundamental if no symmetry of the entire tiling maps it onto itself. If the tile has internal symmetries it is by definition nonfundamental. A nonfundamental tile is obtained by gluing copies of a fundamental tile. A gluing is possible across an edge of a fundamental tile if the edge coincides with a mirror line, or has a centre of twofold rotational symmetry at its midpoint. Alternatively, copies of the fundamental tile may be glued around a vertex if it is the fixed point of a rotational symmetry (i.e. at a cone or corner point of the orbifold).
Following Huson (1993), we enumerate Delaney–Dress symbols in the following order. Our enumeration differs slightly from that in Huson (1993) and in Balke & Huson (1996), because we exclude tiles with less than three edges (for reasons given below).
(1) Start with the tile1transitive fundamental tilings with the given orbifold. These tilings are called Ftilings. Coxeter orbifolds (those of kaleidoscopic groups) have only one such Ftiling; all other orbifold families have more than one.
(2) Determine all additional tile1transitive tilings by applying glue operations to the Ftilings to obtain the FGtilings. Some example FG tilings are shown in Fig. 7.
(3) Determine all tile2transitive fundamental tilings by applying split operations to each Ftiling. The resulting cases are FStilings.
(4) Determine all possible tile2transitive nonfundamental tilings by applying glue operations to the FStilings. The results of a glue operation applied to a single tile class are called FSGtilings. When glue operations are applied to both tile classes, we get an FSGGtiling. See Fig. 8 for an illustration of these split and glue operations.
(5) Continue with tile3transitive fundamental tilings generated by two split operations, followed by up to three glue operations, and so on.
The above procedure forms tilings ordered by tiletransitivity class. In Table 4, we give the number of D symbols for tile1 and tile2transitive tilings within each of our 11 Coxeter orbifolds.

A simple way to generate additional tilings without further split or glue operations is to introduce the dual operation defined by replacing each tile by a vertex and joining these vertices by an edge if the original tiles are adjacent. A number of examples are given in Fig. 9. The dual tiling shares the symmetry of the original tiling. In addition, it has reversed twodimensional topological characteristics: the degree of a dual vertex is equal to the number of edges in the original polygonal tile and a dual tile is a polygon of order equal to the degree of the original vertex. This is why we require our tiles to have at least three edges: so that the dual vertices are at least degree three.
We do not quite double the number of D symbols by appending the vertexktransitive duals to the tilektransitive symbols. This is because some tilings are self dual, while other symbols may be paired as mutually dual; examples are given in Fig. 9. From our lists of tile1transitive tilings in the 11 Coxeter orbifolds we find three selfdual D symbols: a regular {6, 6} tiling (in ), a semiregular (6, 6) tiling (), and a regular {12, 12} tiling ().^{1} Among the tile2transitive tilings we find 17 selfdual symbols. A pair of D symbols from the original tiletransitive enumeration are called `mutual duals' when they are the dual of one another – i.e. the dual symbols are also obtained via a sequence of splits and glues from the fundamental tiletransitive tiling. Such a pair generates just two distinct D symbols, rather than four. We find three pairs that are tile1 and vertex1transitive, 15 pairings of a tile1, vertex2transitive tiling with a tile2, vertex1transitive one, and 69 tile2, vertex2transitive pairs. Thus, from our enumeration of 1450 tile1 and 2transitive D symbols generated via split and glue operations, we obtain a total of 2706 distinct D symbols after considering their duals, corresponding to tilings whose symmetries are those of one of the 131 subgroups. Among those distinct tilings, 95 examples are either vertex1 or tile1transitive.
For ease of reference, we give each of the 2706 D symbols a distinct name of the form QCn, where n = 1, 2, 3,…, characteristic of a tiling. We capitalize letters to indicate a tiling (as opposed to a net, whose name will contain lowercase letters); `Q' refers to the P, D and G family of TPMSs with cubic symmetry; the letter `C' indicates that the tiling comes from the family of Coxeter orbifolds of the (genus3) cubic TPMS. The final part of the name is an index, n, that is assigned according to a ranking of the 2706 distinct D symbols by the following sort key:
Here v is the vertex transitivity, t is the tile transitivity and the D symbol has a natural ordering discussed by DelgadoFriedrichs (2003). Thus, the tilings QC1–QC95 are a complete enumeration of (vertex or tile)1transitive tilings with kaleidoscopic symmetry compatible with the P, D and G surfaces, and QC96–QC2706 are all the 2transitive tilings. Should we choose to continue the split–glue sequence and enumerate tile3transitive tilings, the new symbols generated can be appended to the end of this list, since any duals that happen to have lower vertex transitivity already appear in the first 2706 tilings.
4. Embedding and unfolding Delaney–Dress symbols
In the previous section we described how to enumerate Delaney–Dress symbols that encode the symmetry and topology of tilings. Our next step is to take these abstract representations of tilings and determine explicit realizations in the hyperbolic plane that are compatible with the P, D and G surface covering maps. We call this process embedding.
The embedding operation is a manytomany mapping from tilings in simply connected twodimensional hyperbolic space to tilings on the multiply connected triply periodic minimal surfaces. First, a single D symbol can have multiple realizations that project to distinct surface reticulations. Second, two embedded D symbols (from different subgroups, say) can generate the same surface reticulation. Clarification of the embedding process involves subtleties that require us to go beyond Delaney–Dress tiling theory, since the standard theory is applicable to simply connected spaces only. In order to enumerate distinct tilings of a multiply connected TPMS, we require an explicit representation of the tiling in the translational domain. Finding such a representation is called unfolding. We call these embedded and unfolded tilings Utilings.
Before proceeding further, we must clarify what we mean by `the same surface reticulation'. Since our periodic minimal surfaces have a high degree of symmetry, we say that two surface tilings are equivalent if their Delaney–Dress chamber systems are related via a symmetry of the surface, otherwise they are distinct. Symmetries of the surface pull back (through the covering map) to symmetries in the hyperbolic plane, so two Utilings project to equivalent surface reticulations if their D chambers are related via a symmetry from .^{2}
The process of generating an unfolded D symbol involves a number of steps, whose degree of complexity depends on the symmetry of the original D symbol. To clarify the description, we first discuss the construction for tilings with symmetry (§4.1); tilings with other Coxeter orbifolds are discussed later (§4.2).
4.1. Tilings in
First, we introduce a triangulation of the tritorus, called the chart. This chart forms a bridge between geometry in and the group structure of modulo the translational domain. We then describe how to embed a D symbol in a fundamental domain. Finally, we use the chart as a scaffold to unfold the D symbol into one with symmetry.
4.1.1. The chart
The chart is a triangulation of a tritorus with triangles labelled by the 96 distinct words that represent elements in the quotient group. This chart forms the basis for unfolding a D symbol with symmetry into a D symbol with symmetry. We make an explicit connection between the geometry of the triangulation and the reflection group, using the process described below (Fig. 10 will assist visualization).
(1) First, we list the 96 distinct elements of the quotient group . We do this using the GAP package kbmag (an acronym for Knuth–Bendix in Monoids and Groups), although any enumeration algorithm will suffice (The GAP Group, 2002; Holt, 1998). The quotientgroup cosets are represented by minimal words in the reflections R_{1}, R_{2}, R_{3}. The kbmag enumeration uses a shortlex ordering derived from a given order for the generators. The 96 elements are listed in full in Appendix C, Table 15.
