- 1. Introduction
- 2. Coincidence and symmetry operations acting on a component of the crystal space
- 3. Crystallographic orbits, point configurations and lattice complexes
- 4. Normalizers
- 5. Derivative structures and symmetry relations
- 6. Reticular theory of twinning
- 7. Topology of crystal structures
- 8. Conclusions
- References

- 1. Introduction
- 2. Coincidence and symmetry operations acting on a component of the crystal space
- 3. Crystallographic orbits, point configurations and lattice complexes
- 4. Normalizers
- 5. Derivative structures and symmetry relations
- 6. Reticular theory of twinning
- 7. Topology of crystal structures
- 8. Conclusions
- References

## feature articles

## Does mathematical crystallography still have a role in the XXI century?

^{a}LCM3B UMR-CNRS 7036, Nancy Université, BP 239, F-54506 Vandœuvre-lès-Nancy CEDEX, France^{*}Correspondence e-mail: massimo.nespolo@lcm3b.uhp-nancy.fr

Mathematical crystallography is the branch of crystallography dealing specifically with the fundamental properties of symmetry and periodicity of crystals, topological properties of crystal structures, twins, modular and modulated structures, *International Tables for Crystallography* and is sometimes still enshrined in more specialist texts and publications. To cover this communication gap is one of the tasks of the IUCr Commission on Mathematical and Theoretical Crystallography (MaThCryst).

Keywords: graph theory; mathematical crystallography; OD structures; polytypes; symmetry; theoretical crystallography; topology; twins.

### 1. Introduction

*Mathematical Crystallography* is the title of a classical textbook by Hilton (1903), which represented one of the first efforts of providing a comprehensive presentation of the theory of symmetry in crystals in the English language. Hilton's book was mainly devoted to morphological symmetry and lattice symmetry, with some excursions into the physical properties of crystals and the theories of crystal growth.

Nowadays, the field of mathematical crystallography has greatly enlarged its horizons: not only because of its development towards higher-dimensional crystallography (Yamamoto, 1996) and non-Euclidean spaces (Robins *et al.*, 2004) but also because different types of symmetry relations (such as acting on a subspace of the crystal space and polychromatic symmetry describing physical properties of crystals and mapping of individual orientations in twins) have been discovered which naturally fell into the field of mathematical crystallography. Symmetry, however, is far from exhausting the targets of mathematical crystallography, which also investigates the crystal-chemical problems related to the topology of crystal structures by means of mathematical tools, such as graph theory, whose relation with crystallography has been recognized relatively recently.

The basis of the theory of symmetry in crystals, and thus of crystallography itself as a modern science, is found in mineralogy. The term `crystallography' was introduced by Cappeller (1723) but it was with the studies by Romé de L'Isle (1772, 1783), Bergman (1773), Haüy (starting from 1784) and Delafosse (1840) that a new route opened towards a systematic derivation of the properties of periodicity and symmetry of crystals, developed especially in France (*e.g*. Bravais, 1850; Mallard, 1879; Friedel, 1926) and in Germany (*e.g*. Bernhardi, 1808*a*,*b*,*c*; Weiss, 1809*a*,*b*, 1815; Neumann, 1823; von Groth, 1895). The modern notation of lattice planes and crystal faces is due to the British school (Whevell, 1825; Miller, 1839). The theory of space groups was finally developed by Fedorov (1890), Schoenflies (1891) and Barlow (1894) (for a recent historical review, see Lalena, 2006). A systematic path thorough space groups was presented by Wyckoff (1930), in what can be considered the ancestor of Vol. A of *International Tables for Crystallography*. Nowadays, eight volumes of *International Tables* are published, which may give the impression that exhaustive information on all aspects of crystallography is available there. Actually, some aspects of crystallography, and especially of mathematical crystallography, are inadequately presented in *International Tables*, and several topics are still quite unknown to a large number of structural crystallographers. This partly comes from the use of concepts and language that are unfamiliar to non-specialists. Some efforts to provide the necessary introduction have been made, for example by Hahn & Wondratschek (1994) and with the production of a series of *Teaching Pamphlets* freely available from the IUCr website (http://www.iucr.org/iucr-top/comm/cteach/pamphlets.html ) and, more recently, with the launch of the *IUCr Online Dictionary of Crystallography* (http://reference.iucr.org/dictionary/Main_Page ). At the same time, crystallography, a victim of its own successes, is often considered as a *diffraction technique* rather than a *multidisciplinary science*.

The purpose of this article on *Mathematical Crystallography* for this Special Issue celebrating the 60th Anniversary of the IUCr is to present some of the topics that are often overlooked by the structural crystallographer and to give him a key to navigate the literature on the subjects, as well as to present some hot topics in mathematical crystallography which are currently under-represented in *International Tables*. Needless to say, there is no aim at compiling an exhaustive review: the topics selected for this article are among those of central interest to the MaThCryst commission.

Mathematical methods and probabilistic theories used in the solution and

of crystal structures from experimental data are addressed by the IUCr Commission on Computational Crystallography and are not treated in this article.### 2. Coincidence and symmetry operations acting on a component of the crystal space

The symmetry operations of a *a*) or `global operations' (Sadanaga & Ohsumi, 1979; Sadanaga *et al.*, 1980). These are the operations that normally come to mind when one thinks of the symmetry of a crystal. The fact that they act on the whole crystal space may sound an obvious prerequisite for any coincidence operation in a crystal structure. This is not true because more generally a coincidence operation can act on just part of the crystal space, which we will call a `component', and bring it to coincide with another component. As a consequence, one is led to consider a much wider category of coincidence operations, which nevertheless play an important role in some types of crystal structure.

Let us imagine subdividing the crystal space into *N* components *S*_{1} to *S*_{N}, and let Φ(*S*_{i}) → *S*_{j} be a coincidence operation transforming the component *i* into the component *j*. Such an operation in general is not a coincidence operation of the whole crystal space and therefore is not one of the operations of the of the crystal. It is called a *partial* operation and in general it is not required that it brings *S*_{j} back onto *S*_{i}: more strongly, Φ(*S*_{k}) in general is not defined for any component *k* different from *i* and therefore a partial operation is not necessarily a From the mathematical viewpoint, partial operations are space-groupoid operations, in the sense of Brandt (1927).^{1}

When *i* = *j*, *i.e.* when the operation is Φ(*S*_{i}) → *S*_{i} and brings a component to coincide with itself, the partial operation is of special type and is called *local* (Sadanaga & Ohsumi, 1979; Sadanaga *et al.*, 1980). A local operation is in fact a which is defined only on a part of the crystal space: local operations may constitute a (Kopský & Litvin, 2002), and in particular a diperiodic group (Holser, 1958) when *S*_{i} corresponds to a layer.

On the basis of these very general definitions, we can briefly analyse some examples that will make clear the role of partial and local operations in describing and rationalizing a crystal structure.

#### 2.1. Partial and local operations in `supersymmetric structures'

About 8% of crystal structures contain more than one formula unit in the *Z*′ > 1). In the case of molecular crystals, the crystallization mechanism and the structural relations among the molecules in the is the subject of extensive research because of its direct applications in supramolecular chemistry and crystal engineering (see *e.g*. Steed, 2003). The role of (approximate symmetry) has been emphasized but the panorama obtained is not always fully satisfactory. As a matter of fact, the role of *partial* and *local* operations is of paramount importance.

