research papers
Report of a Subcommittee on the Nomenclature of nDimensional Crystallography. I. Symbols for pointgroup transformations, families, systems and geometric crystal classes†
^{a}Institute of Theoretical Physics, University of Nijmegen, NL6525ED Nijmegen, The Netherlands, ^{b}Department of Physics, City University of New York, New York, NY 10031, USA, ^{c}Institute of Physics, Moscow State University, Moscow 119899, Russia, ^{d}Department of Mathematics, Smith College, Northampton, MA 01063, USA, ^{e}Laboratoire de ChimiePhysique du Solide, Ecole Centrale Paris, Chatenay Malabry, F92295 France, ^{f}National Institute for Research in Inorganic Materials, Namiki, Tsukuba, Ibaraki, Japan, ^{g}Physics Department, Southern Oregon University, Ashland, OR 97520, USA, and ^{h}Institut für Kristallographie, RWTH Aachen, Aachen, D52066 Germany
^{*}Correspondence email: ted@sci.kun.nl
The notation of crystallography in arbitrary dimensions is considered. Recommended symbols for pointgroup transformations,
families and systems are presented.Keywords: Commission Report; nomenclature; pointgroup transformations; geometric crystal classes; families; systems.
1. Introduction
The derivation of all 230 threedimensional space groups by Fedorov (1891a,b) and Schoenflies (1891) became the geometrical and grouptheoretical basis for analysis developed by both Braggs following the discovery of Xray diffraction by von Laue in 1912. Spacegroup information based on these derivations became widely available to users in International Tables for Crystallography (1995) and its predecessor publications, hereafter ITC.
The mathematical formulation of the theory of space groups, for example by Bieberbach (1911), led naturally to their generalization in arbitrary dimensions. An algebraic treatment of the theory was given by Ascher & Janner (1965, 1968). Lists of transformations, point groups, lattices and space groups were derived for specific higher dimensions. Special cases were treated by Heesch (1929), Hurley (1951, 1966) and Janssen (1969) for fourdimensional space. The first complete list of all fourdimensional space groups was established by Brown et al. (1978). Partial results in five, six and seven dimensions have been derived by Plesken & Pohst (1977), Janner et al. (1983) and Plesken & Hanrath (1984).
The increasing use of threedimensional space groups for ). Although a complete overview of all possible applications is not yet available, it is useful to consider a unified nomenclature before the diversity of the fields and usages leads to widely different de facto nomenclatures and notations for the same objects. Concern for this situation led the Commission on Crystallographic Nomenclature of the International Union for Crystallography to establish a Subcommittee to study a possible system of notation.
during the early decades of this century made a nomenclature that would be widely accepted necessary. The notations of Schoenflies and of Hermann–Mauguin successfully filled this need. The former is often used by spectroscopists, the latter by crystallographers. Such a notation for higher dimensions is presently missing, although the number of fields in which the use of specific space groups in higher dimensions has become essential has increased steadily. This situation holds not only for mathematics where, for example, certain groups are associated with spaces of constant curvature but increasingly in describing the symmetry of aperiodic systems such as quasicrystals (Yamamoto, 1996The charge of the Subcommittee was to assess the extent to which the representational symbolism in use at that time (1990) in the field of ndimensional crystallography may have become so nonuniform that it is unacceptably ambiguous. If the results of this assessment so warranted, the Subcommittee was charged further with proposing a set of recommendations for a unified nomenclature and symbolism for use in ndimensional crystallography, following adequate discussion with other leaders in the field.
The first Report of the Subcommittee, presented herein, discusses the notation of pointgroup transformations and
Further, a proposal is made for the standard setting of representative lattices for four, five and sixdimensional crystal families. The notation for centring symbols, Bravais classes, and finally space groups in higher dimensions will be presented in a subsequent Report.2. Symbols and terms
The theory of crystallographic point groups, lattices and space groups may be formulated quite independently of their possible applications by treating it as a mathematical problem. From this viewpoint, it becomes a matter of the determination of all nonisomorphic crystallographic space groups in n dimensions. Applications enter in treating equivalence relations between groups or in introducing a nomenclature. Only 219 classes of space groups need be distinguished in three dimensions if any two space groups are regarded as equivalent if and only if they are isomorphic or, as shown by Bieberbach (1911), to be equivalent if they are conjugated subgroups of the general affine group. However, when the handedness of a structure is of importance, a finer classification becomes necessary, leading to the 230 classes because there are 11 threedimensional enantiomorphic pairs. This extension is a well known example of the effect of physical considerations on the choice of the equivalence definitions. Inferences arising from applications will not be discussed in this Report. Instead, grouptheoretical isomorphism will be used as an equivalence relation. The isomorphism classes of space groups lead naturally to and which can be grouped into Bravais classes, systems and families.