(2) The fundamental domain is a triangle bounded by mirror lines that meet at angles π/2, π/4 and π/6. A fundamental tiling of consists of these domains with 4, 8 and 12 triangles meeting at , and corner points, respectively. We pick an initial triangle and give it the label of the identity element, I.
(3) Since the reflections R_{1}, R_{2} and R_{3} map the identity triangle onto its three neighbours, these triangles are labelled accordingly. The R_{1} triangle is the neighbour opposite the corner point, R_{2} lies across the shortest edge (opposite the corner), and R_{3} is the neighbour sharing the hypotenuse of the identity triangle, opposite the point.
(4) We continue labelling each triangle by the minimal Rword that maps the identity triangle onto it. By continuity, the neighbours opposite the , and corners of a given triangle, V, will be VR_{1}, VR_{2} and VR_{3}. For example, the neighbours of R_{1} are R_{1}R_{1} = I, R_{1}R_{2} and R_{1}R_{3}, and the neighbours of R_{2} are R_{2}R_{1} = R_{1}R_{2}, R_{2}R_{2} = I and R_{2}R_{3}. The process of converting each neighbourword VR_{i} to its minimal version is called word reduction, and is performed using the kbmag package.
(5) The translational periodicity is encoded by performing word reduction within the quotient group, rather than the full group. The diagram in Fig. 10 shows, for example, the neighbour of triangle V, with label VR_{3}, which lies outside the 96 elements of the dodecagon. In fact triangle VR_{3} is the image of triangle W under the translation t_{2}. In the quotientgroup word reduction, we find . Thus, in the compact tritorus, triangles V and W are neighbours along the edge opposite their corners. We attach the t_{2} translation to this adjacency information to mark a boundary or cut line that enables us to convert from the compact surface triangulation to the universal cover in . We also use the cut lines to transform tilings and networks from to surface reticulations in . This is described in §5.1.
There is a nice correspondence, hinted at above, between the I. For example, the that maps the identity triangle onto the W triangle is R_{3}R_{1}R_{3}R_{1}R_{2}R_{3}R_{1}, see Fig. 10. By convention, the operation acts from the left, so the order of successive reflections is read from right to left. In contrast, the path through the triangulation associated with this word is read from left to right. It starts at I, visits its `R_{3}neighbour' (the triangle opposite its corner), then this triangle's `R_{1}neighbour' (the triangle opposite a corner) and so on, ending at the triangle labelled W. Similarly, triangle V is reached by a path starting at I then (reading from left to right) visiting neighbours according to the word R_{1}R_{3}R_{1}R_{2}R_{3}R_{1}, a twofold rotation about the circled vertex in Fig. 10.
that maps the identity triangle onto some image triangle and a path through the triangulation starting atFinally, the translation that maps W onto VR_{3} may be encoded by a path in the triangulation from W to VR_{3} passing through I,
Word reduction to the expression for t_{2} given in equation (1) uses the identity (R_{1}R_{3})^{6} = I which implies that the central part of the word reduces as
4.1.2. Embedding in
We now return to Delaney–Dress symbols and the problem of embedding them in the hyperbolic plane. Readers unfamiliar with Delaney–Dress theory should refer to Appendix B for the relevant definitions.
The embedding of D symbols with symmetry is simpler than other subgroups for two reasons. First, the geometry of the fundamental domain is uniquely determined by the angles of the hyperbolic triangle. Second, the three distinct angles permit only one way to embed a D symbol into this fundamental domain.
Consider, for example, the tile1transitive glued tiling (shown in Fig. 7b). The D symbol for this tiling is given in Table 5 and its embedding into the fundamental domain is shown in Fig. 11. The D symbol has two chambers labelled a and b that form a triangulation of the orbifold. Embedding this symbol amounts to determining how the two chambers sit within the triangle. Each triangular chamber has a 0, 1 and 2vertex that correspond, respectively, to a vertex of the tiling, an edge midpoint and a tile centroid. First note that the D symbol specifies that a and b are adjacent along their 0edge, i.e. the chamber edge opposite a vertex of the tiling, and that a and b are selfadjacent across their 1 and 2edges. This means that the 0edge is internal to the domain and the 1 and 2edges lie along mirror boundaries. The topological indices m_{12} specify the degree of the tile vertex that sits at the 0vertex of each chamber. So the 0vertex of a has degree 6 and the 0vertex of b has degree 4. This tells us that the 0vertex of a must sit at the corner and the 0vertex of b must sit at the corner. Finally, the index m_{01} specifies the order of the tile centred at the 2vertex of a chamber. For this example, the single tile is a quadrilateral and the 2vertex (of both a and b) sits at the corner.

In general, we know that the chambers of the D symbol must form a triangulation of the fundamental domain. We therefore need to identify the chamber edges that lie along mirror boundaries, and chamber walks that trace a circuit around each corner point of the triangle. This information comes from the adjacency information and the topological indices given by the D symbol. Once the mirror boundaries and corner points are identified, we can deduce the embedding of the remaining chambers into the fundamental domain.
4.1.3. Unfolding into the tritorus
Once we know how a D symbol sits inside the fundamental domain, we can use the chart to unfold this D symbol to cover the tritorus and thus obtain a larger D symbol with symmetry. The process involves a straightforward tilerewriting procedure, described below for the fundamental tiling, and illustrated in Fig. 12.
The D symbol for contains six chambers labelled 0 to 5. Since the chart has 96 triangles, the D symbol derived from will have 576 chambers. These chambers are labelled so that chamber n is symmetrically equivalent to chamber n [modulo(6)], see Fig. 12. In this example, the chamber adjacencies opposite 0 and 1vertices are exactly those derived from the symbol. The adjacencies across 2edges map from one tile into its neighbour, so these relations are derived from the chart structure. The cuts defined on the chart are also attached to the appropriate Dchamber adjacencies and recorded as information additional to the standard Dsymbol form. These cuts effectively define the embedding of the D symbol into the T of .
The glue and split operations described in §3 preserve the symmetry group, so their chambers cover the same fundamental orbifold domain. Thus, unfolding D symbols associated with tilings generated via split and glue operations proceeds in almost identical fashion to that for the fundamental tiling described above.
4.2. Tilings in other kaleidoscopic subgroups
Consider next the construction for D symbols whose symmetries are kaleidoscopic subgroups of . The embedding and unfolding of those D symbols proceeds in essentially the same manner as described for . However, we shall see that D symbols for subgroups of can embed in more than one way, further complicating the construction, but giving additional surface reticulations.
4.2.1. Building a chart
The chart is built from knowledge of the quotientgroup structure, i.e. explicit representation of the t_{i}, τ_{i} translations in terms of the R_{1}, R_{2}, R_{3} reflections. Although we constructed the 131 subgroups to have the t_{i}, τ_{i} translations as elements, we do not have explicit expressions for the t_{i}, τ_{i} translations as words in the generators, so it is simpler to use a combinatorial procedure to obtain the charts. For the 14 kaleidoscopic subgroups this is feasible because each fundamental domain is built from k whole triangles, where k is the index of the subgroup.
