On Tilings and Patterns on Hyperbolic Surfaces and Their Relation ...

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On Tilings and Patterns on Hyperbolic Surfaces and Their Relation to Structural Chemistry Reinhard Nesper*[a] and Stefano Leoni[b] Dedicated to Professor Dr. Hans-Georg von Schnering on the occasion of his 70th birthday

Hyperbolic Periodic Nodal Surfaces (PNSs) have been investigated with respect to surface modulations. The resulting patterns and tilings are beautiful examples of texture information on hyperbolic objects. The construction principle, based on short Fourier series, allows for a strict symmetry control of the generated pattern by choosing appropriate sets of structure factors for information coding. Furthermore, a tailor-made design of complexity is achieved based on a proper choice of the structure factor moduli. In some cases, the resulting patterns are instantaneously transferred into hyperbolic framework information just by our visual

perception. In most cases, such correlations can be systematically worked out by utilizing crystallographic and chemical expertise, as shown in this contribution. The presented approach is much simpler than most attempts to generate three-dimensional frameworks because it is subject to boundary conditions like pseudo-twodimensional topology, given base topology, and symmetry control.

ªAs simple as possible but not simplerº ± A. Einstein

concerned with the elucidation of the organization of chemical bonding in three-dimensional space and, with the understanding of general principles of structure formation,[8, 10±14] even reconstructive phase transitions were approached in a novel way based on PNS.[13, 15] Very recently, Andersson and Jacob presented an elegant mathematical scaffolding based on exponential functions.[12] Their approach allows representation of single shapes and forms, however complex, in a single function, where more complex descriptors are achieved by addition of several simpler forms. The use of periodic hyperbolic descriptors, like PNS, as a tool for understanding chemical structures very frequently led to the discovery of hyperbolic networks ªhiddenº in the complex three-dimensional arrangements of atoms and molecules. Networks of atoms, or sequences of molecules, follow more or less closely a central hyperbolic surface, whereby they populate one or both labyrinths partitioned by the PNSs. The topology of the labyrinths, the so-called labyrinth nets, can be illustrated with two periodic equisurfaces (PEs) wrapped around the central surface (PNS).[16] We present here the use of PNS as a basic form that can be modulated systematically to explore tilings on hyperbolic

Introduction A recent article on ªThe Art of Elegant Tilingº in Scientific American stated that the most beautiful overlap of art and mathematics is the concept of symmetry[1] and an extremely convincing expression of this interaction are tilings. Over many millennia artists were fascinated by tilings and worked out periodic motifs to decorate the surfaces of nearly all sorts of entities. Although practically all 17 symmetrically different periodic tilings of the plane were found in this way, it was not before 1891 that Fedorov gave the proof of the limitedness of the set.[2] If periodicity is not a requirement, there is in principle an infinite number of plane tilings. The Penrose tiling[3] became famous in conjunction with the descriptions of quasicrystals,[4] which can be understood as aperiodic projections of higherdimensional periodic lattices[5]Ðup to dimension six (R6). In three dimensions there are 230 different symmetry groups (space groups, SGs) of periodic character, which can be used to form tilings of R3.[6] This in fact is the world of crystals. Necessarily, there must be a three-infinite number of aperiodic threedimensional arrangements, some of which may be periodic in a higher-dimensional space as the abovementioned quasicrystals. About 20 years ago, Andersson[7] started to describe crystal structures by the help of periodic minimal surfaces (PMSs). This concept was cast into a much more practical and powerful tool by deriving topologically equivalent periodic zero potential[8] and finally periodic nodal surfaces (P0PSs, PNSs) as structure descriptors.[9] Since then, much work has been invested in correlating such bicontinuous, curved hyperbolic PMSs, P0PSs, and PNSs with general crystal chemistry. The efforts have been CHEMPHYSCHEM 2001, 2, 413 ± 422

KEYWORDS: crystallography ´ hyperbolic frameworks ´ hyperbolic tilings ´ periodic nodal surfaces ´ solids

[a] Prof. Dr. R. Nesper Laboratory for Inorganic Chemistry ETH Zürich Universitätstrasse 6, 8092 Zürich (Switzerland) Fax: ( 41) 632-1149 E-mail: [email protected] [b] Dr. S. Leoni Max Planck Institute for Chemical Physics of Solids Nöthnitzer Strasse 40, 01187 Dresden (Germany) Supporting information for this article is available on the WWW under http://www.chemphyschem.com or from the author.