This Report deals with symbols for orthogonal transformations, O(n), receive the same symbol, and this is discussed in §4. In the same way, orthogonal transformations, which are elements of O(n), receive the same symbol when they are conjugate in this group. This means that the symbol does not give information about the orientation of the orthogonal transformation. Sometimes, the term type of orthogonal transformation is used for this Symbols for these transformations are treated in §3.
and with the choice of standard bases for lattices. The point groups in the same which are conjugate subgroups of the orthogonal groupThe and Table 3. An explanation of various terms can be found in ch. 8 of Vol. A of ITC. The relation of the various terms used is indicated in Fig. 1.
can be grouped together in pointgroup systems and families, see §4A standard basis may be indicated for an invariant lattice in each family. Up to orientation, such a standard basis is characterized by its g_{ij} = [(a)\vec]_{i}·[(a)\vec]_{j}, where [(a)\vec]_{i} () are the basis vectors. The standard bases are chosen to give a particularly simple form to the matrices of the point groups belonging to the family.
with elements,Some symbols used in this Report (the symbol as used in ITC is given in the second column):
3. Orthogonal transformations
Pointgroup transformations are orthogonal transformations in nD space. They are represented by orthogonal matrices on an orthogonal basis. A pointgroup transformation is crystallographic if there is an nD lattice that is left invariant. On a basis of such an invariant lattice, the pointgroup transformation is represented by an integer matrix.
An orthogonal transformation can be put in matrix form, which is the direct sum of 1D and 2D orthogonal matrices, by a real basis transformation. Any other orthogonal transformation in the same O(n), i.e. of the same type, can be brought into the same block form. The 1D blocks are , the 2D blocks are of the form
ofIf the transformation is of finite order q, the value of is , where p and q are mutually prime integers.
An integer matrix of finite order corresponds to an orthogonal transformation on the basis of an invariant lattice. Therefore, it can be brought into block form with blocks of dimension 1 or 2 by a real basis transformation. It can also be brought into block form by a rational basis transformation but the minimal dimension of the blocks is now generally larger. Rationally irreducible blocks can be written as a sum of blocks with eigenvalues for fixed q by a real transformation. The dimension of the rationally irreducible block is given by , the Euler function; for integer q is the number of integers coprime with q and smaller than q.
This relationship suggests a possible notation. The type of orthogonal transformation of finite order is given by the rational numbers p/q in the 2D blocks and the number of 1's and −1's in the block form. These numbers are unique up to a permutation of blocks. All orthogonal transformations that can be transformed into each other by a real basis transformation give the same numbers. In particular, all elements of one in O(n) are given the same characterizing numbers.
Considering the reduction by rational transformations, each rationally irreducible block of dimension higher than two can be reduced further to a sum of 2D blocks by a real transformation. In that case, the conjugate roots of unity occur simultaneously. They have the same q value. This forms the basis of the proposal by Hermann (1949) for a notation of crystallographic transformations. His notation gives the sequence of orders of the rationally irreducible blocks. For example, the sequence 321 indicates a transformation which in reduced form is the direct sum of a 2D threefold rotation, a 1D inversion (of order 2) and the 1D identity. Therefore, such a sequence corresponds to a 4D transformation. Hermann denotes the eightfold rotation with values p/q = 1/8, 3/8, 5/8, 7/8, which is a 4D operation, by `8'. This nomenclature is only applicable for crystallographic transformations and suppresses the information on the values of p. Therefore, we recommend a slightly different scheme.
The first principle is to use the same symbol for an orthogonal transformation in n dimensions as that in n+k dimensions obtained from the former by adding k 1's. The only exception is the unit element in n dimensions denoted by 1_{n}. The second principle is to retain the symbols in ITC in one, two and three dimensions, namely 1, m in one dimension, 2, 3, 4, 6 in two dimensions, and , , , in three dimensions.