To obtain a fundamental domain for a kaleidoscopic k distinct elements gives us a fundamental domain. For example, , an order4 of , has a fundamental domain built from the four triangles labelled I, R_{1}, R_{2} and R_{1}R_{2}, as shown in Fig. 13. This domain is replicated by reflections in its boundaries, and we use a cumulative algorithm to acquire one fundamental domain at a time. For example, a neighbouring fundamental domain is the set of triangles labelled {R_{3}, R_{3}R_{1}, R_{3}R_{2}, R_{3}R_{1}R_{2}}, i.e., the image under R_{3} of the initial domain. The number of fundamental domains that covers the tritorus is 96/k, so the chart has 24 quadrilaterals.
we use the labelling of triangles by the action. A contiguous set of  Figure 13 chart. A fundamental domain for is built from the four triangles labelled 
The final step in building the chart is to identify new cut lines that follow domain boundaries. Starting from an initial domain, we grow out one domain at a time with the aim of keeping the region as circular as possible in the hyperbolic plane. This region is shown highlighted in Fig. 13. Eventually, the growing region will meet itself on the genus3 surface. The edges where this occurs define new cut lines for unwrapping the tritorus into the hyperbolic plane. The translation associated with each cut line is found by tracing a path through the underlying chart that is constrained to pass through the I triangle and the cutline edge. The path defines a word that reduces to a translation under kbmag word reduction.
4.2.2. Embedding and unfolding in a chart
The embedding and unfolding of D symbols into a
chart uses the same feature identification and tilerewriting procedures described earlier for . However, fundamental domains for the subgroups are not as constrained as the triangle, and this means some D symbols have more than one embedding compatible with the covering map. There are two routes to generating these multiple embeddings: distinct subgroups with the same orbifold and automorphisms of a fundamental domain. We illustrate each of these situations below.First, recall from Table 3 that three orbifolds (, and ) appear as pairs of distinct subgroups of . The different structures guarantee that these pairs have distinct charts. Thus, every D symbol from one of these orbifolds has two distinct embeddings in . An example from is illustrated in Fig. 14. Although the two embeddings have identical topology, they are not related by a symmetry of , and therefore project to different surface reticulations. The difference is also apparent via unfolding in the respective charts: a single D symbol embedded in two distinct charts generates distinct D symbols.
 Figure 14 labelled in Table 3 
Second, automorphisms of the kaleidoscopic subgroups are manifested as abstract symmetries of their fundamental domains, which may or may not be associated with symmetries from . Multiple embeddings of a D symbol are possible when the
domain has automorphisms that are not induced by an element of . If the tiling does not display the corresponding of its D chambers, then it has two distinct embeddings that are not related by a symmetry and therefore map to inequivalent surface reticulations. The kaleidoscopic subgroups with automorphisms that can produce distinct embeddings of D symbols are , , , , and .We illustrate this situation with an example. The subgroups and both have quadrilateral fundamental domains and an ). The of is defined by conjugacy with the R_{3} reflection, so automorphic embeddings of a D symbol will project to equivalent surface reticulations. The geometry of the domain shows that the of its domain is not a conjugacy. Now suppose we have a D symbol that is embedded in the domain, and consider the effect of the Clearly this induces a new embedding of the D symbol, which may or may not be equivalent to the original embedding. Equivalence occurs only when the D symbol has an of its chambers that corresponds to the of the domain. Thus, a split tiling where the extra edge divides the domain into two quadrilaterals (as in Fig. 16) will have two distinct embeddings, but a split between two opposite vertices will not.
that swaps the two opposing corner points (see Fig. 15  Figure 15 swapping the two opposite corners of is generated by a but this is not the case for . This leads to multiple embeddings of some D symbols in . 
We conclude this section by discussing the effects of a T, and the corresponding unfolded D symbols. There are two possibilities: the may or may not induce an of T. If T is preserved, then the two embeddings of a D symbol will unfold to isomorphic D symbols. If T is not preserved, then distinct D symbols are generated. The only kaleidoscopic where we see distinct embeddings of a tiling that unfold to equivalent D symbols is . This is due to an of that swaps the role of the t_{i} and τ_{i} generators of T. This is not generated by a but nonetheless preserves the T so the unfolded D symbols are isomorphic. An example is shown in Fig. 17.
on the translation5. Surface tilings
Our enumeration so far has taken distinct D symbols from each Coxeter orbifold, found all possible embeddings of these symbols into the kaleidoscopic T Since these tilings are obtained via unfolding and are embedded in the universal cover of the triply periodic minimal surfaces, we call them Utilings. In this section we describe how to determine a unique representative for each distinct Utiling. These tilings are compatible with the surface covering maps, and so project onto the P, D and G surfaces giving Etilings. Finally, we also consider tilings of the tritorus, or Otilings, obtained from these Utilings.
domains, and unfolded them into corresponding tritorus charts. The result is 14 lists of tilings and their unfolded D symbols augmented with `cuts' that determine their embedding in the5.1. Utilings
In §4 we described how to generate distinct embeddings of D symbols within each kaleidoscopic of . Each Utiling is represented by a D symbol augmented with cuts and its precursor D symbol. This extra information defines the embedding of the tiling in the domain. We determine which Utilings generate equivalent surface reticulations as follows.
Recall from the introduction to §4 that two tilings of a TPMS are equivalent if their chamber systems are related via a symmetry of the surface. Thus, two Utilings are distinct if they have distinct D symbols, or if they are distinct embeddings of the same D symbol. The first situation is easy to test using combinatorial tiling algorithms. The second requires the additional information about the embedding of the Utilings in the T either from the cuts, or from an understanding of the embedding and unfolding process. By their construction, the Utilings generated within a particular chart are distinct. However, two Utilings defined via different charts may have the same D symbol and equivalent `cuts'. This occurs when the two precursor D symbols and their embeddings are related by symmetry raising or lowering within the lattice (see Fig. 6). An example is shown in Fig. 18, where we see a vertex1transitive tiling with symmetry , and two different symmetry lowerings to vertex2transitive tilings in the subgroups and . The three Utilings have isomorphic D symbols and equivalent embeddings.
We next consider the union of the 14 lists of Utilings, determine equivalence classes by comparing the D symbols as described above, and find that there are 6079 distinct Utilings. We assign them names of the form UQCn, where `Q' and `C' stand for cubic and Coxeter. The running index n is determined by ranking the distinct Utilings by their D symbols (cf. §3). If two distinct Utilings have the same D symbol, as for the example of Fig. 17, they are listed in arbitrary order. In Table 6, we give the number of Utilings found within each kaleidoscopic We count each equivalence class of Utilings just once, in the of the precursor tiling that has the highest symmetry.

5.2. ETilings
We now map each Utiling from the hyperbolic plane onto the P, D and G periodic minimal surfaces, forming Etilings, where `E' refers to Epinet. Each distinct Utiling UQCn maps to exactly three Etilings and we give these the directly corresponding names, EPCn, EDCn and EGCn; see Fig. 19. The Etilings are the bridge between twodimensional and threedimensional structure that our enumeration scheme is built around. They are twodimensional because they are defined almost entirely by the Utiling; threedimensional structure is given by the covering map and the geometry of the particular TPMS. More specifically, we use the covering map action on the hyperbolic translations, (as summarized in Table 2 of §2) to map the cuts attached to the D symbol into Euclidean translations. The Etiling, then, is represented by the same pair of tiling and D symbol as the Utiling, but the cuts are written as Euclidean translations, rather than hyperbolic ones.