WILEY-VCH-Verlag GmbH, D-69451 Weinheim, 2001 1439-4235/01/02/07 $ 17.50+.50/0

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R. Nesper and S. Leoni surfaces in any of the 230 space groups. As the mathematical number of dimensions does not suffer from limitations imposed by the physical space, this procedure is not restricted to the latter, but may easily be expanded to any space Rn by constructing the corresponding Fourier polynomial. In addition, we will show how such tilings correlate to real crystal structures. As a space partitioner, PNSs are very often able to show the spatial segregation of different chemical forces in different spatial domains. So, for example, if a covalent network closely follows the course of a PNS, major ionic interactions in general develop across the surface. As structural relations in 3 D space are very difficult to perceive, our approach may become a novel strategy to support the intuitive understanding of symmetry relations in space.

Characteristics of the Hyperbolic Parent Surfaces One of the most effective methods for the generation of periodic hyperbolic surfaces is provided by short Fourier summations under strict symmetry control.[9] This provides a straightforward way to define the simplest topological descriptor of a space group[15] and a consecutive series of more and more complicated ones by the systematic addition of structure factors (SFs) with increasingly higher moduli (reciprocal Miller indices hkl). Any such summation is characteristic for a specific space group, dependent on a) the set of equivalent reciprocal vectors and the permutation of their phases and b) the way of combining different sets of structure factors. The latter means that individual SF sets, say SFa and SFb , may be defined to belong to the same or to different symmetry groups. In the latter case, the resulting surface shows the symmetry of a subgroup and the group ± subgroups relations can be traced in a rigorous and elegant way. Reciprocal space is strictly ordered, such that with increasing length of the vector h (hkl) or increasing Bragg angle q, respectively, increasing complexity is simultaneously contained. Thus, an ordering of PNS according to complexity is at hand. For cubic symmetry in a cell of unit length a, the relation to the lattice plane spacing D is given in Equation (1) and shown in Figure 1. The term l is the wavelength applied in a corresponding scattering experiment. q h2 k2 l2 1 l (1) D jhj 2 sin q a Figure 2 shows three PNSs which belong to the space groups Ia3Åd, Pn3Åm, and Im3Åm, respectively. These cubic surfaces, of lowest genus as discussed more fully later, have been termed Y**, D*, and P* (Y-shaped, diamandoid, and primitive; Figure 2 a ± c) according to the names of the lattice complexes[17] by which the enveloped skeletal graphs are defined. Due to their symmetry properties, they corre-

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Figure 1. Portion of reciprocal space.

spond to the PMSs G, D, and P. However, the SF sets can be assigned to distinct (for example, colored black and white) labyrinth subgroups I4132, Fd3Åm, and Pm3Åm (Figure 2). Because of the most simple symmetry coding in reciprocal space, real-space symmetry is generated from the SFs in the form of a physically informationless, pristine density space. However, from such density spaces, periodic, orientable, equidense surfaces (PESs) can be obtained. Among the PESs, the PNSs may bear symmetry elements that allow for a mapping of the two labyrinths on both sides of the PNS onto each other. In this case, the surface is said to be ªbalancedº. This holds for Y**, D*, and P*. Accordingly, their symmetry must correspond to the normalizers of the generating space groups, which are Ia3Åd, Pn3Åm, and Im3Åm.[18] That means that an uncolored, balanced PNS must transform according to the symmetry of the normalizer of the space group, in other words the PNS has a higher symmetry than all PESs. In Figure 5 a, the borderlines of the tiling on D* correspond to a set of twofold axes, allowing for an exchange of

Figure 2. a ± c) The PNS Y**, D*, and P* of symmetry Ia3Åd, Pn3Åm, and Im3Åm (space groups 230, 224, and 229). d ± f) The respective PNSs obtained from conformal higher moduli, I4132, Fd3Åm, and Pm3Åm (space groups 214, 227, and 221).

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Tilings and Patterns in Structural Chemistry the two sides of the surface. If the labyrinths are distinct, the space group of the surface is a black-and-white subgroup of the grey or uncolored surface. Figures 2 d ± f display PNSs generated from higher symmetry conformal moduli, namely (330, ÿ p/2), (333, 0), and (300, 0), respectively.

Generation of the Tilings on Periodic Equisurfaces We develop tilings on surfaces by using the mathematical scaffolding of PNSs, and we map two PNSs on top of each other: One is the fundamental topological form corresponding to a given symmetry, and the other a modulated derivative of the first. For this task, a short Fourier summation f(x) is used according to Equation (2). X j Sh j cos(2p(h ´ x) ÿ a(h)) (2) f(x) h

The term Sh is a geometric structure factor, h refers to a set of reciprocal space vectors given by the Laue symmetry of the space group being considered, and a is a set of phases defined by a(PTh) a(PTh) ÿ 2phTt,[19] in which P and t are the rotation and translation parts of the symmetry operations of the space group. To achieve a hyperbolic tessellation, a second short summation is required. This one contains the same SF set as for the fundamental PNS, plus at least a second set Sh' of higher order, Equation (3), weighted as to slightly modulate the shape of the fundamental surface. The Sh' are called modulating structure factors (MSFs). The tilings are then produced by the intersections of the fundamental and modulated PNSs. Of course, intersection lines or points of a tiling will satisfy both Equations (2) and (3) simultaneously. X j Sh j cos(2p(h x) ÿ a(h)) f(x) h