Molecular crystals characterized by *Z*′ > 1 have been called `supersymmetric structures' by Zorkii (1978) because the independent molecules in the can be brought to more or less exact superposition by a screw rotation about a direction relating the molecules, and this direction has some special orientation with respect to the lattice basis. The `supersymmetry operations' evidently act only on part of the crystal space and may be either local or partial, depending on whether a component is mapped to itself or to a different component. The cases of picric acid and phenol were analysed by Sadanaga *et al.* (1980). In the first case, two molecules of picric acid exist in the of *Pca*2_{1}, *a* = 9.254 (2), *b* = 19.127 (4), *c* = 9.704 (2) Å, which has a pseudo-tetragonal mesh in (010) (Duesler *et al.*, 1978). The two molecules in the are related by a pseudo partial 4_{2} axis parallel to the crystallographic *b* axis. Fig. 1 shows the two molecules in the in the original orientation (top) and after ±90° rotation about the crystallographic *b* axis (centre and bottom). Let us indicate by *M*_{N,ϑ} the molecule No. *N* (*N* = 1 or 2) after a rotation of ϑ° (ϑ = 0, +90 or −90°). *M*_{2,+90} has almost the same orientation as *M*_{1,0}, whereas no relation can be found between *M*_{1,+90} and *M*_{2,0}. In the same way, *M*_{1,−90} has almost the same orientation as *M*_{2,0} whereas no relation can be found between *M*_{2,−90} and *M*_{1,0}. We can therefore conclude that a (pseudo) partial 4_{2} axis exists parallel to **b**, as shown in Fig. 2, where the three parts of Fig. 1 are overlapped. It is partial because 4_{2}^{+} acts on *M*_{1,0} producing *M*_{2,+90}, but it does not act on *M*_{2,0}. At the same time, 4_{2}^{-} acts on *M*_{2,0} producing *M*_{1,−90} but it does not act on *M*_{1,0}. The pseudo-character of this partial coincidence operation is shown by the imperfect overlap of the molecules in Fig. 2. Fig. 3 (modified after Sadanaga *et al.*, 1980) shows the idealized unit-cell contents, where each molecule of picric acid is represented by a paper kite, the smaller higher-angle side indicating that the corresponding part of the molecule is directed towards the observer. Corresponding molecules along the *c* axis in the same row differ in their *y* coordinate by about ¾ (row I) or ¼ (row II). The symmetry operations of the map solid kites on one side and dashed kites on the other side. The pseudo partial [.4_{2}.] rotation maps a solid kite onto a dashed kite. The combinations of these two types of operations map a solid kite to a dashed kite located in different positions in the (010) projection: these are partial pseudo-[.4_{1}.] and partial pseudo-[.4_{3}.] axes. These partial operations can be seen as space-groupoid operations or, alternatively, as cosets of the of the crystal (Grell, 1998) obtained by composing the [.4_{2}.] partial operation with the operations of the However, as seen above, [.4_{2}.] and [.4_{2}^{-1}.] do not possess the same domain and therefore they cannot be composed with themselves, as would be the case of a local operation. The crystal structure as a whole has to be described in terms of groupoids instead of groups, although each operation in itself is a group operation, namely from the group generated by the *Pca*2_{1} and the 4_{2} operation.

In the case of phenol (Fig. 4), three molecules exist in the of the which is of type *P*2_{1}. The lattice parameters (*c*-unique setting) are *a* = 6.050 (1), *b* = 8.925 (2), *c* = 14.594 (3) Å, γ = 90.36 (2)° (Zavodnik *et al.*, 1988). The structure is thus metrically pseudo-orthorhombic. The three molecules in the are related by a pseudo local 3_{2} axis parallel to the crystallographic *a* axis. The structure, however, is not pseudo-hexagonal because, despite the almost exact orthohexagonal relation between the *b* and *c* parameters (*c* ≃ *b*3^{1/2}), the lattice type is primitive and not *A*-centred, as would be required to obtain a pseudo-hexagonal metric symmetry. In this case, the `supersymmetry axis' is local because it acts on the same part of the crystal space, defined by the three phenol molecules in the Both the direct and the inverse operations are defined. The pseudo-character of the operation is shown by the *x* coordinates in Fig. 4, which are *slightly* displaced with respect to the ideal ±*a*/3 screw motion.

#### 2.2. Partial and local operations in OD structures

The OD theory (Dornberger-Schiff, 1964*a*, 1966) specifically deals with structures in which partial operations act on layer structures, although extensions to rod and block structures have been suggested too (see *e.g*. Dornberger-Schiff, 1964*b*; Belokoneva, 2005). The OD theory distinguishes two types of partial operations: **λ**, which transform a layer into itself, and **σ**, which transform a layer into an adjacent layer (Dornberger-Schiff & Grell-Niemann, 1961). **λ** operations correspond to local operations in Sadanaga *et al.* (1980); **σ** operations are partial operations as defined above.

OD structures are polytypic, namely structures built by stacking layers in different orientations/positions; the opposite may or may not be true depending on the degree of idealization one adopts in describing the layer structure (for a critical discussion, see Zvyagin, 1993). OD stands for order–disorder, has no relation with the chemical order–disorder phenomena but indicates that the stacking of layers may produce both periodic (ordered) and non-periodic (disordered) structures. The crystal chemical reason for is that adjacent layers (two-dimensionally periodic units) can be linked to each other in more than one translationally non-equivalent way, which however preserve the nearest-neighbour relationships. The operations interchanging the layers of a pair of adjacent layers are partial operations: they act on a part of the crystal space (consisting of the layer pair) and, in general, they are not the same for each layer pair. Moreover, the local **λ** operations Φ mapping layer *i* onto itself – Φ(*S*_{i}) → *S*_{i} – and Φ′ mapping layer *j* onto itself are, in general, not the restriction of a common operation mapping both components. In the OD language, one says that these operations do not have a `continuation' in the rest of the crystal structure: Φ acts on *S*_{i} but not on *S*_{j}, Φ′ acts on *S*_{j} but not on *S*_{k}, and so on.

It must be emphasized that the choice of layers is made precisely to locate the components of the crystal space on which the local operations act and the layers located in this way do not necessarily coincide with the classical crystal-chemical layers defined by cleavage properties. This is why one speaks of `OD layers', to emphasize the choice criterion, even in the cases when the result coincides with the crystal-chemical layers. Moreover, there may exist more than one possibility of dividing the crystal space into layers so that local operations are defined: this is why one says that the choice of OD layers is in general not unique (Grell, 1984).

Among the infinitely many possible *etc*. of layers are geometrically equivalent (or, when this equivalence is not possible, the number of different triples *etc*. is minimal) are called *maximum degree of order (MDO) polytypes*. In a class of compounds, they are normally the most frequent a fact suggesting that the of layers is actually somehow related to a thermodynamic stability, although it is hardly conceivable that long-range interactions like those existing between the second or third layer may play a fundamental role in discriminating the stability of different polytypes.

The OD interpretation of polytypic structures is not only an elegant way of rationalizing a series of structures within a general framework but also an extremely powerful way of interpreting the diffraction pattern of these structures and to model unknown structures. This becomes possible once the concept of *family* is introduced.