The spacegroup symmetries of the Etilings can be found by considering the covering map action in more detail. Recall from §2 that the covering map induces a , where S is the of the nonoriented surface. The map is an explicit relationship between the hyperbolic reflections R_{1}, R_{2} and R_{3}, and Euclidean isometries from S. If we now restrict the action of to one of the kaleidoscopic subgroups, , then the image is a of the Euclidean S, giving an elegant correspondence between twodimensional hyperbolic (discrete) groups and threedimensional Euclidean space groups. This procedure will be explored in detail in a future publication. We give the surface space groups that correspond to each kaleidoscopic in Table 6.
5.3. OTilings
Our hyperbolic tilings also define tilings of the tritorus, which we call Otilings. These tilings are formed by wrapping Utilings onto the tritorus, whose structure is explained in §4.1.3. We distinguish Otilings by the distinct D symbols. In our current enumeration we find 5912 Otilings. This is less than the number of distinct Utilings (6079) because of the examples discussed at the end of §4.2.2.
6. Nets from tilings
In this section we consider the nets derived from the vertices and edges of tilings. Various definitions of the word `net' can be found in the literature; we use the term here to denote a graph embedded in a metric space. Specifically, we investigate nets embedded in the hyperbolic plane, the tritorus and the periodic minimal surfaces, called hnets, onets and enets, respectively. Finally, we discard possible edge entanglements present in the enets to study their topology and maximalsymmetry embedding in , and call the resulting structures snets. hnets remain embedded in their parent space – the hyperbolic plane – while o, e and snets are embedded in threedimensional Euclidean space.
There is a simple onets and enets to include all graph embeddings related by deformations that do not involve edges crossing. Accordingly, one of our goals is to determine equivalence classes of nets under ambient isotopy: i.e., two nets are equivalent if their embedding space can be continuously deformed in such a way as to map one onto the other without allowing edges to pass through each other. This leads to the distinction between variously entangled versions of a net as inequivalent o or enets, akin to the distinction between inequivalent entanglements of a loop in space as different knots. Finally, the equivalence class of snets includes all nets with identical graph topology. As for hnets, there is a for snets that allows most distinct examples to be readily identified.
for hnets that allows different net topologies to be readily distinguished in the hyperbolic plane. Some care must be taken, however, in defining the equivalence class of threedimensional embedded nets, namely the enet, onet and snet categories. Trivial geometric deformations of a particular embedding are of little interest to us. Nontrivial deformations may include those that change the entanglement of edges within the graph, or the graph topology. We define equivalence classes for6.1. hNets
As illustrated in Fig. 20, many tilings can carry the same net topology. Since a Delaney–Dress symbol encodes both topology and symmetry, such tilings have different D symbols. Clearly then, the number of distinct hnet topologies will be less than the number of distinct tilings enumerated in Table 4 of §3.
We adopt the convention that an hnet should be represented by a tiling with maximal possible symmetry. The process of finding the highestsymmetry version of a given tiling is rather simple using combinatorial tiling theory – it amounts to computing the minimal image^{3} of the D symbol (DelgadoFriedrichs, 2003). Since minimalimage D symbols classify nets up to homeomorphism in the hyperbolic plane, they are a for hnets.
We obtain a list of 2451 distinct hnet topologies by applying the minimalimage algorithm to the 2706 . The results are presented in Table 7 and are organized by the orbifolds of the minimalimage D symbols. Although we start with D symbols from kaleidoscopic subgroups of , the resulting hnets have a broader range of symmetry types, for example , a hat orbifold with a single cone point and mirror boundary. Further, the hnet symmetry need not be one of the 131 subgroups of , for example, . This is because the minimal image of a D symbol may form a tiling whose symmetries are a of the symmetries of the initial tiling. This catalogue of hnets will allow us to compare reticulations across different classes of TPMSs.
tilings referred to in Table 4

6.2. eNets
We call the edge skeletons of the Etilings of the threeperiodic minimal surfaces enets, where `e' again refers to Epinet. Recall that distinct enets are characterized by embeddings in threedimensional space which cannot be deformed to each other without edge crossings or changes in the underlying graph topology. Thus, in particular, different enets may have identical topology. The situation is analogous to distinct knottings of a loop, characterized by ambient isotopy (Adams, 2004).
Since an enet inherits the embedding of an Etiling in the topologically complex TPMS, its edges may exhibit complex entanglements in threedimensional space induced by windings about surface channels; for example, enets can be selfcatenated (as in Fig. 24).
enets may also be multigraphs, with more than one edge linking a pair of vertices. These multigraphs can arise from two sources. The distinction between these two scenarios lies in the different types of edge cycles that emerge in Etilings: those that bound a tile and lie wholly in the surface, called `nullhomotopic rings', and those that do not bound a patch of the surface. The former cycles are found in the twodimensional universal cover, the latter are not. Therefore the first examples are caused by hnets that are themselves multigraphs; these are generated as follows. Our Delaney–Dress tiling enumeration allows tilings with adjacent tiles that share more than a single edge; examples are readily generated by gluings across mirror lines incident to vertices. An example of such a tiling is shown in Fig. 21(a). The duals of these tilings are necessarily multigraphs, with a pair of edges linking the vertices corresponding to multiedge sharing faces in the original tiling (Fig. 21b).
Alternatively, some hnets that are themselves simple graphs give rise to enet multigraphs as a result of the covering map action. The simplest examples occur when two edges lying on opposite sides of a channel join the same two vertices. The resulting enet therefore contains a double bond.
We label enets epcN, edcN and egcN, where `e' stands for Epinet, `p', `d' and `g' denote one of the three TPMSs, and `c' refers to the Coxeter orbifold family. The value of the index, N, depends on the net embedding in , up to edge entanglement. Consider, for example, an enet generated from the Etiling EDCN. The name of this enet is determined as follows. First, we compare the candidate enet with the accumulated list of distinct enets generated from simpler tilings, viz epcJ, egcJ and edcJ, where . If our candidate net is equivalent to a previous enet, say , where , the tiling does not result in a new enet. Formally, the enet name is remapped to and the label edcN is unused. If the candidate is distinct from the list of `lower' enets, it is labelled by the new name edcN, where N is exactly the same index as that of the corresponding E (and U) tiling. Thus the maximum index number used is less than or equal to the total number of enets, due to unfilled labels wherever the Etiling skeleton is ambient isotopic to a previous enet. This schema allows the indexing of enets to remain in register with the indexing for E and Utilings, thereby explicitly retaining links to precursor tilings of enets.
An uncharacteristically degenerate example is given by regular and semiregular (4, 6) and (6, 6) tilings of the P and D surfaces. The Utilings and Etilings are illustrated in Fig. 22. Those four cases, derived from four distinct tilings (see Table 8), generate equivalent enets whose simplest embedding (identical to the snet, sqc947) is illustrated in Fig. 23. Since the lowest Etiling index found among these examples is EDC1, all four enets map to edc1. This example demonstrates the phenomenon of collapse of distinct U and Etilings to a single enet.