X

b j Sh' j cos(2p(h' ´ x) ÿ a(h'))

(3)

h'

The modulations presented are characterized by modulating structure factors that have shorter wavelengths than the fundamental ones, namely their h vectors are longer than those of the fundamental. With respect to the symmetry of the basic PNS, the MSF may be chosen in two different ways: a) They preserve the symmetry of the PNS they modulate, in which case they have to be characteristic for the same space group, or b) they reduce the symmetry to a well-defined subgroup. The question whether an MSF is or is not characteristic has been investigated for a large number of space groups.[20] We find that characteristic MSFs generate an unchanging tiling pattern, independent of variations of their amplitudes. In other words, in the latter case the common nodes of basic and modulated PNSs remain unchanged. Consequently, MSFs which are not characteristic for the symmetry of the basic PNS generate changes of the tiling with changing MSF amplitudes. In the examples given herein we chose j Sh j 1/b j Sh' j 10 j Sh' j . The effectiveness of the approach is illustrated in Figure 3, where we present some selected tilings of the plane generated CHEMPHYSCHEM 2001, 2, 413 ± 422

Figure 3. Tilings of the plane of symmetry a) p4mm, b) c2mm, c) p31m, and d) p6mm (space groups 11, 9, 15, and 17).

by this method. Figure 3 a ± d shows patterns of symmetry p4mm, c2mm, p31m, and p6mm, respectively. A fundamental PNS from a characteristic set of structure factors can be calculated for each of the symmetry groups we are considering. In the following, we will refer to a structure factor in the form SF (hkl, a), where hkl represent the set of vectors equivalent under a particular symmetry and a the phase of the first vector of the set. For all three surfaces we find, among many others, a fundamental tiling, a regular tiling constituted by patches of the same type which have the general multiplicity of the space group in question. Consequently, such patches are the fundamental patches of the PNS. They are also directly related to those of PMS and P0PS. They carry the necessary information to construct the surfaces upon application of symmetry elements. Other tilings may as well be regular in the respect that there is only one kind of patch by which the surface can be constructed but these patches are smaller or larger than the fundamental patch. We like to refer to the latter as semifundamental tilings. In addition, there are patterned surfaces which do not have a continuously linked network of tiles but of (partially) isolated patches. Both tiled and patterned surfaces can be very effectively used to derive hyperbolic chemical networks. The number of simple tesselations and patterned surfaces increases strongly from high to low symmetry groups because the number of symmetry constraints is released until finally only translational symmetry is left for the trigonal space group P1. However, it is important to point out that the continuous forms of PNS, which we base our investigations on, give rise to additional boundary conditions beyond just considering symmetry as a set of symmetry elements.

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Tilings on Y** Y** is generated from the SF set (110, ÿ p/2) in space group I4132 and its symmetry is Ia3Åd,[9] as shown in Figure 4. It is balanced and partitions space into two labyrinths, each of which is a three-connected network and enantiomorphic to the other, as inversions centers are embedded in the PNS Y**. In the language of lattice complexes,[17] the Y-shaped nodes of the labyrinths correspond to a point configuration of a Y* lattice complex, from which the name Y** is derived. If spheres are located on the Y* sites and are chosen such that they touch, an offene Kugellagerung, that is, a nondense but continuous spherical packing is formed.[21] A tiling of most general appearance results if the MSF set is (222, ÿ p/2), as shown in Figure 4 a. Consisting of patches of the same shape, it is a regular tiling. The MSF (222) is not characteristic for the group of the PNS and thus the inversion centers, marked on the surface by small yellow spheres, do map the patches onto each other while the color is changed. The group of this tiling is I4132. Consequently, for this two-color partitioning, tiles of the same color are equivalent in SG I4132, a direct subgroup of the group of Y** (Ia3Åd). Tiles of different color on the same side of the PNS are enantiomeric pairs. The tile centers belong to the Wyckoff set 48g, and thus this is the total number of tiles per unit cell, the same as the number of general positions of the SG I4132. Each set of enantiomers is a manifold of 24 per unit cell. (We conjecture that for all hyperbolic periodic surfaces with two chiralÐenantiomericÐtunnel graphs on each side, the fundamental set can only be formed by two sets of enantiomeric tiles of half the general multiplicity). The form of the tile can be described as a hyperbolic rhomboid. The two sets of opposite corners of a tile correspond to the Wyckoff positions 16a and 24d in SG 230 (yellow and green points in Figure 4 a), respectively. The corresponding network notation reads [4644], which means that there are six and four fourcornered patches meeting at the first and second network positions, respectively. A similar tiling, but with curved borderlines, can be achieved if MSF (330, ÿ p/2), instead of MSF (222, ÿ p/2), is used, as shown in Figure 4 b. The network notation is again [4644]. The squarelike patches of the pattern resulting from the application of SF (220, p) are corner-connected along the paths limited by position 24d. They are part of a partitioning also comprising six-ring loops (Figure 4 c), which relates to a known