If one takes two or more identical copies of the same polytype, translated by a vector corresponding to a submultiple of a translation period – what is commonly called a *superposition vector* – a fictitious structure is obtained, which is termed a *superposition structure*. Among all possible superposition structures, one plays a special role: it is the one in which the superposition vectors correspond to all possible positions of each layer. It is called a *family structure* and it exists only if the shifts between adjacent layers are rational, *i.e*. if they correspond to a submultiple of lattice translations. Because it is built by superposing *all* possible positions of a layer, the is common to *all* of the same family (Dornberger-Schiff, 1964*a*; Ďurovič, 1994). From a group-theoretical viewpoint, building the corresponds to transforming (`completing') all the operations of a space into the global symmetry operations of a (Fichtner, 1977, 1980). From the symmetry viewpoint, pairs of adjacent layers remain *geometrically equivalent* in all of the same family.^{2}

The OD character of a polytype appears also in its diffraction pattern. The group of translations of the *family sublattice*), which corresponds to the Fourier transform of the and one or more cosets. The family is again common to all of the same family. This means that all of the same family, normalized to the *same volume of scattering matter*, have a weighted reciprocal in common.^{3} The diffractions that correspond to the family are termed *family diffractions* (or, more commonly, *family reflections*). When indexed with respect to the basis vectors of any of the of the same family, the family may show several non-space-group absences, which is a clear indication of the existence of (space-groupoid) operations. The family reflections are always sharp, even in the case of non-periodic (disordered) In fact, in the *all* the layer-stacking operations are completed to global operations so that the as well as its Fourier transform, is common to both periodic and non-periodic of the same family (Ďurovič & Weiss, 1986; Ďurovič, 1997, 1999). The remaining diffractions, which correspond to the cosets of the weighted with respect to the family are termed *non-family diffractions* (*non-family reflections*, *characteristic reflections*) and are instead typical of each polytype: they can be sharp or diffuse, depending on whether the polytype is ordered or not.

When inspecting the diffraction pattern of an unknown structure, the regular sequence of reciprocal-lattice rows showing and not showing non-space-group absences is a strong indication of the OD character of the structure. The disorder in the stacking sequence appears as streaking along the non-family rows; the coexistence of reciprocal-lattice rows that are not affected by streaking (family rows) suggests at a glance the OD character. Frequent *e.g*. Takeda, 1967).

### 3. Crystallographic orbits, point configurations and lattice complexes

The symmetry of an atomic structure can be seen as the intersection of the symmetries of the spatial distribution of each crystallographic type of atom. The structure of a crystal containing *N* crystallographically different types of atoms can be ideally decomposed into *N* distributions of points in space, similar to what is commonly done in the study of crystal morphology, when the external shape of a crystal is analysed in terms of its face forms. Each point, under the action of the symmetry operations of the *G* of the crystal, generates an infinite set of symmetrically equivalent points, called a *crystallographic orbit* (Matsumoto & Wondratschek, 1979, 1987). The *G* is called the *generating space group of the orbit*. Three features of each have to be considered: the *inherent symmetry* (eigensymmetry) *E*, which may coincide with *G* or be a of it; the *site-symmetry group* *S*; and the *translation subgroup* *T*. The intersection of the inherent symmetries *E*_{i}, *i* = 1, *N*, of the *N* crystallographic orbits gives back the *G* of the crystal.

A *G* is an infinite group that can be seen as an extension of an infinite group of translations *T*, representing the lattice, by a *P*. *P* then is isomorphic to the *G*/*T*^{4} (Hahn & Wondratschek, 1994).

The *site-symmetry group* *S* of a is the of *G* that maps a point of that position onto itself: in the language of abstract algebra, it is the *stabilizer* of the point. The site-symmetry group *S*_{max} of the highest-symmetry lowest-multiplicity is isomorphic with *P* for or with a of *P* for non-symmorphic space groups. The site-symmetry groups of the other Wyckoff positions are subgroups of *S*_{max}.

The *inherent symmetry* *E* of a corresponding to a in *G* is at least *G*. If *G* = *E*, the orbit is called a *characteristic crystallographic orbit*, otherwise it is called a *non-characteristic crystallographic orbit*. *G* can be a *translationengleiche* of *E* (same translation group *T*: , *T*_{G} = *T*_{E}): the atoms sitting in the corresponding contribute to the diffraction pattern a symmetry higher than that from the whole crystal. *G* can also be a *klassengleiche* of *E* (same , *T*_{G} < *T*_{E}): *E* then contains translations additional to those of the generating *G* and the orbit is called an *extraordinary crystallographic orbit*.^{5} The atoms sitting in a corresponding to an extraordinary orbit do not contribute to some classes of reflections: this information appears under the `special tabulated in *International Tables for Crystallography* (2002), Vol. A (ITA for short), but, as we are going to show, it is actually incomplete. The case when *G* is a general of *E* (*G* < *E*) includes the features of both and .

The concept of *point configuration* but differs from it by the fact that point configurations are detached from their generating space groups. As said above, from a given position in a certain *G*, a is obtained by the actions of the symmetry elements of *G*. The result is a spatial distribution of points with an inherent symmetry *E*, which is a fundamental feature of the spatial distribution of points and does not depend on the having generated it. This same spatial distribution may occur in space groups of different type and takes the name of *point configuration*. There exists evidently a surjection of point configurations onto crystallographic orbits because a depends on the *G* where it occurs and, consequently, corresponds to a well defined site-symmetry group, whereas point configurations do not. In other terms, a set of points is called a if there exists at least one that generates it as an orbit of one of its points. The relation between crystallographic orbits and point configurations in has a close analogy in to the relation between the face form attached to the that has generated the form and the face form detached from its generating There again, the same form may occur in different point groups. For example, the tetragonal prism has inherent symmetry *E* = 4/*mmm* and may occur in 4/*mmm*, where it is a characteristic form, but also in all the other six tetragonal point groups, where it is a non-characteristic form.

In each individual *types*, which are called *lattice complexes*. The concept of is actually older than that of having been introduced by Niggli (1919) and fixed by Hermann (1935). A rigorous definition, however, was provided much later by Fischer & Koch (1974) and by Zimmermann & Burzlaff (1974). The same may occur in different types of space groups of the same For example, the set of six points ±*x*,0,0; 0,±*x*,0; 0,0,±*x* in *Pm* corresponds to the 6*e* of site-symmetry group *mm*2.. but it occurs also in *Pm**m* where it corresponds again to the 6*e*, which now has site-symmetry group 4*m.m*. The inherent symmetry *E* of this set of points, which forms the vertices of an octahedron around the sites of a cubic primitive lattice, is *Pm**m* independently of the *G* where it occurs. Taken as such, it defines a and *Pm**m* is the characteristic space-group type of the (*G* = *E*). When instead it is considered together with the *G* from which it has been generated, it is a Evidently, 6*e* in *Pm**m* is a characteristic orbit because the inherent symmetry *E* coincides with the generating group *G*. Instead, in *Pm*, 6*e* is a non-characteristic orbit because the inherent symmetry (*Pm**m*) is higher than the generating group (*Pm*). It is not an extraordinary orbit, however, because *Pm**m* contains no additional translations. In the lattice-complex approach, the phenomenon of extraordinary orbits is treated in analogy to the concept of limiting forms in crystal morphology: a *L*_{1} is called a *limiting complex* of another *L*_{2} if the set of its point configurations forms a (true) subset of the set of point configurations of *L*_{2}. In this sense, the of all cubic primitive lattices is a of the set of all tetragonal primitive lattices.