All information on the number of eigenvalues +1 is hence suppressed in the symbol, except in the case of the unit element 1_{n}. If it is desired for some reason to indicate explicitly the dimension of the space in which a transformation acts, it is possible to add a number of digits 1. Then, 411 is clearly a 4D transformation, to be distinguished from the 3D 41. If there are only eigenvalues +1 and exactly one eigenvalue 1, then the symbol is m according to the second principle above. Pairs of eigenvalues 1 can be combined to give 2D twofold rotations written as 2. We call the number of eigenvalues different from unity the effective dimension of the orthogonal transformation.
A 3 (4 or 6) fold rotation in two dimensions can be written as 3 (4 or 6, respectively). The rotations 3 and 3^{2}, as well as the pairs 4, 4^{3} and 6, 6^{5} denote the same rotation type because 3 and 3^{2} can be transformed into each other by a real transformation. This is generally true: orthogonal transformations R and R^{1} have the same eigenvalues and, therefore, are of the same type.
A 5, 8, 10 or 12fold rotation is not crystallographic in two dimensions (there is no invariant lattice). Moreover, there are two different rotations with the same order: 5 and 5^{2}, which must be distinguished. The same holds for 8 and 8^{3}, 10 and 10^{3}, and for 12 and 12^{5}.
There are three different choices for 7, 9, 14 or 18fold rotations in two dimensions: 7, 7^{2} and 7^{3} and the triplets 9, 9^{2}, 9^{4}; 14, 14^{3}, 14^{5}; 18, 18^{5}, 18^{7}.
In general, for a qfold rotation in two dimensions, there are different rotations. For arbitrary dimensions, the rotation can be written as a sequence of numbers. For example, 75^{2}3 is equivalent to the direct sum of rotations 7, 5^{2} and 3. It exists in spaces of dimension 6 and higher and is of order 105. Its effective dimension is 6 and it can be written as
The same rotation in higherdimensional spaces is denoted by the same symbol since the digit 1 is omitted. If q becomes 10 or larger, it should be separated from the other digits by a thin space. If q becomes 22 or larger, there can be confusion whether this is 22 or two digits 2. Then one has to put the number in curly brackets: {22} is a tendimensional rotation, but 22 is a fourdimensional rotation, usually denoted by .
In the case of a crystallographic transformation, an eigenvalue is always accompanied by its conjugates, the eigenvalues with coprime with q. This allows the symbol for crystallographic transformations of dimension 4 or higher to be shortened. For example, the rotation 7 7^{2} 7^{3} is crystallographic and minimally of dimension 6. Because the six eigenvalues always come together, it is sufficient to give only one of them (with the smallest denominator value). The symbol is chosen as [7], standing for the longer symbol 7 7^{2} 7^{3}. This allows the crystallographic transformation [5] = 55^{2} to be distinguished from the noncrystallographic transformation 55, another 4D transformation.
A transformation with determinant 1 contains an odd number of eigenvalues 1. It is customary in three dimensions to denote the product of the inversion and a qfold rotation () by an overbar: is a 3D transformation obtained as the product of and the rotation 4. Its matrix is the direct sum of the 2D matrix 4^{3} (which is of the same type as 4) and the number 1.
An exception is the product of a twofold rotation and , because this is a mirror, denoted by m , not by . It should be noted that the transformation with determinant 1 denoted by is the product of and the rotation 3, but at the same time it is the sum of the rotation 6^{5} and 1. On the other hand, is the sum of 3^{2} and 1. (In ITC, the corresponding group is the same as 3/m.) In the past, one has chosen a symbol using multiplication () rather than summation (3/m = 3+m).
This notation is not different in higher dimensions: an odddimensional transformation with determinant 1 is the product of a rotation and the inversion in the odddimensional space and can be denoted by a bar above the digits corresponding to the qfold rotation ().
This transformation is of order 10. The same symbol can be used for a (5+k)dimensional transformation, obtainable from this 5D operation by adding k diagonal terms 1. This is equivalent to obtaining from the rotation 55^{2} on multiplication by the total inversion in a space of dimension equal to the effective dimension of 55^{2} (which is four) plus one.
A general nD rotation (with determinant +1) is denoted by a series of k digits. This is the symbol already in use for a 2kdimensional rotation. The number of omitted digits 1 is n2k. The symbol for a general orthogonal transformation of determinant 1 is , again with k digits. This is the symbol for a (2k+1)dimensional operation. In nD space, there are still n2k1 omitted digits 1.
Inversion is a special case. It is either a rotation (in even dimensions) or has a determinant equal to 1 (in odd dimensions). For even dimensions, this element may be written as a series of 2's. To stress its special role, the transformation in n dimensions is denoted as .