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Figure 4. Tilings on Y**. The MSFs for the different tilings are a) (222, ÿ p/2), b) (330, ÿ p/2), c) (220, p), d) (211, 0), e) (321, 0), f) (332, 0), g) (431, 0), h) (521, 0), i) (532, 0), k) (611, 0), l) (541, 0), m) (631, 0), n) (543, 0), o) (552, 0), p) (633, 0), q) (651, 0), r) (653, 0), s) (655, 0).

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structure, namely that of the mineral analcime, and we will come Tilings on D* back to this in the next Section. If we analyze the SFs used so far with respect to the SG of the The D* or diamond surface is generated from the basic SF basic PNS Y**, we find that MSF (220, 0) is not characteristic of (111, p) or (111, 0) and has the symmetry of the normalizer of the Ia3Åd but instead of Im3Åm;[20] the same holds for MSF (330, 0), diamond group, Pn3Åm,[9] as shown in Figure 5. The symmetry of Å which is even extinct in Ia3d. MSF (222) as well is extinct and not the colored surface is the same as that of diamond, namely characteristic of Ia3Åd. Fd3Åm. As it was the case for the PNS Y**, it is also possible to find In the next series we will discuss only tilings and patterns of tilings on this surface which are composed only of tiles of the MSF that are characteristic for the group Y**. With MSF (211, 0), same shape (Figure 5 a). However, in this case there is no chirality only common points of intersection are found, which are the condition and thus all the tiles can be mapped onto each other inversion centers (Figure 4 d). The next characteristic ones, with without inversion or mirror. As there are 192 tiles, this is a increasing length h, are the MSFs shown in Figures 4 e ± 4 s. They fundamental tiling with the network notation [4644] (Figure 5 a). may have the allowed phases 0 or p without changing the The MSF (333, p or 0) has the same symmetry as the basic SF tilings. Clearly, there is the tendency of continuously increasing (111). The corners of the saddlelike patches belong to the complexity of the tilings with increasing h length. This allows us, Wyckoff sites 32e, 48f twice, and 96g, and are represented by to a certain extent, to determine the complexity of the tiling if yellow, blue, and red spheres, respectively. An expanded netonly a very limited number of theÐin principle infiniteÐset of MSF is chosen. As mentioned before, the latter series of patterns is independent of the choice of the amplitudes of the MSF, and the patterns strictly preserve the underlying SG symmetry. The restriction of preserving symmetry is a very limiting one, as one can deduce from observing the environmental structure of the 3Åm points (yellow spheres in Figure 4 a). The repetition of certain local pattern characteristics is related to orbits around such special symmetry points. Some of the tilings exhibit seemingly flowerlike or meandering structures, while others make for a more well-balanced tiling, in which the overall areas assigned to different colors are about the same. Except those tilings in Figures 4 a and 4 b, which we discussed already, the ones in Figures 4 h, 4 i, 4 o, and 4 q are especially noteworthy. Without any great knowledge of structural chemistry, they can be used to construct more or less homogeneous networks, that is, networks that are based on only one specific distance between contiguous network points. In more general terms, the principle difference that we identify here is that between tilings with a continuously connected border structure, like in Figure 4 a, 4 b, 4 g, 4 h, and 4 q, and those with locally isolated borders around the patches, respectively. It is important to note that the analysis of these pattern correlations in general turns out to be very useful for deriving continuous framework structures that follow the course of a PNS. They allow for an intimate mutual coding of symmetrical and topological requirements of a SG. For example, the pseudosixfold patterns around 3Åm (yellow spheres in Figure 4 a) may rotate with an axis perpendicular to the surface but induce an inverse rotation of the pseudofourfold patterns around 4Å points (green spheres in Figure 4 a). Any construction of a corresponding hyperbolic (chemical) framework has to obey these Figure 5. Tilings on a ± f) D* and g ± m) P*. The MSFs for the different tilings are a) (333, 0), relations and is thus quite bound in its geometrical b) (222, 0) (400, p), c) (444, 0), d) (555, 0), e) (800, 0), f) (12 00, 0), g) (111, 0), h) (321, p), possibilities. i) (200, 0), k) (400, 0), l) (222, 0), m) (430, p). CHEMPHYSCHEM 2001, 2, 413 ± 422

