International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 1.10, pp. 245247
Section 1.10.2. Symmetry^{a}Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands 
Because the embedded periodic structure in n dimensions has lattice periodicity, it has ndimensional spacegroup symmetry as well. It is not a priori clear that such a symmetry group in the unphysical ndimensional space is relevant for the physical structure, but we shall show here that the superspace description is indeed useful for the description of quasiperiodic systems. First we shall discuss some of the structures of these higherdimensional space transformations.
Suppose the diffraction pattern has rotational symmetry. Consider for example an orthogonal transformation R that leaves the diffraction pattern invariant. In particular, any basis vector of the module is transformed into an element of the module, i.e. an integral linear combination of the basis vectors.Because the matrix M depends on R and acts in reciprocal space, we denote it by . The matrix has integer entries. Because the intensity of the diffraction pattern is not constant on circles around the origin (that would imply that one can not distinguish separate peaks), the orthogonal transformation R is of finite order. Then a theorem from group theory states that is similar to an ndimensional orthogonal transformation . The latter certainly has an invariant subspace: the physical space. Therefore, one can find a basis transformation S such that the matrix is conjugated to the direct sum of an orthogonal transformation R in and an additional orthogonal transformation in :We denote this orthogonal transformation in as or as a couple (). Clearly, the transformation leaves the embedded reciprocal lattice invariant. Moreover, this transformation leaves the direct lattice invariant as well. As always, the action of on the basis of for which form the reciprocal basis is then given byThis is the usual relation between the action on a basis and the action on the reciprocal basis.
By construction, the orthogonal transformation leaves the lattice invariant, and can therefore belong to the point group of a periodic structure with this lattice. In general, such a pointgroup element does not leave the periodic structure itself invariant, just as a point group in three dimensions does not leave a crystal with a nonsymmorphic space group invariant. One then has to combine the orthogonal transformation with a translation that in general does not belong to the lattice. Here a translation has components in physical as well as in internal space. A translation can be denoted by (). Then a general solid motion can be written asThe action of such a transformation on a point in superspace is given byIf such a transformation leaves the periodic array of atomic surfaces in superspace invariant, it is a symmetry transformation. In particular, the elements () of the translation group are such symmetry transformations.
The orthogonal transformations that leave the diffraction pattern invariant form a point group K, a finite subgroup of O(d), where d is the dimension of the physical space. All elements act on the basis of the Fourier module as in (1.10.2.1) and the matrices form a representation of the group K, an integral representation because the matrices have all integer entries, and reducible because the physical space is an invariant subspace for . Because K is finite, this representation is equivalent with a representation in terms of orthogonal matrices. Moreover, by construction leaves the ndimensional reciprocal lattice invariant. It is an ndimensional crystallographic point group. The components R of form a ddimensional point group , which is not necessarily crystallographic, and the components form an ()dimensional point group .
Consider as an example an IC phase with orthorhombic basic structure and one independent modulation wavevector along the c axis. Suppose that the Fourier module, which is of rank four, is invariant under the point group mmm. Then one has for the three generators Therefore, the corresponding matrices arewhich implies that the three generators of the fourdimensional point group are (), () and ().
The diffraction pattern of the standard octagonal tiling has rank four, basis vectors of the Fourier module areand the pattern is invariant under a rotation of and a mirror symmetry. The action of these elements on the given basis of the Fourier module isBy a basis transformation, one may bring these transformations into the formTherefore, the rotation in physical space is combined with a rotation in internal space in order to get a transformation that leaves a lattice invariant.
A threedimensional example is the case of a quasicrystal with icosahedral symmetry. For the diffraction pattern all spots may be labelled with six indices with respect to a basis with basis vectors with and . The rotation subgroup that leaves the Fourier module invariant is generated byMoreover, there is the central inversion . The sixdimensional representation of the symmetry group, which is the icosahedral group , is reducible into the sum of two nonequivalent threedimensional irreducible representations. A basis for this representation in the sixdimensional space is then given bywhich projects on the given basis in .
The pointgroup elements considered here are pairs of orthogonal transformations in physical and internal space. Orthogonal transformations that do not leave these two spaces invariant have not been considered. The reason for this is that the information about the reciprocal lattice comes from its projection on the Fourier module in physical space. By changing the length scale in internal space one does not change the projection but one would break a symmetry that mixes the two spaces. Nevertheless, quasicrystals are often described starting from an ndimensional periodic structure with a lattice of higher symmetry. For example, the icosahedral 3D Penrose tiling can be obtained from a structure with a hypercubic sixdimensional lattice. Its reciprocal lattice is that spanned by the vectors (1.10.2.8) where one puts c = 1. The symmetry of the periodic structure, however, is lower than that of the lattice and has a point group in reducible form. Therefore, we shall consider here only reducible point groups, subgroups of the orthogonal group O(n) which have a ddimensional invariant subspace, identified with the physical space.