A major issue remains unresolved here, which we hope to explore in detail later. In contrast to h and snets, we have yet to produce an algorithm for determining equivalence or otherwise of enets. The problem is related to that of identification of distinct knots, a central topic of knot theory. We suspect that our problem can be approached by forming a canonical embedding for enets. For now, we can only determine `by inspection' whether enets are ambient isotopic. Equivalent enets are also necessarily topologically equivalent; we therefore first check for graph isomorphisms by comparing the enets' topological structure. In many cases, that can be done by forming a barycentric embedding, giving an snet, described in the next section. If the snets differ, the enets are necessarily distinct. If they are equivalent, we then determine – currently by eye – whether the enet embeddings are ambient isotopic to the canonical embedding of their common snet. If they are, both enets are equivalent. If one differs, they are inequivalent; if both differ we must look further to establish whether they can be mutually deformed without edge crossings to a common intermediate embedding. Some nets cannot be analysed via this method due to vertex `collisions' that occur in the barycentric embedding. In those cases, comparison with an snet is impossible. We must then resort to other measures to compare net topologies, including coordination sequences (Brunner & Laves, 1971) and comparison of subgraphs, implemented in the Topos package (Blatov, 2006).
We close this section with an example, shown in Fig. 24, that illustrates many of the issues discussed here. The two enets shown in Fig. 24(a) are generated by tilings of the gyroid (that lie outside the Coxter class) and exhibit self catenation. The enets cannot be compared via their snets, due to vertex collisions in their respective barycentric embeddings. However, they share identical coordination sequences (to shell 24) and certain finite subgraphs strongly pointing to their topological equivalence. Their embeddings are not ambient isotopic, evidenced, for example, by the distinct link types highlighted in Fig. 24(b). These are the (2, 4) and (2, 6) torus links (Adams, 2004), distinguished by their distinct threadings through each other. These are thus distinct enets, despite their topological equivalence.
6.3. sNets
We have noted above that different Etilings can define the same enet (in Fig. 22) and that different enets may share a common periodic net topology (Fig. 24). In general there is no effective algorithm to determine when two crystal nets are topologically identical. However, for a large class of `collisionfree' Euclidean nets, the Systre algorithm furnishes a for the quotient graph of a net (DelgadoFriedrichs & O'Keeffe, 2003). We call these canonical forms snets.
The Systre algorithm [available as part of the GAVROG package (DelgadoFriedrichs, 2006)] is based on finding the barycentric embedding (or equilibrium placement) for the net: each vertex is located at the centre of mass of its edgeadjacent neighbours. This produces the highestsymmetry embedding of a periodic net, and permits the computation of the associated when there are no collisions. A collision occurs when two vertices have the same coordinates in the equilibrium placement (configurations that generate collisions are described later in this section). The final output of the algorithm is a systre key – a canonical representation of the labelled quotient graph for the periodic net that uses its smallest translational The systre key provides a unique signature for topologically isomorphic crystal nets (DelgadoFriedrichs & O'Keeffe, 2003).
We derive systre keys and barycentric embeddings for our enets as a way to identify the distinct periodic net topologies generated by our enumeration. First, a quotient graph for the enet is derived directly from an Etiling via its cut D symbol. The enet topology is adjusted slightly by coalescing any multigraph edges. We then compute the equilibrium placement and systre key for the net where possible. Most of the nets enumerated via our hyperbolic tiling approach are collisionfree: from 18 285 Etilings (6095 Utilings on each of the P, G and D surfaces) we found 2247 nets with equilibrium collisions. Thus, the systre key is an effective, although occasionally inadequate, tool for identifying the set of distinct network topologies derived from our surface reticulations. Indeed, most snets have a single enet antecedent: of the 14 532 distinct systre keys, only 954 are obtained in more than one way.
The snets are ranked by their systre key and assigned distinct names of the form sqcN, where s, q and c have the same meanings as above, and the index N is derived from the systrekey ranking. This ordering is intuitively appealing, with a gradual increase in net `complexity' (to first order, the number of vertices and edges in a primitive unit cell) as the index rises. For example, the firstranked net according to this scheme, sqc1, is the standard net of the simple cubic lattice; the diamond net (whose enet antecedent is a multigraph) is sqc6 etc.
We finish this section with a discussion of some problems that arise when analysing nets using barycentric embeddings. A minor cautionary example – a degree6 (4, 6) tiling with orbifold (UQC42) derived from a reticulation of the D surface, forming the (degree6) enet edc42 – is shown in Fig. 25. Here, the barycentric embedding of the net is free of vertex collisions but a pair of edges intersect, inducing apparent vertices of degree four: a situation that is due solely to the edge embedding in . Barycentric embedding of edc42 gives the snet sqc900, which contains virtual degree4 edge crossings in addition to the degree6 vertices. These edge crossings in the snet are problematic when determining the ambient isotopy classes of associated enets.
Vertex collisions are even more problematic, and can involve either small subsets of the net or triply periodic subgraphs. An example of the former are tilings that contain kites with a single interior edge like those in Fig. 21. The barycentric embedding of an associated enet collapses the interior edge of the kite and places its two vertices at the same point. A more serious collapse is induced by `ladder' graphs: examples composed of identical periodic nets linked by runglike edges. In those cases, the identical components collapse onto themselves entirely. An example is the graph illustrated in two distinct entangled forms in Fig. 24. The graph topology of both nets is a ladder graph built from a pair of regular degree3 triply periodic graphs. Each component of the ladder is a graph known to chemists as the srs graph (O'Keeffe, 2008). As mentioned in §6.2 we must resort to other topological signatures such as coordination sequences to characterize nets that contain vertex collisions.
6.4. oNets
Our enumeration of tilings of the tritorus, known as Otilings, can be extended to produce embeddings of finite graphs in , just as Etilings induce enets. Recall from the introduction to this section that – like enets – onets are distinguished by their ambient isotopy class. Unlike the E–e map, a single Otiling can induce an infinite number of distinct onets, due to the flexibility in forming a tritorus in . The nuances of the construction of onets from Otilings are complex and remain to be explored in detail. Some discussion of the problem will be presented in a forthcoming paper which explores the construction of entangled polyhedral nets from tilings of the (genus1) torus, to which we direct the interested reader (Castle et al., 2009). In addition to the onetomany map in going from Otilings to onets, a manytoone collapse is also possible: onets derived from distinct Otilings may be equivalent, in the sense that one can be deformed into the other without any edge crossings. The O–o map is therefore manytomany. We have yet to explore the taxonomy of onets in detail, but flag their presence here due to their strong links to the other surface reticulations.
7. A worked example
The path described in this paper – from Delaney–Dress symbols and their embedding within subgroups of (Utilings), then to Etilings on the TPMS, giving enets and their canonical embeddings as snets – is one that traverses aspects of geometry, group theory and tiling theory. It is therefore rather tortuous to navigate. With the formalities in place, we offer a detailed worked example – starting with a D symbol, and finishing with the triply periodic Euclidean nets obtained from reticulations of the P, D and G surfaces – to illustrate the connections between structures generated by our enumeration procedure.
We begin with a sequence of hyperbolic tilings in Fig. 26 that show the progression from the fundamental tiletransitive tiling for , to a tile2transitive split and glued tiling, to its vertex2transitive dual (cf. §3). We adopt the vertex2transitive tiling as the starting point for our example. This tiling contains one vertex of degree 3 and one of degree 5, with four distinct tiles (three quadrilaterals and an octagon) arranged around each vertex to give the twodimensional Schläfli symbol (4.8.8), (4.4.4.8.8). The combinatorial description of this tiling requires eight chambers and is given by the Delaney–Dress symbol in Table 9 (cf. Appendix B). This symbol is given the label QC643 according to the algorithm explained at the end of §3. Since this symbol is minimal – i.e. it has the highest possible symmetry for a tiling with this topology – it defines an hnet, labelled hqc583 (cf. §6.1).