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R. Nesper and S. Leoni work notation thus reads [(46)32e (44)48f (44)96g]. The connectors between these positions belong to the set of twofold rotation axes (sites 24h and 24j) of the normalizer of the diamond group (Pn3Åm), which are symmetry elements of the uncolored (grey) surface. These axes generate the geometrical framework in which a corresponding PMS would form. MSF (222, 0) generates a fairly simple, unconnected pattern that is found in a similar form for MSF (400, p). Consequently, they can be combined to develop the averaged smoothed pattern shown in Figure 5 b. This pattern is very much related to the Y** (220, p), Figure 4 c, and both can be transferred into each other by a continuous transformation. This point will be briefly illustrated in the last Section. Moving on in the series of MSF with the indices hhh, the tilings in Figures 5 c and 5 d for MSFs (444, 0) and (555, 0) are yielded, respectively. The primer has four different tiles, three of which are centered by the corner sites of the tiles of Figure 5 a. Undoubtedly Figure 5 d looks more harmonic, probably because it constitutes only two different tiles. If we investigate modulations induced by h00-type MSFs, we find two unconnected patterns with (800, 0) and (12 00, 0), displayed in Figures 5 e and 5 f.

Tilings on P* The P* surface (Figure 2 c) is generated in SG Pm3Åm from the basic SF (100, 0). The colored surface has symmetry Pm3Åm; the grey surface transforms according to the normalizer Im3Åm.[9] The simplest tiling, due to MSF (111, 0), is shown in Figure 5 g and consists of 24 tiles. The corners of the saddlelike tiles are the Wyckoff sites 8g and 12h of Pm3Åm. Their connectors belong to the 48i set of twofold axes, which are symmetry elements of the normalizer Im3Åm. The fundamental tiling with 48 tiles is generated by MSF (321, p) with a tile of four vertices, which have a particular geometrical meaning (Figure 5 h). They are the points of the intersections between lines connecting sites 1a, 1b, 3c, and 3d in Pm3Åm and the surface. As these lines divide the unit cell in 48 tetrahedranes, each single tile corresponds to what in the realm of minimal surfaces is called a stationary minimal surface inscribed in a tetrahedron.[22] MSF (200, 0), the higher harmonic of the basic SF leads to a [(64)2] tiling with planar four-rings and six-rings in a chairlike conformation (Figure 5 i). If MSF (400, 0) is used instead of (200, 0), a rounded patch occurs at the center of the sixfoldsymmetric tiles, which results in an unconnected pattern (Figure 5 k). Modulation by (222, 0) yields a [436] tiling as shown in Figure 5 l. Finally, if MSF (430, p) is used, a [4644] tiling is generated, the tiles of which are rhombuslike (Figure 5 m). This tiling is in fact a development of the tiling obtained with MSF (111, 0), where each one of the 24 equal tiles has been subdivided in two pairs of equal tiles. The number of nonequivalent vertices amounts to four, and an explicit notation would be [46444444]. We note further that the vertices of the tiles are located on the set of twofold axis of SG Im3Åm.

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Networks Derived from Hyperbolic Tilings Perceiving the beauty of such hyperbolic tilings, different observers may imagine very different motifs or find different correlations to their own experienceÐmaybe even hyperbolic ªEschersº. We will concentrate here on our knowledge of crystal structures, although we believe that many more relations to the micro- and the macroscopic cosmos may be found. During our investigations of hyperbolic tilings, it became immediately clear that there is an intimate topological relation between some of the tilings and real framework structures with the same or a related symmetry. Such relations seem to apply in very general terms and they may be used to derive in a systematic way frameworks of increasing complexity for any given symmetry group. In the following, we will not just focus on known crystal structures[23] but derive and analyze hyperbolic networks from tilings and patterns on the PNSs Y**, D*, and P*, regardless whether or not they lead to known structures. In two recent papers,[24, 25] Hyde and Oguey make a serious attempt of tracing a route towards an enumeration of translationally symmetric graphs. Their method is based on mapping tilings of the hyperbolic plane onto triply periodic minimal surfaces. Our contribution is very much committed to a similar task, although on a quite different way.

Framework Structures Related to Y** The pattern Y** (220, 0) corresponds to the structure of the mineral analcime (NaAlSi2O6 ´ H2O). The silicon and aluminum positions of the real structure are located at the corners of the squares (Figure 6 a). The pattern obtained from SF (220) may further be refined by application of MSF (620, 0). This provides the same network type but the hexagonal and square patches are changed in their internal part, so that the network is now matched by a continuous single stripe on the surface: The latter is depicted in red in Figure 6 b. Use of a MSF (440, p) allows for the design of a network that develops along the path connecting equivalent positions of Wyckoff site 16a. This tiling can be translated into a framework of eight- and six-rings [682]. The former are located on more curved regions, the latter sit on flat points, around positions of label 16a. Mackay and Terrones proposed a hypothetical [682] network of graphitic heritage[26] and named it g688. This is isotypic to ours (Figure 6 c). The position defining the network [682] is of a general type (site 96h) and lies on a path connecting sites 16a. Combining MSFs (220, p) and (420, p) gives the location and mutual orientation of six- and eight-rings even more clearly (Figure 6 d). The networks of analcime [624] and [682] can be compared with respect to their placement on the gyroid. Superposing the two networks reveals their structural and topological relation: Rotation of the dumbbells at the cross-point of the two nets provides the mutual transformation in a continuous way (Figure 6 e). There seems to be an orthogonal relation between the two. Table 1 collates some of these relationships. CHEMPHYSCHEM 2001, 2, 413 ± 422