The contribution of atoms in a primitive cubic *Pm**m*. Besides, when this defines a with respect to the axial setting of the – in other words, it forms an extraordinary orbit in *G* – special occur too. This becomes clear when the example of the primitive cubic is worked out.

The primitive cubic *Pm**m*, which is the characteristic of this It occurs in the Wyckoff positions 1*a* and 1*b*, which differ by a shift of ½½½. In this space-group type, it does not define any and no special are observed. The contribution to the diffraction symmetry is that of *Pm**m*. In the language of the crystallographic orbits, this corresponds to two characteristic orbits.

The primitive cubic *P*3*m, P*432, *Pm* and *P*23, again corresponding to the Wyckoff positions 1*a* and 1*b*. The situation is essentially the same as in *Pm**m* but, because the inherent symmetry is higher than the the diffraction symmetry from atoms in this is higher than that from the whole crystal. In the language of the crystallographic orbits, this corresponds here to two non-characteristic (*G* < *E*), non-extraordinary (same translation lattice: *T*_{G} = *T*_{E}) orbits.

The primitive cubic *Im**m*, *Fm**m*, *I*432, *Im* (Wyckoff position 8*c*), *Fm**c*, *F*3*c* and *Ia* (Wyckoff positions 8*a* and 8*b*). In all these space-group types, the primitive cubic corresponds to a primitive with halved translations along each of the three translation directions: for this reason, the symbol *P*_{2} is used (*cf*. Chapter 14 in ITA). As a consequence, atoms in this only contribute to diffractions with even values of *h*, *k* or *l*. In the language of the crystallographic orbits, this corresponds here to extraordinary orbits (*G* < *E*, *T*_{G} < *T*_{E}).

Although the above example seems well representative of how lattice complexes can occur in different space groups and of the consequences on the diffraction symmetry, there is another, even more specialized, category, which is not explicitly taken into account in ITA. Wyckoff positions with one or more free coordinates can be specialized by making these free coordinates take a rational value. This specialization may result in switching to another which therefore has a different inherent symmetry and a different site-symmetry group. If, however, the result remains in the same but the inherent symmetry *E* of the orbit is enhanced, two cases have to be distinguished: (i) the orbit is extraordinary because it defines a (different) with respect to *G*, to which additional special necessarily correspond; (ii) the orbit shows no additional translations, *i.e.* it is non-extraordinary. Then it nevertheless may give rise to additional although not necessarily. These additional conditions can be obtained, *e.g*. with the aid of the list in Engel *et al.* (1984). Two examples will make the situation clear.

The symmetry of the diffraction pattern from a crystal is finally the intersection symmetry of the diffraction symmetry from each

(the effect of Friedel's law, when respected, can be taken into account as the union of an inversion centre and the diffraction symmetry of the crystal). This makes a close parallel with the situation in the where the of the crystal corresponds to the intersection of the inherent symmetries of the lattice complexes occupied by the atoms in the crystal.The occurrence of limiting-complex relations between space groups of different crystal families depends on the realization of special metric conditions and may be coupled with special coordinates. This means that only part of the space groups within one type contains such non-characteristic orbits. A striking example is given by space groups of type *P*2_{1}2_{1}2_{1}, position 4*a xyz*. Here, simultaneous specialization of the lattice parameters to *a* = *b* = *c* and of the coordinates to *x* = *y* = *z* = 1/8 result in a non-characteristic (non-extraordinary) orbit with inherent symmetry *E* = *P*4_{3}32 and the complicated addition for *h*, *k*, *l* (permutable): *h*, *k* = 2*n*+1 or *h* = 2*n*+1, *k* = 4*n* and *l* = 4*n*+2 or *h*, *k*, *l* = 4*n*+2 or *h*, *k*, *l* = 4*n*. So far such relations have been systematically worked out only for the cubic limiting complexes of tetragonal and trigonal lattice complexes (Koch & Fischer, 2003; Koch & Sowa, 2005). It must be noticed that these additional are valid for all space groups of type *P*2_{1}2_{1}2_{1} only if *x* = *y* = *z* = 1/8. This is because structure factors are independent of the cell metric.

When heavy atoms occupy non-characteristic orbits and light atoms are in characteristic orbits, the symmetry of the diffraction pattern is closer to that of a higher-space-group type and this

makes the more complex. Furthermore, when the positions occupied by heavy atoms correspond to extraordinary orbits, some classes of diffractions receive contributions only from light atoms, and in the diffraction pattern one can see strong diffractions, contributed by all atoms, and weak diffractions, contributed by light atoms only.The literature on the subject is huge. Fundamental texts on crystallographic orbits are: Wondratschek (1976), Lawrenson & Wondratschek (1976), Matsumoto & Wondratschek (1979, 1987) and Engel *et al.* (1984). Chapter 8 in ITA introduces the concept of crystallographic orbits, without making a detailed analysis, however. An extension to polychromatic orbits has been introduced by Roth (1988). About point configurations and lattice complexes, besides the literature quoted above, the book by Fischer *et al.* (1973) and Chapter 14 in ITA cover the subject with full details. An exhaustive discussion on the difference between crystallographic orbits and point configurations can be found in Koch & Fischer (1985). The application of lattice complexes to the classification of crystal structures is treated in several articles, for example Hellner (1965).

### 4. Normalizers

Normalizers are a mathematical concept extensively used in the solution of crystallographic problems, such as the choice of the origin and of the gives an extensive presentation of the application of normalizers.

in the comparison of equivalent descriptions of crystal structures, the choice of a setting for indexing a diffraction pattern, the choice for indexing morphological faces of a crystal, the interchangeability of Wyckoff positions, and the definition of lattice complexes. Chapter 15 in ITATo understand what a

is and how it works, the concepts of conjugacy and of have to be recalled first.A group *G* and one of its supergroups *S* are uniquely related to a third, intermediate, group *N*_{S}(*G*), called the *normalizer of G with respect to S*. *N*_{S}(*G*) is defined as the set of all elements *s* of *S* that map *G* onto itself by conjugation, *i.e*. all the elements *s* of *S* such that *sgs*^{−1} = *g*′, for all *g* and *g*′ belonging to *G*; this condition is synthetically written as *sGs*^{−1} = *G*. Two limiting cases may exist, namely: (ii) the *N*_{S}(*G*) coincides with *G*, *i.e*. the elements of *S* that map *G* onto itself are just the elements of *G*; (ii) the *N*_{S}(*G*) coincides with *S*, *i.e*. all elements of *S* map *G* onto itself. Evidently, *G* is always a of its a fact that explains the name `normalizer' itself.

Two types of normalizers are useful in crystallographic problems: *Euclidean normalizers* (also initially known as *Cheshire groups*) and *affine normalizers*. The difference between them is easily understood after recalling the corresponding types of mappings (transformations or functions).

The Euclidean and affine normalizers of a *G* are the normalizers obtained by taking as *S* the group of all Euclidean or affine mappings, *E* or *A*, respectively. They are the set of all elements *e* of *E* or *a* of *A* that map *G* onto itself by conjugation.