A consequence of this system of notation is that a property sometimes used in three dimensions is not generally true: if a qfold rotation belongs to the Laue group, then the transformation also belongs to it. Indeed it is true that, if A is a rotation without twofold component and of dimension 2k, then is a (2k+1)dimensional orthogonal transformation and . In particular, in three dimensions, of course. However, this is no longer true in an arbitrary dimension. In general, q is not equal to . In four dimensions, the transformation 4 is actually 411. Then its opposite 411 = 42 is also a rotation and belongs to the same Laue group as 4. But in four dimensions is actually and has determinant 1, which cannot be changed by multiplication with . The 4D operators 411 and , in short notation 4 and , leave different lattices invariant. It should be noted that one can make the remark already in two dimensions. There, 4 is not but 4 itself.
It is noteworthy that, in some cases, the cyclic groups generated by q and (for integer q) belong to the same (family preserving ) and, in other cases, to different families (family breaking ). Wondratschek (1998) observed that, in odd dimensions, the groups generated by q and belong to the same if and only if there is exactly one constituent +1 (and no constituent 1) in the operation q and, hence, one 1 and no +1 in . In all other cases, is family breaking.
The Subcommittee has discussed this point extensively. The reasons for recommending the use of the overbar in this Report are, in short, the following.

Recommendation 1. An orthogonal transformation of finite order that can be written after a suitable real basis transformation as the direct sum
should be written in full as
with m/2 2's when m is even,
with (m1)/2 2's when m is odd, and with , but it is
when l = n (only eigenvalues +1), and
when m = n (only eigenvalues 1). The order is such that
Recommendation 2. For consistency with current practice, m should be used for , 2 is used exceptionally instead of , and is used instead of .
Recommendation 3. In the case of a crystallographic orthogonal transformation, a short form is obtained by combining conjugate blocks with the same q into the symbol [q]. This represents the sequence
where are the integers that are coprime with q and smaller than q/2.
Recommendation 4. The symbol does not depend on the number of eigenvalues +1. However, if there are only eigenvalues +1 and no others, then the transformation should be denoted by 1_{n}. If it is desirable to indicate explicitly the dimension of the transformation, as many digits 1 may be added as there are eigenvalues equal to +1.
The symbols for elements of crystallographic point groups in dimensions up to 6 are given in Table 1. The sequence mentioned in the heading is the Hermann symbol.

4. Geometric crystal classes
An nD is a of the orthogonal group in n dimensions. If it leaves a subspace invariant, it can, by a suitable real basis transformation, be put into reduced form, i.e. all elements are simultaneously in the form of direct sums of blocks having the same dimension for all elements. If the dimensions of the invariant subspaces are , then the is said to be ()reducible or ()Rreducible. All point groups in the same have the same reducibility character. As in the case of point transformations, point groups are denoted by a symbol for the space in which they act effectively. In particular, the symbols in two and three dimensions should remain the same as those used in ITC. These symbols remain the symbols for the For higherdimensional spaces, the symbols can, to a large extent, be based on the reducibility of the just as for point transformations. Some of these ideas follow those of Weigel et al. (1987, 1993), who have proposed symbols for point groups in four, five and six dimensions.
A special case for a m blocks if it is the of m subgroups consisting of unit matrices, except for one of the m blocks. An example is the 3D group 4/mmm with generating matrices
occurs in reduced form withThis is the 4mm of order eight generated by the first two matrices and the group m of order two generated by the third matrix. In this situation, the group is the of subgroups acting only in one of the m mutually orthogonal invariant subspaces. The symbol for such a group can be related to those for lowerdimensional point groups:
of theIn the example, group 4/mmm would have the alternative symbol . However, preserving the principle that standard notation be retained in dimensions up to three, the preferred symbol remains 4/mmm. Choosing a basis of an invariant lattice that makes the simple structure evident is very useful because the matrices remain simple.
An (m_{1} + m_{2})reducible H is always a of an (m_{1} +m_{2})dimensional , where K_{i} is an m_{i}dimensional The elements h of H can be written as pairs (h_{1}, h_{2}) of elements of K_{1} and K_{2}, respectively. The elements h_{1} generate K_{1} and the elements h_{2} generate K_{2}. H is then said to be a subdirect product of K_{1} and K_{2}.