ARTICLES fantastic [(672) (725) (752) (54)] framework is easily derived (Figure 6 h). This contains 3 96 three-nodes and 1 24 four-nodes, corresponding for example to 96 sp2 and 24 sp3 carbon atoms per unit cell. Applied to carbon, it would be a mixture of 92.3 % graphitic and 7.7 % diamondlike contributions. It is interesting to note that the [(672) (725) (752) (54)] net is a systematic expansion of the [(672) (73) (724)] frame by replacement of the four-ring by four fiverings according to an orthogonality condition. If we center the path connecting the two corners of site 24d in tiling Y** (222, p) with a dumbbell, a [1224] net is formed (Figure 6 k). The connection with the network of analcime is easily recognized: By contracting the square patches, six-rings can be enlarged to twelve-rings. Contraction of the squares until they collapse to a point leads to a limiting network of type [64]. Based on MSF (222, 0), a [643] net may be found which occurs on the other basic PNSs as well (Figure 6 l). This hints to a transformation pathway between the PNS by which the network type is not changed. Such a transformation has already been described.[27]

Framework Structures Related to D* Structures D* (222, 0), D* (400, 0), and D* (222, 0) (400, 0) have already been discussed in connection and with respect to Y** (220, p). From D* (400, 0) a [6442] net is derived which is the basic part of the structure of the mineral sodalite (Na4Al3(SiO4)3Cl), namely a framework of tetraFigure 6. Networks on Y**. a) Network of analcime, in which the Si and Al positions operate a tiling on hedrally coordinated aluminum and silicon the Y** surface. b) The pattern obtained from the combination of MSFs (220, 0) and (620, 0). c) The pattern atoms (Figure 7 a). obtained form MSFs (220, 0) and (440, 0), corresponding to the hypothetical network g688.[26] If the isovalue for D* (333, p) is changed, a d) Refinement of the pattern by addition of MSFs (220, 0) and (420, 0). e) Comparison of the networks of point pattern is yielded which can be conanalcime and g688. f) Tiling obtained from MSF (444, p). g) Three-node network derived from (444, p). h) Hypothetical framework derived from Y** (873, 0). i) Tiling obtained from MSF (666, ÿ p/2). k) A [1224] nected to form a homogeneous network network derived from the Y** (222, p) tiling. l) A [6464] network obtained from the same tiling. m) A tiling (Figure 7 b). This is a three-connected net obtained from MSF (888, 0). that is part of the clathrate II structural type (Figure 7 c), found for example in Na24Si136 . More complex frameworks with characteristics similar to [682] This defect clathrate network is found to play an important role are easily produced for example by application of hhh-type MSFs in those structures, where atoms are placed at the interface like (444, p), (666, ÿ p/2), and (888, p). They are described in between other structural units.[13] Furthermore, it may be Table 1 and displayed in Figures 6 f ± 6 i. MSF (444, p) sets up a another template for three-connected sp2 carbon modification beautiful tiling in which a net of three-, four-, five-, and six-rings forming a [692] framework. can be directly seen (Figure 6 f). If one centers the trigons by By addition of centered tetrahedra X5 at the centers of both points, the parallelograms by pairs, and the squares by smaller labyrinths (yellow spheres in Figure 7 d), a four-connected squares, a [(672) (73) (724)] net is deduced (Figure 6 g). Choosing network occurs which is exactly the X136 net of the clathrate II MSF (873, 0) a seemingly complex pattern occurs from which a type with the notation [(56)8a (56)32e (556)96g] for the three-dimenCHEMPHYSCHEM 2001, 2, 413 ± 422

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R. Nesper and S. Leoni Table 1. Topological description of networks related to PNS Y**, D*, and P*. Tiling

a[a]

b[b]

Network

1 1 1 1 5 3 3 5

2 2 2 2 3 3 4 5

[(4 )16a (4 )24d] [(1224)96h] [(436)48f] [(46)16a (44)24d] [(6353) (3545) (454'5)] [(672) (73) (724)] [(672) (725) (752) (54)] [(44) (433) (4332) (435) (36)]

D* (333, 0) D* (333, 0) Clathrate II D* (444, 0) D* (555, 0) D* (660, p) D* (400, 0) D* (800, 0)