Each operation of the Euclidean *N*_{E}(*G*) maps the group *G* onto itself, and thus also the symmetry elements of *G*: it represents the *symmetry of the symmetry pattern*. Fig. 5 shows the symmetry elements of a *G* of type *P*222 as well as the in (001) projection. If we think of the symmetry of the distribution of these symmetry elements, it is easily shown that they repeat with symmetry *N*_{E}(*G*) = *Pmmm*. Moreover, the lattice translations in *N*_{E}(*G*) are all halved with respect to *G*. The Euclidean becomes however more symmetric in the case of the specialized metric. If two of the lattice parameters of *G* are equal (for example, *a* = *b*), then a fourfold axis arises in *N*_{E}(*G*) that relates the symmetry elements of *G*: *N*_{E}(*G*) is now of type *P*4/*mmm*. Finally, if *a* = *b* = *c*, *N*_{E}(*G*) becomes of type *Pm**m*. The existence of more than one type of Euclidean for the same type of depending on the metric, concerns the triclinic and monoclinic space groups as well as 38 types of orthorhombic groups, where two or three lattice directions may interchange if the corresponding lattice parameters become equal.

The affine *N*_{A}(*G*), being defined by the group of affine mappings, never depends on the metric and, when more than one type of Euclidean *N*_{E}(*G*) exists for the groups of type *G*, the affine *N*_{A}(*G*) always corresponds to the highest-symmetry *N*_{E}(*G*). In the above example, *N*_{A}(*P*222) = *Pm**m*, *a*/2,*b*/2,*c*/2. One could say that the symmetry of the symmetry elements of *P*222 for a general metric becomes more symmetric under geometric contraction, expansion, dilation, rotation …, *i.e*. under an that corresponds to modifying the lattice parameters towards a specialized metric. This process is repeated until the highest symmetry is obtained and the result is the affine Evidently, when only one Euclidean exists for a space-group type, it coincides with the affine normalizer.

For monoclinic and triclinic space groups, the affine normalizers are not symmetry groups and have to be described by a matrix–column pair and the corresponding restrictions on the coefficients.

Crystals belonging to space groups *G* whose *G*/*T* is isomorphic to a pyroelectric cannot have their origin fixed with respect to some appropriate The origin may be chosen at any point along one direction (for space groups of crystal classes 2, 3, 4, 6, *mm*2, 3*m*, 4*mm* and 6*mm*), in a plane (crystal class *m*) or anywhere in space (crystal class 1). The Euclidean normalizers for these space groups are not space groups themselves but contain continuous translations in one, two or three independent directions.

As an example of the use of normalizers and of their relation with the lattice complexes described in the previous section, we consider the position 12*h x*,½,0 in *P*3*m*, already discussed in dealing with lattice complexes. The point configurations corresponding to this position, which has inherent symmetry *Pm**m*, form cube-octahedra around the sites of a cubic primitive lattice. The Euclidean (which in this case coincides with the affine normalizer) is *N*_{E}(*P*3*m*) = *Im**m* with identical translation subgroups. The additional generator *t*(½½½) of *N*_{E}(*G*) with respect to *G* generates a second in the same and the two are said to be Euclidean-equivalent (they are evidently also affine-equivalent). There exist therefore two point configurations by which the same atomic distribution can be described, and only the other atoms in different Wyckoff positions define which of the two is actually occupied in the structure under investigation or, if both are occupied, by which atoms they are occupied. If however *x* takes the value ¼, the two point configurations coalesce in one, with inherent symmetry *Im**m*, *i.e.* the symmetry of the normalizer.

When, for each crystallographically independent type of atom there exist two or more Euclidean-equivalent point configurations, the same crystal structure can be described differently. To verify whether two apparently different crystal structures actually simply differ in their description, it is sufficient to verify whether the corresponding point configurations are Euclidean-equivalent or not. The case of crystals composed of only one type of atom is the simplest example: when the only independent atom occupies one *i.e.* a set of Wyckoff positions having a site-symmetry group that is conjugate under the Euclidean then there are *N* − 1 alternate equivalent descriptions, where *N* is the number of Wyckoff positions in the Evidently, in the case of crystals composed of only one type of atom, the that it occupies must correspond to a characteristic orbit (*E* = *G*).

An exhaustive presentation of normalizers, with several examples, is given in Chapter 15 of ITA and references therein. A didactic text has been published by Koch & Fischer (2006).

### 5. Derivative structures and symmetry relations

A *derivative structure* is any structure derived from another structure (*basic structure*) by the suppression of one or more sets of operations of the (Buerger, 1947). Basic structures are also known as *aristotypes* and derivative structures as *hettotypes* (Megaw, 1973). Two important kinds of derivative structures exist: *substitution structures* and *distortion structures.* In the former, two or more different kinds of atoms replace one kind of atom in the basic structure and consequently the space-group symmetry decreases; furthermore, some atomic sites that were equivalent in the basic structure may be divided into two or more different sites in the derivative structure. Distortion structures correspond to displacive phase transitions: the space-group type of the derivative structure, often called *daughter phase*, is a of the space-group type of the basic structure, often called *parent phase*.

The structure relationship between two structures whose space groups *G* and *H* are group–subgroup related (*G* *H*) can be analysed in terms of maximal subgroups. *H* is called a *maximal subgroup* of *G* if one cannot find an intermediate group *K* that is a of *G* and a of *H*. The relations between *G* and *H* can be classified in the following way:

1. *H* is a *translationengleiche* of *G* (*t* subgroup);

2. *H* is a *klassengleiche* of *G* (*k* subgroup),

2.1 a special case of *k* is when *H* belongs to the same type as *G*: it is called an *isomorphic* of *G* and is sometimes indicated as an *i* subgroup;^{6}

3. *H* can finally be a general of *G*, *i.e*. its translation is a of the translation of *G* and it also belongs to a crystal class that corresponds to a of the crystal class of *G*. In this case, Hermann's (1929) theorem shows that there exists a unique intermediate group *M*, which is a *t*-subgroup of *G* such that *H* is a *k*-subgroup of *M*, and suggests a privileged path from *G* to *H*.

The relation between *G* and *H* can eventually be subdivided into a number of steps *G* → *H*_{1} → *H*_{2} → … → *H*, each step involving either a *k*-subgroup or a *t*-subgroup. At each step, a can either split into several symmetry-independent positions, which keep the original or have its reduced; both changes may also happen simultaneously (Wondratschek, 1993). In a substitution structure, the may split, whereas, in a distortion structure, the in general is reduced when going from *H*_{j} to *H*_{j+1}, unless it is already low enough (Müller, 2005). Vol. A of *International Tables for Crystallography* gives part of the information necessary to build the relation, namely the maximal non-isomorphic subgroups and the isomorphic subgroups of *lowest* index. *International Tables for Crystallography* (2004), Vol. A1, which was published recently, gives the complete information: besides the above subgroups, it also gives the series of isomorphic subgroups, the origin shift relating the axial settings of *G* and *H* and the transformation of each Wyckoff position.

The structural relation is best represented in the form of a tree, introduced by Bärnighausen (1980), where each node consists of a group and the Wyckoff letter of the occupied positions, with the numerical values of the general coordinates, and the branches are arrows relating pairs of groups, labelled by the type of (*t*, *k*, *i*), the order of the followed – when these are not trivial – by the basis vectors of the in terms of those of the and by the origin shift.

The information necessary to build a Bärnighausen tree can be obtained either from Vol. A1 of *International Tables for Crystallography* or by means of the Bilbao Crystallographic Server at http://www.cryst.ehu.es/cryst/ (Aroyo, Perez-Mato *et al.*, 2006; Aroyo, Kirov *et al.*, 2006), in particular, using the routines *SUBGROUPGRAPH*, *HERMANN* and *WYCKSPLIT*.