Our tiling can be embedded into two distinct subgroups of , as shown in Fig. 27. The next step is to unfold these tilings in their respective charts to obtain the corresponding Utilings: UQC1346 and UQC1345 (cf. §5.1). The translational domain for the example is shown in Fig. 28.
We pause for a moment to look for other D symbols from our enumeration that generate the same Utilings as our example. We find that UQC1346 is also the unfolding of a distinct D symbol, labelled QC1442, in the . Further, UQC1345 is generated by QC1442, but via an embedding in . The initial tiling (QC643, with symmetry ) is related to the tiling QC1442, of symmetry , by an additional symmetry that halves the area of the fundamental domain. Thus, QC1442 has the same hnet as QC643. If we now step sideways, we find that QC1442 has two distinct automorphic embeddings in the , giving two additional Utilings. And so we see that this hnet topology (hqc583) is in fact associated with two tilings, five embeddings in four kaleidoscopic subgroups and three Utilings. These relationships are illustrated in Fig. 29.
We now return to the example (UQC1346) and the periodic nets generated by its projection onto the P, D and G surfaces (cf. §6). First look at the periodic net as it sits in the hyperbolic plane. The net carried by UQC1346 has 16 translationally distinct vertices and 32 distinct edges, as shown in Fig. 28. With the vertex labels marked as shown in this figure, we build the labelled quotientgraph description of the net given in Table 10. Edges between vertices that lie in different copies of the translational are labelled by the relative translation between the cells. We map this hyperbolic periodic net directly onto three enets (one on each TPMS) using the covering map actions defined in Table 2 of §2. The resulting triply periodic nets are also given in Table 10. Translational unit cells for the P and D surface tilings are shown in Fig. 30.

Lastly, we apply the Systre algorithm to the three enets to find a for the associated snets (cf. §6.3). Both the P and D enets reduce to crystal nets with only eight vertices per rather than the 16 found in the enets. This means that the crystal nets can be symmetrized to display an additional translational symmetry, absent in the surface embedding (in fact, this translation is one that swaps sides of the surface). In contrast, the snet derived from the enet embedded in the G surface retains the full 16 vertices (because the G surface does not have an extra translational symmetry that swaps sides of the surface). Crystallographic descriptions of the resulting snets are given in Tables 11–13 and unitcell images are shown in Figs. 31 and 32. Note that each snet has two symmetrically distinct vertices and four distinct edges – an equivalent multiplicity to that of the tiling.



8. Future directions
The work presented here has focused on tilings and surface reticulations derived from the kaleidoscopic subgroups that are compatible with the genus3 translational unit cells of the P, D and G minimal surfaces. There are many directions to extend our enumeration, including other subgroups compatible with the P, D and G surfaces; other triply periodic minimal surfaces; and further generalizations discussed below.
In the near future we intend to study nets derived from nonkaleidoscopic subgroups compatible with the P, D and G surfaces. These orbifold families include mixed reflection–rotation examples that we call `hat' orbifolds, and pure rotational `stellate' orbifolds (Hyde et al., 2009). Given that there are 29 and 21 distinct subgroups of with hat and stellate orbifolds, respectively, the profusion of results is expected to be significant. Other orbifold classes, including the nonorientable `projective' examples, may be explored at a later date. We also plan to extend the project to explore nets generated as reticulations of other TPMSs, specifically the remaining genus3 TPMSs: the hexagonal H and tetragonal CLP surfaces. We also plan to investigate this process on the genus4 cubic IWP surface. The complications associated with extending to genus4 examples are likely to be outweighed by the novelty of the examples. We have already determined the hyperbolic crystallography and covering maps for these surfaces and derived the relevant compatible orbifolds (Robins et al., 2004b; Robins, 2006).
A parallel effort will be directed at generalizing the remarkably powerful Delany–Dress apparatus to allow enumeration of other tilings that are commonplace in the hyperbolic plane but which have no analogue in the Euclidean plane. Such tilings contain infinitesided hyperbolic polygons arranged in ribbon or treeshaped patterns which we propose to call `free tilings'. Specific examples that are commensurate with the P, D and G surfaces have been shown to lead to multiple intergrown nets (Hyde & Oguey, 2000; Hyde et al., 2003); other examples are summarized online (Ramsden et al., 2004). We intend to explore free tilings using an extension to Delaney–Dress tiling theory and thus achieve a systematic enumeration of complex net intergrowths.
In addition to the triply periodic structures described here, our techniques readily adapt to the enumeration of finite (molecular) nets via reticulations of compact surfaces. We have already mentioned these Otilings and onets in §§5.3 and 6.4. Specific embeddings of these nets are a rich topic, barely explored to date. By defining a covering map from the hyperbolic plane onto an explicit embedding of the tritorus in Euclidean space, we can use the structure of the surface reticulation to define an embedding of the onet. Generic examples will be knotted, linked and ravelled, governed by the wealth of possible cycle homotopies on the tritorus. A sample of the possibilities can be seen in a study of knotted toroidal polyhedra, generated via reticulations of a torus (Hyde & SchröderTurk, 2007; Castle et al., 2009), as well as a recent exploration of ravelled graphs (Castle et al., 2008).
9. Discussion of results
As discussed in §6, our approach enumerates a range of nets from infinite symmetric twodimensional hyperbolic hnets and infinite triply periodic Euclidean e and snets, to finite onets.
The Euclidean snets formed from lowtransitivity tilings of the hyperbolic plane afford an interesting variety of canonical, symmetric embeddings of distinct network topologies in three dimensions. These data are directly comparable with complementary enumeration schemes pursued by (principally) structural chemists, including O'Keeffe's Reticular Chemistry Structural Resource (RCSR) database of threedimensional nets and tilings (O'Keeffe, 2008; O'Keeffe et al., 2008) and the Hypothetical Zeolite database of Treacy and colleagues (Foster & Treacy, 2008). These collections comprise nets in threedimensional Euclidean space constructed using threedimensional tilings and symmetry principles. We note that our 14 532 embedded snets derived from tilings of Coxeter orbifolds include circa 150 examples found in the RCSR database, which contains more than 1500 distinct structures. Among those common examples are 14 known zeolite structures. Therefore, this first pass has already furnished around 10% of nets considered to be of relevance to reticular chemistry. More than 200 examples of our snets also appear in the Hypothetical Zeolite database (Foster, 2008).
Many more structures of actual and/or potential chemical interest are certain to emerge by moving beyond Coxeter orbifolds and by reticulating noncubic surfaces. However, we hasten to point out that our motivation for this work goes beyond enumeration of chemically interesting patterns; rather we are primarily interested in exploring the variety of nets that emerge without imposition of chemical constraints. Evidently, we are forced to impose filters to avoid the inevitable combinatorial explosion associated with such a search; we have opted to filter the nets according to their twodimensional hyperbolic symmetries.
Our approach is governed by the simplicity of tiling enumeration in two dimensions as opposed to three, along with the suspicion that symmetric patterns in twodimensional hyperbolic space yield symmetric patterns in threedimensional space. The latter feature has been (and will be) discussed elsewhere; some statistics on the variety of threedimensional symmetries and vertex transitivity of snets can be read in Hyde et al. (2006). Our goal, however, has not been to offer a circuitous route to what is already enumerated elsewhere. Rather, we hope that this route generates new examples not readily deduced by more conventional threedimensional approaches.