1 1 2 2 1 2 2

3 1 3 2 4 1 1 1

[(46)32e (44)48f (44)96g] [(692)96g] [(56)8a (56)32e (556)96g] [(6364) (636'3)] [(36) (3252) (54) (54)'] [(436)24k] [(6442)12d] [(6442)12d]

P* P* P* P*

3 2 2 5

1 1 1 4

[436]48I [682]48k [4262]24h [(63) (625) (568) (582)]

Y** Y** Y** Y** Y** Y** Y** Y**

(222, (222, (222, (330, (444, (444, (873, (666,

(330, (440, (440, (660,

ÿ p/2) ÿ p/2) ÿ p/2) ÿ p/2) p) p) 0) ÿ p/2)

0) 0) 0) 0)

[d]

6

SG of the net 4

I4132[c] Ia3Åd Ia3Åd I4132[c] Ia3Åd Ia3Åd Ia3Åd I4132[c] Fd3Åm[c] Pn3Åm Fd3Åm Pn3Åm Pn3Åm Pn3Åm Im3Åm Im3Åm Im3Åm Im3Åm Im3Åm Im3Åm

[a] Number of tiles. [b] Number of vertices per tile. [c] This SG is the one of the colored tiling. [d] 3D network.

sional net. The two-dimensional net, on the surface, can be written [(53)32e (54)96g] (Figure 7 d). The centers of the other set of equivalent tiles (white patches in Figure 7 b) necessarily connect to a framework of the same type but ªrotatedº by 608 with respect to the other one. In the hh0 series, MSF (660, 0) generates a set of patches on D* which connect into a [436] net (Figure 7 e). The strain of the fourrings and the high connectivity of the nodes may be suited for a metal ± metal bonded net and we would expect this to occur in intermetallic compounds. If we expand the summation by MSF (440, 0), the result is a [6242] network which correlates beautifully with the projection of the framework of the zeolite faujasite (Mg0.5 ,Ca0.5 ,Na,K)3±4[Al3±4Si9±8O24] ´ 16 H2O) onto D* (Figure 7 f). The network of faujasite is of the [634] type and develops in one of two equivalent labyrinths of the D* surface. The difference in the network notation is again due to the fact that not all framework atoms of faujasite are on the surface. The basic faujasite structure contains 633 atoms, of which 576 belong to the framework. The combination of MSFs (660, 0) and (400, 0) yields a [682] net (Figure 7 g), which again is very much related to the [682] net on Y** due to the abovementioned transformation path between Y** and D*. Consequently, the quasiorthogonal relation between the [682] and the [4262] network on the Y** surface exists on D* as well. This is shown in (Figure 7 h), where the yellow connections of the [4262] net cross the red patch structure with a [682] topology in an orthogonal fashion. The [4262] linkage corresponds to the silicon framework of the mineral sodalite. The pattern in Figure 7 h is generated through a combination of MSFs (222, 0) and (400, p). Narrowing the square rings to a point results in a [64] network, which is a limiting case for the [4262] type. The six-rings of the latter are in a chairlike conformation.

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If the indices of the MSF of the last example are doubled to (444, 0) and (800, 0) a collection of spots arises, which belong to the crystallographic positions 48f and 32e, the latter being the center of the flat regions. It should be noted that such a collection of positions represents a noncharacteristic orbit of this space group.[28] The position 48f matches precisely the silicon positions of sodalite (Figure 7 i).

Framework Structures Related to P* In the series hh0, the MSF (330, 0) induces a pattern on P* which corresponds to a [436]-type network. This is a match of the framework of the zeolite RHO, such that the edges of the tiling correspond to the silicon/aluminum positions (Figure 7 k). For P* (440, 0) again a net of type [682] is found, which corresponds to a hyperbolic carbon structure proposed some years ago by O'Keeffe (Figure 7 l).[29] The quasiorthogonal [4262], which is a limiting case of the more general [4(12)2] net, contains cubooctahedral building elements. The former is identical with the [4262] net on D*. This is due to the fact that this frame exhibits common symmetry characteristics of both Pn3Åm and Im3Åm, the normalizers of Fd3Åm and Pm3Åm, as is easily seen by comparison of the site symmetries of 12f (Pn3Åm) and 12d (Im3Åm). Clearly, Pn3Åm has an intrinsic body-centered characteristic (see also site 2a). Application of MSF (660, 0) leads to a more complex pattern of the planar six-ring motives around the flat regions of P* from which coronenelike building elements can easily be deduced on replacing the six-rings of [682] (Figure 7 m). The notation reads [(63) (625) (568) (582)].