The fact that the spaces and are usually taken as mutually perpendicular does not have any physical relevance. One could as well consider oblique projections of a reciprocal lattice on . What is important is that the intersection of the periodic structure with the physical space should be the same in all descriptions. The metric in internal space follows naturally from the fact that there is a finite group .
The quasiperiodic function in d dimensions can be embedded as lattice periodic function in n dimensions. The symmetry group of the latter is the group of all elements g (1.10.2.4) for whichThis group is an ndimensional space group G. It has an invariant subgroup of translations, which is formed by the lattice translations , and the quotient is isomorphic to the ndimensional point group K. However, not every ndimensional space group can occur here because we made the restriction to reducible point groups. For example, the ndimensional hypercubic groups do not occur in this way as symmetry groups of quasiperiodic systems.
The product of two superspace group elements isOn a lattice basis for , the orthogonal transformations and are integer matrices and the translations and are column vectors. The orthogonal transformations leave the origin invariant. The translations depend on the choice of this origin. For a symmorphic space group there is a choice of origin such that the translations a are lattice translations.
The pointgroup elements are reducible, which means that in the physical space one has the usual situation. If then the only intrinsic nonprimitive translations are those in screw axes or glide planes. An ndimensional orthogonal transformation can always be written as the sum of a number r of twodimensional rotations with rotation angle different from , a pdimensional total inversion and a qdimensional identity transformation. The integers r, p, q may be zero and . The possible intrinsic nonprimitive translations belong to the qdimensional space in which the identity acts. For the three examples in the previous section, the internal component of the nonprimitive translation for and in the first example can be different from zero, but that for in the same example is zero. For the octagonal case, only the second generator can have an intrinsic nonprimitive translation in the fourth direction, and for the icosahedral case the two generators have one twodimensional invariant plane and one pointwise invariant line in .
In the diffraction pattern of an IC phase one can distinguish between main reflections and satellites. A symmetry operation cannot transform a main reflection into a satellite. This implies that for these structures the reciprocal lattice of the basic structure is left invariant by the point group, and consequently the latter must be a threedimensional crystallographic point group. Therefore, the point groups for IC phases are the same as those for lattice periodic systems. They act in superspace as a representation of a threedimensional crystallographic point group. This is not true for an arbitrary quasiperiodic structure. The restriction in the general case comes from the requirement that the threedimensional point group must have a faithful integer matrix representation in superspace. There is a mathematical statement to the effect that the lowest dimension in which a pfold rotation can be represented as an integer matrix is given by the Euler function, the number of integers smaller than p that do not divide p. For example, for a prime number p this number is p − 1. This implies that if one restricts the rank of the Fourier module (i.e. the dimension of the superspace) to six, only values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14 and 18 are possible for p. The values 7, 9, 14 and 18 only occur for twodimensional quasiperiodic structures of rank six. Therefore, the allowable threedimensional point groups for systems up to rank six are limited to the groups given in Table 1.10.2.1. The possible superspace groups for IC modulated phases of rank four are given in Chapter 9.8 of Volume C of International Tables (2004). Superspace groups for quasicrystals of rank are given in Janssen (1988).

The notation of higherdimensional symmetry groups is discussed in two IUCr reports (Janssen et al., 1999, 2002).
References
International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Janssen, T. (1988). Aperiodic crystals: a contradictio in terminis? Phys. Rep. 168, 57–113.
Janssen, T., Birman, J. L., Dénoyer, F., Koptsik, V. A., VergerGaugry, J. L., Weigel, D., Yamamoto, A., Abrahams, S. C. & Kopsky, V. (2002). Report of a subcommittee on the nomenclature of ndimensional crystallography. II. Symbols for arithmetic crystal classes, Bravais classes and space groups. Acta Cryst. A58, 605–621.
Janssen, T., Birman, J. L., Koptsik, V. A., Senechal, M., Weigel, D., Yamamoto, A., Abrahams, S. C. & Hahn, Th. (1999). Report of a subcommittee on the nomenclature of ndimensional crystallography. I. Symbols for pointgroup transformations, families, systems and geometric crystal classes. Acta Cryst. A55, 761–782.