We find, for example, a vertex2transitive (12, 4) tiling generated in the orbifold within the FSGG class of tilings. Projection of this via the Utiling UQC104, gives an enet, egc104, whose barycentric form is a rhombohedral () vertex1transitive (uninodal) snet in threedimensional Euclidean space, sqc906, labelled usf by O'Keeffe (2008). This structure has been identified in molecular frameworks (Moulton et al., 2003), yet threedimensional tiling theory cannot find a `proper' tiling (Blatov et al., 2007) for this pattern.
tiling onto the gyroid,A second example is the sphere packing 3/10/h4 (Sowa & Koch, 2006), named wiw by O'Keeffe (2008). This structure too emerges as an snet, sqc3054 via the degree3 FSGG tiling with symmetry illustrated in Fig. 33. That tiling unfolds to the Utiling UQC262, which maps onto the gyroid to form an Etiling, whose enet, egc262, is topologically equivalent to sqc3054, also illustrated in Fig. 33. The threedimensional tiling algorithm TOPOS (Blatov, 2006) fails to find a simple tile for this structure (Blatov, 2007), due to the threading of short rings in the net by other edges. This threading precludes the possibility of those rings spanning faces of a simple threedimensional tile.
We expect the most powerful aspect of our enumeration technique to emerge over time: namely the possibility of finding distinctly tangled embeddings of topologically isomorphic nets. It is that possibility that has encouraged us to retain enets as a distinct class, since their edges wind according to the tiling of the TPMS, in contrast to barycentric snet embeddings. The latter nets relax to a canonical embedding that may lose the knottedness and generic entanglements of the original enet.
The complete results of our enumeration are collated in an online database, accessible at https://epinet.anu.edu.au , which we urge interested readers to explore at their leisure. Ultimately, we expect this growing database of nets to provide a substantial foundation for a range of investigations into the physical features of nets, extending current work on percolation, transport and elastic responses (Gibson & Ashby, 1997; Roberts & Garboczi, 2002; Durand, 2005; Durand & Weaire, 2004). This suite of examples will allow detailed exploration of possible correlations between physical, topological and geometric features of crystal nets.
APPENDIX A
Orbifolds
An orbifold encodes the symmetric properties of an infinitely repeating twodimensional pattern with the topological features of a compact, connected surface. The concept has its roots in the theory of discrete groups, but the name was coined by Thurston and his students by combining orbit and manifold, since each point of an orbifold represents the entire orbit of a point under the group of symmetry operations of the pattern. Alternatively, one can invert that process and generate a symmetric pattern in the universal cover of the orbifold by `rolling' the inked orbifold in all possible directions on the relevant twodimensional homogeneous space – the sphere, , Euclidean, , or hyperbolic, , planes – printing the pattern as it goes. Technically, the orbifold is the quotient space of the pattern by its symmetry group, and the pattern is the universal cover of the orbifold.
Twodimensional symmetries include rotation about a point, reflection in a mirror line, translation and glide reflection. In the orbifold a reflection produces a boundary component, a rotation induces a cone point, while translations and glides that do not arise from other symmetries of the pattern are encoded by global topological features such as rings, handles and crosscaps. These relationships are described in more detail below.
A cone point is just what its name suggests. Rolling a cone about its apex will generate a pattern with rotational symmetry at that point. For the pattern to repeat exactly, the angle swept out by the cone must be an integer subdivision of 2π. An orbifold containing just three cone points looks like a samosa, or triangular pillow. If the only orbifold features are two cone points, then each must be of the same order, as they represent opposite poles of a spherical pattern. There are no compact orbifolds consisting of a single cone point.
Now suppose the orbifold has a puncture, and therefore a boundary edge. When rolling the orbifold surface over its universal cover, in order to continue beyond the edge the orbifold side in contact with the covering space must flip so the pattern is locally reversed. Boundaries therefore encode mirror reflections. A boundary must be a closed loop, which may contain corners. A corner in the orbifold is formed by two mirror lines meeting at an angle less than π. As with cone points, for the pattern to unwrap and meet itself exactly, only corner angles that are integer subdivisions of π are allowed. Note that cone and corner points are local geometric features that distinguish orbifolds from true manifolds.
Translational symmetries may be induced by combinations of reflections or rotations. Those translations that are not induced by other symmetries are encoded by a nontrivial loop in the orbifold such as formed by a ribbon with its ends glued together, or around the two axes of a torus. For example, a finite cylinder with two boundary loops can be rolled across the Euclidean plane mapping out a translationally periodic pattern in one direction normal to its axis. Its boundaries represent mirror lines so extension in the other direction of the Euclidean plane is induced by parallel reflection lines.
A glide reflection is the (irreducible) composition of a translation with a reflection, encoded within an orbifold by a nonorientable surface feature such as a Möbius strip or crosscap. A crosscap has the topology of the real projective plane, and can be modelled by a disc with opposite points on the boundary identified. For example, one traversal along the centre of a Möbius strip brings you to the same point on the opposite side of the ribbon (the flipped image of the initial pattern); a second traversal is needed to come back to the starting point on the original side. This doubled traversal is identical to the translation induced by applying a glide reflection twice.
We can encode independent symmetry operations of any twodimensional pattern on the sphere, Euclidean and hyperbolic planes by distinct surface features of its orbifold; namely boundaries, cone and corner points, crosscaps and handles. This result depends on the central theorem of 2manifold topology that every closed, compact, twodimensional manifold is topologically equivalent to either the sphere, a sphere with n handles attached, or a sphere with n crosscaps.
A simple notation for orbifolds is due to John Conway (Conway, 1992; Conway & Huson, 2002). From it we can reconstruct the orbifold surface topology and all its associated symmetry features. The notation is unique up to certain rearrangements of elements. The global topological features are the handles, crosscaps and boundaries, and are denoted by the symbols , × and . A handle transforms to two crosscaps in the presence of another crosscap (). Thus, by convention, handles and crosscaps are quarantined and an orbifold symbol will either be prefixed by 's or suffixed by ×'s. The nonmanifold features (cone and corner points) have specific angles that are represented by their integer fractions of 2π and π, respectively.
As there is no inherent ordering to nonboundary points of a manifold, the cone points may be listed in any order. Thus, the orbifold symbol 457 is equivalent to 754 or even 475 – in fact any permutation is valid. We typically use lexical ordering to make a nice but this is not intrinsic to the notation. Corner points, however, have a distinct ordering based on their sequence around a boundary component. A mirror string is given by a single , representing the boundary component, followed by a list of its corner points (if any). Here the order matters, is different to . However, because there is no intrinsic start or end to a mirror boundary, any cyclic permutation is equivalent, i.e. and represent the same orbifold as . Also, because a punctured surface has no intrinsic inside or outside, one can reverse the order of the corner points: i.e. is equivalent to . Entire mirror strings correspond to punctures and, like cone points, there is no way to intrinsically order these features. Therefore the order of the mirror strings does not matter.
The kaleidoscopic symmetry groups considered in this paper are generated by reflections only, and are therefore examples of Coxeter groups. Their orbifolds have the form of a single mirror string, .