Surface Genus and Network Topology With respect to its topology, a surface in three-dimensional space is completely characterized by its orientability and its genus. The genus is an expression of the number of holes or handles a surface happens to have: A sphere has a genus of zero, a torus a genus of one. In principle, the genus of a periodic surface with handles must necessarily be infinite. However, there is a well-defined genus for the unit cell or the asymmetric unit, respectively. The tilings presented in the previous Sections allow for a simple calculation of the genus of a surface. Starting from the Euler relation, the Euler ± PoincareÂ characteristic c(P) of the surface P can be defined as V ÿ E F c(P) for V vertices, E edges, and F faces. The relation of the latter with the genus g(P) of a surface is given by g(P) [2 ÿ c(P)]/2. For the platonic solids, c(P) is two and the genus is zero, relating them topologically to the sphere. As a topological characteristic, c(P) can be calculated from any tiling of a surface. This will be illustrated by calculating the genus of the three basic cubic PNSs, which we focus on in this work. Considering the most regular tilings, for SF (222, p/2) on Y** (Figure 4 a) with V 40, E 96, and F 48, the Euler ± PoincareÂ characteristic is c(x) ÿ 8. Thus, the genus for the body centered cell is five and with respect to a primitive cell choice, c(x) ÿ 4, CHEMPHYSCHEM 2001, 2, 413 ± 422


ARTICLES on one surface to another. This is shown for Y** (220, 0) , D* (222, 0) in the series in Figure 8 a ± c, where the genus remains unchanged.

Conclusions We understand the PNS, which are the nodes or the roots of the first fundamental plane waves of the symmetry groups (standing waves of the unit cells), as trajectories of symmetry space. The construction of hyperbolic tilings and the derivation of framework structure models are based on these trajectories. They combine the beauty of coding necessary symmetry information whichÐby our procedureÐis automatically cast into topological forms and the possibility of gradually increasing the information content. In the increasing complexity of the patterns, indications for scaling symmetry occasionally become visible. They may already hint at the necessity of a repetition of structural pattern in real structures, in sense of a hierarchical development of structural themes. The method presented here serves a threefold purpose, namely a) the generation of tilings on hyperbolic surfaces, b) the understanding of chemical networks as a two-dimensional, hyperbolic form embedded in R3, and c) the prediction of hitherto Figure 7. Networks on a ± i) D* and k ± m) P*. a) Basic part of the structure of the mineral sodalite, correlated to the unknown hyperbolic nets. The distiling obtained with MSF (400,0). b) Point pattern obtained from MSF (333, p) for a different isovalue. c) Threecrete nature of reciprocal space alconnected network related to the previous pattern, of type [692]. d) Addition of a centered tetrahedron to the lows for the systematic generation of previous network results in a four-connected network, clathrate II. e) The [436] network related to the point pattern obtained from MSF (660, 0). f) The network related to faujasite, obtained by a further addition of MSF (440, 0) to known and hypothetical networks (660, 0). g) The network of type [682] resulting from a pattern generated from MSFs (400, 0) and (660, 0). from simple to complicated patterns. h) Comparison of the network of sodalite, represented as a continuous stripe, and the previous (400, 0) and (660, 0) The approach applies equally well to network. i) The network of sodalite, matching a point pattern obtained from MSFs (444, 0) and (800, 0). k) Tiling of surfaces of higher genus, allowing MSF (330, 0) matched by the network of the zeolite RHO. l) Hyperbolic carbon structure, proposed by O'Keeffe,[28] corresponding to a pattern obtained from MSF (440, 0). m) The MSF (660, 0) leads to a pattern containing also for a quick determination of this coronenelike motifs, which are part of a [(63) (625) (568) (588)] network, outlined in blue. important topological measure, and in particular for those surfaces which g(x) 3. For the tiling D* (333, p), we find V 176, E 384, F can be realized by the nodal approximation only, that is, having 192, c(x) ÿ 16, and g(x) 9. Consequently for the primitive cell, no corresponding minimal surface. c(x) ÿ 4 and g(x) 3 (Figure 5 a). In P* (111, 0), a regular Another important point, which will be extensively illustrated tesselation (Figure 5 g) with V 20, E 48, F 24, c(x) ÿ 4, in a subsequent paper, is the feasibility of network transformaand genus g(x) 3 occurs. Finally, for the fundamental tiling P* tion of the skin of a hyperbolic surface, where the supporting (321, p/2) (Figure 5 h), there is V 44, E 96, F 48, and c(x) surface of a tiling can be continuously modified[13, 27] and the networks deformed. ÿ 4, and thus the genus is g(x) 3. We feel that there are enormous possibilities based on this If we recall the possibility of transforming the basic PNS[13, 27] or PMS[30] into each other according to P* , Y** , D*, there is a approach for understanding and classification as well as well-defined way of transforming tilings, patterns, and network predictions in structural chemistry. However, our artistic intuition CHEMPHYSCHEM 2001, 2, 413 ± 422

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Received: November 23, 2000 [F 154] Revised: March 12, 2001

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