#### 5.1. The example of the diamond–sphalerite–chalcopyrite substitution derivative structure

From the *Fd**m*) to that of chalcopyrite (*I*2*d*), two shortest paths exist, *Fd**m* → *F*3*m* → *I**m*2 → *P**n*2 → *I*2*d* and *Fd**m* → *I*4_{1}/*amd* → *I**m*2 → *P**n*2 → *I*2*d*. The first one passes through the of sphalerite (*F*3*m*). As a matter of fact, the structural relation can be separated into two steps: first, by replacing half of the C atoms by Zn atoms and the other half by S atoms, one obtains the structure of sphalerite; then, by further ordered replacement of Zn by Fe and Cu atoms, one gets the structure of chalcopyrite.

The first passage is straightforward, involving just a *t*-subgroup of order 2. Diamond has lattice parameter *a* = 3.566990 (3) Å (Hom *et al.*, 1975). With respect to `origin choice 1', the C atoms occupy the position 8*a* (0,0,0), whose site-symmetry group is 3*m*. The transformation to *F*3*m* needs an origin shift by (−1/8,−1/8,−1/8). The position 8*a* of *F*3*m* splits to 4*a* (0,0,0) and 4*c* (1/4,1/4,1/4) of *F*3*m*, but keeps the site-symmetry group, 3*m*. The lattice parameter of sphalerite is 5.415 (9) Å (Jumpertz, 1955); the large difference in the lattice parameters comes evidently from the size difference between the C atom on one side and the Zn and S atoms on the other side.

To obtain the chalcopyrite structure, a three-step pathway is needed.

The chalcopyrite lattice parameters are *a* = 5.2864 (8) and *c* = 10.4102 (8) Å (Kratz & Fuess, 1989), close to those calculated by the group–subgroup transformation; the difference comes obviously from the different size of the atoms which substitute in the derived structure. The atomic coordinates are Cu: 0,0,0 (4*a*), Fe: 0,0,½ (4*b*) and S: 0.257(1),¼, (8*d*). The generic coordinate *x* in position 8*d* is fairly close to the value ¼ obtained *via* the transition pathway. The Bärnighausen tree for the complete group–subgroup is shown in Fig. 6.

Detailed examples of applications of Bärnighausen trees are given in Müller (2004, 2005). Application to molecular crystals can be found in Gruber & Müller (1997) and Müller (1978, 1980). Rutherford (2001) presented the first application to organic crystals.

#### 5.2. Structural relationships between structures with no group–subgroup relations

In the case of structures with no group–subgroup relations, the structure of one phase can still be related to that of the other phase *via* a common (see *e.g*. Bärnighausen, 1980; Hoffmann & Pöttgen, 2001), without necessarily implying the existence of a transition pathway. The possibility of using a common instead has been suggested too (see *e.g*. Capillas *et al.*, 2007). The group–subgroup relation would then concern *G*1 and *H*1, as well as *G*2 and *H*2, where *H*1 and *H*2 are space groups of the same type as *H*. Finally, an affine transformation relating *H*1 and *H* on one side, and *H*2 and *H* on the other side, closes the structural relation.

### 6. Reticular theory of twinning

A twin is a modular structure at the crystal level (Ferraris *et al.*, 2004), *i.e*. a heterogeneous edifice consisting of the oriented association of two or more homogeneous crystals (individuals). The operation mapping the orientation of an individual onto that of another individual is called a *twin operation* and the lattice element about which this operation is performed is called a *twin element*. Mallard's law (Friedel, 1926) states that a is always a direct-lattice element, although it may sometimes be useful to use a reciprocal-lattice element instead, especially when unravelling the diffraction pattern of a twin.

Twinning is often simply regarded as a problem in the process of structure solution and *e.g*. Hahn & Klapper, 2003 and Grimmer & Nespolo, 2006). Here we give just a brief summary of the aspects that from the viewpoint of mathematical crystallography are more directly related to the above problems.

#### 6.1. The reticular theory of and the occurrence frequency of twins

In order for two individuals of the same compound to form a twin, the structure at the interface between them should fit as well as possible. Holser (1958) proposed to describe this interface by means of the 80 diperiodic (layer) groups: the intersection of the space groups of the individuals in the respective orientations should be a diperiodic group realized in the thin interface between them. This approach requires the knowledge of a thin section of the structure for the given orientation and a way to judge the goodness of fit, almost reducing the study of twins to a case-by-case analysis.

The reticular theory of twins, originally developed by the so-called `French school' (Friedel, 1926) takes the lattice as basic criterion to judge the goodness of fit and, consequently, to evaluate the probability of occurrence of a twin. Clearly, the use of the lattice instead of the complete structure results in a certain degree of idealization; on the other hand, it has the advantage of a much greater generality, while keeping the specificity of the individuals, through their lattice parameters. Concretely, the reticular theory of twins affirms that the probability of occurrence of a twin is directly related to the goodness of fit of the lattices of the individuals in the respective orientations (see also Hahn & Klapper, 2003). Because the structure of a crystal is a periodic repetition of the unit-cell content with the periodicity of the lattice, a good fit of the latter implies a good structural fit. The lattice nodes that are common, exactly or approximately, to the individuals in their respective orientations define a of the individual, which is called *twin lattice*. As `goodness of fit' is taken the degree of lattice overlap in the as measured by two parameters:

On the basis of these parameters, twins are classified in the following categories, where δ replaces ω in older classifications:

1. *n* = 1, δ = 0: twins by merohedry

2. *n* > 1, δ = 0: twins by reticular merohedry

3. *n* = 1, δ > 0: twins by pseudo-merohedry

4. *n* > 1, δ > 0: twins by reticular pseudo-merohedry.

Twins by ^{7}

The occurrence of twins with *complete* overlap of the lattices is not necessary: a *partial* overlap is sufficient. The occurrence of twins with obliquity ω > 0 (and thus δ > 0) shows that for a twin to occur an *exact* overlap of the lattices is not necessary: an *approximate* overlap is sufficient. Nevertheless, in general, the probability of occurrence of a twin is inversely related to the and to the obliquity, and empirical limits were also given, based on a large number of study cases: a of 6 and an obliquity of 6° were taken as borderline between `normal' (`Friedelian') twins and `exceptional' (`non-Friedelian') twins (Friedel, 1926).

That the above criteria are not absolute is shown by the existence of non-Friedelian twins that, although far less frequent than low-index low-obliquity ones, are nevertheless well represented. The reticular theory seemed unable to explain their occurrence in the same framework as Friedelian twins. An extension of this theory has however been proposed that rationalizes a number of non-Friedelian twins as *hybrid twins*. For *n* > 1 twins, the is a of the lattice of the individual whose cell is defined by the (axis, plane) and the lattice element (plane, direction) quasi-perpendicular to it, where `quasi' means within an acceptable obliquity, usually taken as the Friedelian value of 6°. For large there may exist more than one lattice element satisfying this criterion, and therefore more than one that may be chosen as The overall degree of lattice overlap should therefore take into account the lattice nodes defining *all* these sublattices because the restores all these nodes, although within a different degree of approximation, measured by the obliquity of each The ratio of the lattice nodes contained in the cell of the lowest-obliquity largest-index and the number of lattice nodes corresponding to all the sublattices defined in this way is a better estimation of the degree of lattice overlap and is termed *effective twin index*. By means of this approach, several high-index twins whose existence was previously difficult to explain on the basis of the reticular theory can now be rationalized as well (Nespolo & Ferraris, 2006).