We give the complete lexical specification of an orbifold symbol to summarize the above discussion: [some number of 's][cone points in any order][mirror strings in any order][some number of ×'s] written as
Note that any string that meets the above specification corresponds to a genuine orbifold except for those of the form c, c_{1} c_{2} where , , and where . Note also that we contract strings of identical numerals using power notation to improve readability: e.g. 22222 will be written 2^{5}.
Orbifolds have an associated scalar value, computable from the symbol elements, called the curvature index (also called the cost or characteristic). For kaleidoscopic groups it is
The general formula is given in Conway & Huson (2002). Its value determines which of the three twodimensional geometries the symmetry pattern must belong to: positive implies a spherical symmetry group, zero implies and negative implies . In the case of spherical or hyperbolic geometry, the curvature also quantifies the area of a single copy of the orbifold in the relevant plane of unit Gaussian curvature. Euclidean space is unique in that shapes may be scaled arbitrarily whilst preserving angles, so a Euclidean orbifold has no associated area. The majority of all twodimensional orbifolds are hyperbolic, thus the variety of hyperbolic twodimensional tilings far exceeds that of Euclidean or spherical space.
APPENDIX B
Delaney–Dress symbols
We discuss and illustrate tilings of the hyperbolic plane here, see Fig. 34 for example, but the concepts apply to the sphere and Euclidean plane with only minor changes, and the underlying theory generalizes to higherdimensional spaces.
The following four conditions form the definition of a tiling of the hyperbolic plane:
(1) The tiles are closed topological discs.
(2) Tiles intersect only along their boundaries. The intersection of two tiles defines an edge, the intersection of three or more tiles defines a vertex.
(3) The tiles are uniformly bounded in size.
(4) The tiles cover the whole hyperbolic plane.
To describe a tiling pattern we subdivide the tiles into triangles, called flags or chambers, and then record the neighbour relations of the different symmetry classes of these chambers. To generate a chamber system from a tiling, we make a barycentric subdivision by placing a 2vertex in the centre of each tile, a 1vertex at the midpoint of each edge and a 0vertex at each tiling vertex, then form 012triangles within each tile. The edges of the chambers are also labelled 0, 1 and 2edges according to the type of vertex they face. The neighbour relations are formally described by the action of three maps, σ_{0}, σ_{1} and σ_{2}, that map each chamber to its neighbour across the corresponding edge (or equivalently, opposite the corresponding vertex). For obvious geometric reasons, these maps are involutions (they are their own inverse).
The topology of the tiling is encoded by describing what happens on repeated application of pairs of the neighbour maps. There are three orbits to consider:
(1) The (σ_{0}σ_{1}) orbit maps around a 2vertex, and so visits the chambers in a single tile. If the tile has r edges, then is the identity map on each chamber of the tile. The index r is also called m_{01}.
(2) The (σ_{1}σ_{2}) orbit maps around a 0vertex, so walks around the chambers that meet at a vertex of the tiling. If the vertex has degree p then is the identity for each chamber incident at that vertex. The index p is also called m_{12}.
(3) Finally, (σ_{2}σ_{0}) maps around a 1vertex, i.e. an edge of the tiling. Since exactly four chambers meet at a 1vertex, we have that is the identity for every chamber.
The chamber system described so far is an infinite complex because infinitely many bounded tiles are needed to cover the hyperbolic plane. We obtain a finite description of the tiling by forming equivalence classes of chambers under the action of the symmetry group. If the group has a compact orbifold, then there will be a finite number of chamber classes. The σ_{i} maps preserve these symmetry classes, so only a finite number of neighbour relations need to be recorded. These chamber classes, their neighbour maps, and the topological indices r and p defined above are all the information needed to form the Delaney–Dress symbol.
We illustrate the above definitions with an example, shown in Fig. 34. This hyperbolic tiling is built from two types of pentagonal tile, with vertices of degree 3, 4 and 8, and has symmetry . On the upper left of Fig. 34 each pentagonal tile is shown subdivided into triangular chambers. Labels are assigned so that symmetrically equivalent chambers have the same letter. There are ten symmetrically distinct chambers and these cover a single copy of the orbifold (shown upper right). The chamber vertices are labelled 0, 1, 2, according to whether they lie on a tile vertex, edge or centre, respectively. The neighbour maps σ_{i} that encode the adjacency of chambers opposite a vertex of type i are also shown in Fig. 34, upper right.
The complete set of adjacency relations for the chambers and the topological indices may be given in tabular form, see Table 14, or in two visual formats shown in Fig. 34. At the lower right of Fig. 34 the D symbol is shown as a graph where the nodes represent each chamber class and coloured edges denote the σ_{i} involutions. The topological indices (r, p) for each chamber are also attached to each node.

A more concise depiction is a crankshaft diagram shown in Fig. 34, lower left, another elegant notation due to John Conway. In this representation, each chamber is a broken horizontal line, and the adjacencies between chambers are given by vertical connections in the appropriate σ_{i} column (selfadjacency is indicated by an open endpoint). The topology of the σ_{i}σ_{j} chamber orbits is immediately apparent from this crankshaft diagram. The σ_{0}σ_{1} tile orbits are represented by connected components in the left part of the crankshaft, and the σ_{1}σ_{2} vertex orbits on the right. The σ_{0}σ_{2} edge orbits are only implicitly represented in this diagram. Each symmetrically distinct tile and vertex is defined by a single connected component of the crankshaft, so the corresponding orbit numbers (r, p) are listed only once. The transitivity is visually clear – in our tile2transitive example there are two connected σ_{0}σ_{1} components, while the four different orbit components in the σ_{1}σ_{2} column tell us the tiling is vertex4transitive. A further advantage of the crankshaft diagram is that the dual tiling is obtained simply by reflecting the diagram about the central axis (in our twodimensional case, exchanging the σ_{0} and σ_{2} columns).
The power of Delaney–Dress symbols is that any two tilings with identical topology and symmetry will have isomorphic symbols, and the tiling can be completely reconstructed from this finite amount of information. In addition, the orbifold symbol and curvature index can be computed directly from the Delaney–Dress symbol; see DelgadoFriedrichs (2003) for further details.
APPENDIX C
The chart
Here we give the complete description of the chart that forms the basis from which we derive all the unfolded D symbols; see §4 for further details.
The full labelling of triangles and their neighbours in the chart is given in Table 15. This list is generated from the kbmag word enumeration with shortlex ordering. The generator order here is R_{1}, R_{3}, R_{2}, since this gave a slightly more balanced region in the hyperbolic plane than the ordering R_{1}, R_{2}, R_{3}. The domain covered by the elements given here is not the semiregular dodecagon illustrated in Fig. 10, but is equivalent to it. The triangles with the 96 words from Table 15 define an irregular shape illustrated in Fig. 35. The underlying group and triangulation structure on the tritorus is exactly the same as that given by the dodecagon with opposite sides glued.

Footnotes
^{1}The Schläfli symbol (p, q) denotes a tiling by pgons meeting at vertices of degree q; the symbols {p, q} denote tilings by regular polygons.
^{2}There is an important exception to this when considering chiral tilings projected onto the G surface; see Robins et al. (2005) for details. (in ) is not an issue here, since we are working with kaleidoscopic symmetry groups.
^{3}Here the term `minimal' refers to the length and ordering of the symbol, and should not be confused with minimal surfaces.
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