If the elements of the subdirect product of K_{1} and K_{2} are denoted by (h_{1}, h_{2}), then the elements (h_{1}, e) form an invariant H_{1} and the elements (e,h_{2}) an invariant H_{2} of K_{1} and K_{2}, respectively. The quotient groups K_{1}/H_{1} and K_{2}/H_{2} are isomorphic. All subdirect products of K_{1} and K_{2} are obtained by considering all invariant subgroups H_{1} and H_{2}, taking those pairs for which K_{1}/H_{1} and K_{2}/H_{2} are isomorphic and considering all isomorphisms from one to the other.
A special case of a subdirect product is that in which K_{1} = H_{1} and K_{2} = H_{2}. In this case, the subdirect product is simply a of H_{1} and H_{2}, and is denoted by .
Suppose a group is the subdirect product of an ndimensional K_{1} and an mdimensional K_{2}. The elements are pairs (h_{1}, h_{2}), where the elements h_{1} form the group K_{1} and the elements h_{2} the group K_{2}. To each element of K_{1}, there corresponds at least one element of K_{2} and vice versa. Its symbol may be taken as that of K_{1} with corresponding elements of K_{2} placed between parentheses. For example, the of generated by the pairs (4,m) and (m,1) is the subdirect product of 4mm and m and can be denoted by 4mm(m1m). The second m in 4mm is the product of 4 and the first m. Correspondingly, the second m in the parentheses is the product of m and 1. Because the symbols for subdirect products tend to become rather long, one writes preferentially 4m(m1) instead of 4mm(m1m). If the H_{1} of the elements of K_{1} associated with the identity in K_{2} is not trivial, then the corresponding symbols in parentheses are 1's and analogously for H_{2}. The example 4/mmm could be written as 4mm1(111m), but in this case the shorter notation is .
The subgroups 4 and 4mm of the 3D group are 2D and are denoted by 4 and 4mm, respectively. The 4/m has the same reducibility character as 4/mmm and is written as . The other subgroups are more complicated. For the group 422, the elements in the first 2D block form the group 4mm but the three elements indicated (4, 2 and 2) are associated with 1, 1 and 1, respectively. The symbol is therefore 4mm(1mm). The group is generated by
Similarly, the group may be written as 4mm(mm1). Generators are
However, because the notation for threedimensional point groups should be retained, this procedure is only used in higher dimensions and 422 is written instead of 4mm(1mm) and instead of 4mm(mm1).
The symbols for one, two and threedimensional groups are given in Table 2, both with their current ITC symbol and according to the rules of higherdimensional crystallography. The symbols for the Rreducible 4D point groups are given in Table 3. They are (3+1), (2+2), (2+1+1) or (1+1+1+1)reducible. The are combined into pointgroup systems, the ordering of which is discussed in the next section.


More than one symbol is occasionally proposed because there are arguments in favour of various options, just as in three dimensions has been chosen instead of 3/m, although there are good arguments for both. It should become clear in practice which choice is more convenient. Another example is the choice between 4/mmm, the present standard choice in three dimensions, and , which is more in agreement with the nomenclature we propose for higher dimensions. A third example is the case of the of an octagonal 4D structure, which in projection gives an octagonal tiling. Its is generated by the transformation [8] and a twofold rotation. The transformation [8] is the same as 88^{3} and can be written in block form. The twofold rotation acts in both subspaces as a mirror. The may hence be seen either as a subdirect product 8m(8^{3}m) of 8m and 8m or as a group generated by [8] and m(m), in which case the notation becomes [8]m(m). Again, another possibility would be to indicate the generators, which act in the whole xyzu space and the 2D yz plane, respectively. The is spanned by the generated by [8] and a generated by a transformation 2. It is the smallest group containing both subgroups. This can be denoted by giving the symbols for the generating groups, separated by a dot. An alternative symbol for the group 8m(8^{3}m) in this scheme is [8].2. This cannot be used for the 3D point groups: one still writes 622 (instead of 6.2) and 432 (instead of 4.3). In general, if the K is the smallest group containing its subgroups K_{1} and K_{2}, it is denoted by K_{1}.K_{2}. In the particular case that K is the of its subgroups K_{1} and K_{2}, it is denoted by K_{1}×K_{2}. Whether [8].2 or 8m(8^{3}m) or [8]m(m) is used is a matter of taste and the most convenient notation should again become clear in practice. Finally, it might be convenient to denote a cyclic by the symbol for its generator. Here, one should be careful with symbols of transformations already in use as symbols for threedimensional point groups (222, 32, 422, 622, 432). The cyclic groups based on these transformations should have another notation: in standard notation, , , , and , respectively. A number of alternatives are mentioned in Table 3. Synoptic lists relating the entries in Table 3 to other notations have been compiled by several authors. In particular, Veysseyre (1998) and, separately, Wondratschek (1998) have related the entries in Table 3 to the crystal classes in Brown et al. (1978).