#### 6.2. Twin point groups and the polychromatic symmetry of twins

The symmetry of twins is in

exactly like the morphological symmetry of crystals or the symmetry of their physical properties. Therefore, the symmetry of a twin is described by a (vector) which however is not just one of the 32 crystallographic point groups but rather an extension of them.The symmetry of a twin is the group formed by the intersection of the *oriented* point groups of the individuals augmented by the twin element(s). In the intersection group, only those symmetry elements are retained that, in the respective orientations of the individuals, are parallel. This gives a proper or trivial, of the group of the individual. The twin element(s), as well as those produced by the combination of the latter(s) with the symmetry elements of the intersection group, are then added and the complete symmetry of the twin is obtained. In this group, the operations no longer have the same nature: part of them map an individual onto itself, the others map an individual onto another one. This behaviour has an evident parallel in the polychromatic point groups, where some operations (`achromatic') exchange only parts of the object characterized by the same colour; others (`chromatic') exchange instead parts that have different colours; finally, a third type of operation (`partially chromatic') may exist, which exchanges a subset of the colours, leaving unchanged the others. The theory of polychromatic point groups can therefore be directly applied to twinned crystals; a polychromatic applied to a twin takes the name `twin (Nespolo, 2004).

In the case of merohedric twins, the twin ^{8} because all the symmetry elements of the individual are retained in the intersection group. In the general case, however, this is no longer true, the twin often being isomorphic to a *subgroup* of the crystal or even to a that is not in group–subgroup relation with it. In fact, depending on the orientation of the individuals, few or none of the symmetry elements of the individual may be retained, and the may be of a type not present in the of the individual. For example, the Japan/Verespatak twin of quartz, , has twin *m*′: none of the symmetry elements of the 321/622 of quartz is retained, and there are no mirror planes in it. A further specialized case is that when the is of the same type as the lost in the intersection group. In this case, the twin is isomorphic to the crystal the two groups being however differently oriented. For such a special case, the term of *twinning by reticular polyholohedry* was introduced (Nespolo & Ferraris, 2004).

#### 6.3. The effect of on the diffraction pattern

The diffraction pattern from a twinned crystal is typically the superposition of the diffraction patterns of the individuals. In other words, the intensities from each individual sum up without any phase relation. This of course depends on the crystal size and the wavelength used, and the diffraction behaviour cannot be used as an absolute criterion to distinguish between a twin and a modular structure (see the discussion in Nespolo *et al.*, 2004). Nevertheless, in a large majority of cases, non-interfering diffraction from the individuals is what one observes from a twinned crystal. This means that, in order to solve and refine the structure of a twinned crystal, the diffraction pattern must first of all be unravelled for its components.

Twins by ; Nespolo & Ferraris, 2000).

have their direct and reciprocal lattices completely overlapped. They are classified into three classes (Catti & Ferraris, 1976**Class I**: the is an inversion centre (or any other operation that is equivalent to an inversion centre under the of the individual). Within the limits of the validity of Friedel's law, the diffraction pattern from the twin cannot be distinguished from that of a single untwinned individual and the structure can be solved without taking into account, *provided that the correct space group has been chosen*; assignment of a crystal to a centrosymmetric when instead it lacks the centre of symmetry probably results only in apparent disorder or abnormal displacement parameters if the structure is to some extent pseudosymmetric. When the volume ratio of the individuals is sufficiently different from 1, the presence of inversion can be investigated by means of the Bijovet intensity ratio (Flack & Shmueli, 2007).

**Class IIA**: the does not belong to the Laue class of the crystal. The presence of may hinder a correct derivation of the symmetry from the diffraction pattern. In particular, when the number of individuals coincides with the order of the and the volumes of the individuals are equal, the symmetry of the diffraction pattern is higher than the Laue symmetry of the individual. A wrong can thus be assumed in the initial stage of the structure solution. The presence of can be investigated by statistical analysis of the intensities (Rees, 1980).

**Class IIB**: the situation is similar to that of class IIA but now the crystal has a specialized metric and the belongs to symmetry operations of the lattice corresponding to this specialized metric, not to that of the crystal For example, in a monoclinic crystal with β = 90°, the is a twofold axis parallel to [100] or [001] or a mirror plane normal to one of these directions. This type of has been called `metric (Nespolo & Ferraris, 2000) or `higher-order (Friedel, 1926).

Common to these three classes is that, following the perfect overlap of reciprocal lattices, the diffraction pattern from twins does not show special *Pa* (Koch, 1999).

Twins by reticular

are characterized by partial overlap of the lattices of the individuals. As a consequence, special corresponding to non-space-group absences, are commonly observed in the diffraction pattern. This is among the strongest alerts for the presence of although it does not uniquely come from OD structures, for example, also give non-space-group absences, typically along family rows indexed in the polytype axial setting.Twins by (reticular) pseudo-merohedry show diffraction splitting, which is more pronounced at high angles. The degree of splitting depends on the twin misfit and for very low values it may not be observable, resulting at most in a slight enlargement of the diffractions.

### 7. Topology of crystal structures

The term *topology* is used with different meanings in crystal chemistry. For example, an affine transformation connecting the structure of two polymorphs related by a displacive transformation can be described in topological terms, although the connectivity is modified because the number of bonds is not necessarily the same in the two structures. Here we use the term topology in a narrower sense, with reference to the connectivity of a crystal structure, *i.e*. the way in which the atoms are connected to each other. The latter, reduced to its minimal terms, is a set of atoms joined in a more or less complex way along privileged directions that we call chemical bonds. In the case of completely ionic structures, the bond itself is not directional but the packing of ions determines directions of minimal distances between ions of opposite charge, and these can be taken as `privileged directions'. A crystal structure can therefore be seen as a set of vertices (atoms) and edges (bonds) and is describable as an infinite undirected^{9} graph embedded into the three-dimensional point Euclidean space: it is called a *crystal structure graph*. There exists an isomorphism between the elements of the of the crystal structure (isometries) and the group of automorphisms of the crystal structure graph.^{10} Graph theory is commonly applied in chemistry to molecules. Its powerfulness in the analysis, description and foresight of crystal structures was realized only later.

The famous study of the Seven Bridges of Königsberg is regarded as the first paper in the history of graph theory (Euler, 1736). Euler's formula relating the number of edges, vertices and faces of a convex polyhedron was studied and generalized by Cauchy (1813) and L'Huillier (1861) and is at the origin of topology. Cayley (1875) developed the study of trees and linked his results with the contemporary studies of chemical composition. The fusion of the ideas coming from mathematics with those coming from chemistry is at the origin of a part of the standard terminology of graph theory. In particular, the term *graph* was introduced by the chemist Sylvester (1878*a*,*b*).

An instructive example of the application of graph theory to the analysis of crystal structures is the determination of isostructural compounds. As mentioned in the section about normalizers, the same crystal structure may sometimes be described in different ways, and to verify whether two apparently different crystal structures actually simply differ in their description it is sufficient to verify whether the corresponding point configurations are Euclidean-equivalent or not. We can generalize this concept to the case of different compounds whose crystal structures have the same topology. To reveal this relation, one has first of all to establish a mapping between atoms or groups of atoms, then verify whether the corresponding point configurations are affine-equivalent or not. One of the structures compared may turn out to be a derivative of the other, as is the case of the diamond–sphalerite–chalcopyrite described in the previous section. Graph theory is a powerful tool in such a task: once the mapping of atoms has been established, the bond pattern of the structures under consideration is translated into the corresponding crystal-structure graph. It is therefore the mapping of a graph onto another graph that can reveal whether two structures are isomorphic, derivative or unrelated. Two types of mappings between graphs should be considered:

1. *homeomorphism*, which is an isomorphism between spaces that respects topological properties; in particular, it maps neighbour points and distant points in one space on neighbour points and distant points, respectively, in the other space; in other words, homeomorphism is a continuous invertible bijective deformation between two topological spaces;

2. *homomorphism*, which is a relation between graphs preserving the operations; the special case of isomorphism, or bijective homomorphism, is of particular interest here.

Through homeomorphism, it can be shown that crystal graphs are topologically equivalent to a potentially enumerable set of fundamental nets (see *e.g*. Delgado-Friedrichs & O'Keeffe, 2003). Then, the search for isostructurality reduces to establishing a graph isomorphism, which can be obtained by finding some graph invariants. A typical example of such an invariant is the *quotient graph*. A *finite* quotient graph is obtained from an *infinite* structure graph by finding translationally equivalent points (atoms) and lines (bonds) and preserving their incidence relations. The result is called a quotient graph because it is analogous to the quotient (factor) group *G*/*T* between a and its translation The quotient graph retains the whole connectivity of the structure graph, namely the information about which point lattices are connected to each other by how many lines; however, the information about which individual point of a given lattice is joined to which individual point of another lattice is lost. As a consequence, different non-isomorphic graphs may have the same quotient graph (Klee, 1987). A system of labels can however be assigned to the edges in such a way that it uniquely determines the infinite structure up to isomorphism *via* the so-called *vector method* (Chung *et al.*, 1984). This opens the possibility of an algebraic representation of the quotient graph, which makes it possible to translate the process of comparing crystal structures into a symbolic computer language.

A graph can be represented by its *adjacency matrix*, which is a real square symmetric matrix with a row and a column for each vertex of the graph. The element corresponding to the *i*th row and the *j*th column gives the number of edges by which the two corresponding vertices are joined. Because the numbering of vertices is not unique, the adjacency matrix can be written in different ways, which differ by permutation of rows and/or columns, but do not affect the invariants of the matrix, such as determinant, trace and its eigenvalues. The latter are said to form the *spectrum of the graph*. The adjacency matrix is however not uniquely determined by the spectrum of its eigenvalues and, as a consequence, non-isomorphic graphs with the same spectral values may be found, which are called *cospectral* (Klee, 1987). The comparison of the spectra of the graph of two crystal structures is, alone, in general not sufficient to state that the two structures are isomorphic but it is sufficient to exclude that they are: a number of examples can be found in Eon (1998). Other graph invariants as possible means to establish isomorphism are presented in Eon (2002).

It must be emphasized that, in order to build a structure graph, the ) and generalizations (Hoppe, 1979; O'Keeffe, 1979). To judge up to what distance an atom still coordinates with its neighbours, or the closely related question of accounting for the bond-length distribution in irregular coordination polyhedra, different methods have been proposed, such as bond valence (see a review in Urusov, 1995), resonance (Boisen *et al.*, 1988; Rutherford, 1991, 1998*a*) and charge distribution (Hoppe *et al.*, 1989; Nespolo *et al.*, 1999, 2001).

Several applications of graph theory to crystal-chemical problems can be mentioned, like the analysis of possible polymorphs by means of Schlegel diagrams (Hoppe & Köhler, 1988) and the problem of finding atomic configurations satisfying neighbour-avoidance rules. The aluminium-avoidance rules in tetrahedra of aluminosilicates (Löwenstein, 1954) is an empirical rule subject to exceptions (Depmeier & Peters, 2004). Although far from being a general law of Nature, it is respected in a large number of structures. A graph-theoretical approach to the study of the distribution of Al and Si in a given structure according to this empirical rule was presented by Klee (1974*a*,*b*). A similar problem was addressed by Rutherford (1998*b*) to show that, in the one-dimensional ionic conductor [(CH_{3})_{2}N(CH_{2}CH_{2})_{2}O]Ag_{4}I_{5}, the energy of the system is minimized when Ag ions do not occupy neighbouring tetrahedra.

An exhaustive presentation of the applications of graph theory to crystallography and crystal chemistry will soon be available (Eon *et al.*, 2008).

### 8. Conclusions

Mathematical crystallography, far from having exhausted its task with the development of the space-group theory, not only represents one of the foundations of crystallography but is still a very active field of research, which nowadays extends its interests to cover several branches that were previously seldom considered in their interactions with crystallographic and crystal-chemical problems. This brief survey was limited to only some of the main topics of mathematical crystallography, but quite a few others, mentioned only *en passant*, like higher-dimensional crystallography and topology of non-Euclidean spaces, would have deserved a presentation as well. The reader is encouraged to follow the activities of the IUCr Commissions like MaThCryst and the Commission on Aperiodic Crystals to obtain a wider view of current and future developments in mathematical crystallography.

### Footnotes

^{1}The alternative meaning of `groupoid' introduced by Hausmann & Ore (1937), namely a set on which binary operations act but neither the identity nor the inversion are included, is nowadays called a magma (see *e.g.* Bourbaki, 1998).

^{2}The must be fulfilled not necessarily by the real layers but by their archetypes, *i.e.* the slightly idealized layers to which the real layers can be reduced by neglecting some distortions occurring in the true structure. The notion of becomes thus unequivocal only when it is used in an abstract sense to indicate a structural type with specific geometrical properties.

^{3}The weighted is obtained by assigning to each node of the a `weight' that corresponds to *F*(*hkl*) (Shmueli, 2001).

^{4}The translation *T* is a of *G*: the or quotient group *G*/*T* is the set of all cosets of *T* in *G*.

^{5}The role of extraordinary orbits was first addressed by Sándor (1968), who suggested extending the concept of `special positions' to positions having translational symmetry higher than that of the general position.

^{6}The as described here is limited to crystallographic equivalence. Two space groups of type *P*6_{1} and *P*6_{5} are not considered isomorphic, although they are affine equivalent.

^{7}It is emphasized that the expression `merohedral twins' often appearing in the literature is inappropriate: `merohedral' indicates the symmetry of an individual, not that of a twin (Catti & Ferraris, 1976).

^{8}We say that the twin is isomorphic with a given crystallographic not that it coincides with it because, although it has the same type of symmetry elements, some of these are chromatic. Only by neglecting the chromatic nature of its elements would a twin `coincide' with a crystallographic point group.

^{9}A graph can be undirected (a line from point *A* to point *B* is considered to be the same as a line from point *B* to point *A*) or directed, also called a digraph (the two directions are counted as being distinct arcs or directed edges).

^{10}An is an isomorphism (bijective mapping) from a mathematical object to itself.

### Acknowledgements

I wish to express my deep gratitude to M. I. Aroyo (Universidad del Pais Vasco), B. Souvignier (Radboud University Nijmegen), W. Fischer and E. Koch (Marburg University), J. Rutherford (NSTU Zimbabwe), S. Ďurovič (Slovak Academy of Sciences) and G. Ferraris (Torino University) for several useful discussions and for their help in improving the clearness and readability of this manuscript. The constructive remarks of three anonymous reviewers are gratefully acknowledged.

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