Irreducible point groups and point groups having invariant subspaces of dimension higher than three cannot use only the symbols of one, two and threedimensional point groups. Recommendations for these cases will be given in a subsequent Report.
Recommendation 5. Symbols for are symbols for representative point groups. The following symbols should be used in point groups that are the subdirect product of point groups in at most three dimensions.

5. Families
The set of nD lattices can be partitioned into Bravais classes, lattice systems (Bravais systems in ch. 8 of ITC) and families. Lattice systems and families may likewise be characterized by the of the holohedral point groups.
Lattices for a certain , together with the metric tensors for a chosen standard basis. 6D families are given in Table 8, but their metric tensors are given for a selected set only in Table 9. The others can be read off from the preceding tables. If the holohedral is , the matrix for the is the direct sum of the metric tensors of the lowerdimensional groups. For example, in the 6D family 6D_32 (i.e. number 32 in Table 8), the holohedral is . The for 4(4) can be found in Table 4, and 4m is a twodimensional group. Therefore, the for is
have a with a number of free parameters. This number is the dimension of the subspace of the space of secondrank symmetric tensors that transform with the identity representation. This dimension is the same for all point groups in the same geometric class. A specific choice of standard basis can show the reducibility of the space. Such a standard basis can be used for the definition of a For each family, a lattice may be chosen such that each other lattice of the family may be considered as related by centring. As examples, all orthorhombic lattices in three dimensions can be obtained by adding centring translations to a primitive orthorhombic lattice; the lattices of the trigonal/hexagonal family can be obtained similarly from a hexagonal lattice. The choice of standard basis is very important for the description of structures. Recommended symbols for four, five and six dimensions are given in Tables 4 to 9There are two types of new families in six dimensions. First, those involving nfold rotations for which the Euler function has the value 6: the dihedral groups of order 28 and 36. Second, the four Rirreducible families. The latter are the families of the hypercubic lattice (in the series 2D_4mm, , 4D_23, 5D_32) with holohedral of order 46 080 = 2^{6} 6!, the rhombohedral lattice (in the series 2D_6mm, 3D_bcc, 4D_21, 5D_31) of order 10 080 = 2×7!, a lattice in the series 2D_6mm, 4D_22 with holohedral of order 10 368 = 3!×12^{3}, and a symmetrized version of lattice 6D_78 with holohedral of order 240 (Plesken & Hanrath, 1984), respectively.
‡ order of the maximal holohedral §The order of the holohedral of the lattice spanned by the standard basis is 384. 





The order in which the families are presented is preferably such that families having properties in common are grouped together. Unfortunately, there are conflicting criteria. There is no natural order. Because there is a hierarchy: family – system – geometric
– arithmetic – the numbering could be based on this hierarchy and in that case a numbering of the families is necessary. Simple criteria are connected to the number of free parameters in the and to the order of the holohedral The most general lattice has the maximal number of free parameters and the smallest holohedral whereas a lattice becomes increasingly special if these numbers decrease and increase, respectively.This order is related, but not identical, to that based on the reducibility of the arithmetic holohedral via rational basis transformations and that via real basis transformations. The type of reducibility is given by writing the dimension as the sum of the dimensions of the irreducible components. In three dimensions, the holohedral of a cubic lattice is 3irreducible, that of an orthorhombic lattice is 1+1+1reducible. In four dimensions, a lattice left invariant by a generated by the transformation [8] is 2+2reducible if real transformations are considered, but if rational transformations only are allowed then it is irreducible. Further, there is a difference between rational and integer equivalence as well. In three dimensions, the orthorhombic P lattice is fully 1+1+1reducible if integer matrices are used but the orthorhombic F lattice is reducible but not fully reducible. However, this difference becomes relevant only if centrings are considered. This point will be discussed in a following Report. An alternative ordering of the families takes this reducibility as the starting point. Families with the same dimensions as the irreducible components of the holodral point groups are then grouped together. A comparison of the two schemes is given in Fig. 2 for the 4D families.
The holohedral is reducible if it is equivalent to a direct sum of irreducible representations. A distinction is necessary between equivalenceThe reducibility scheme lists the dimensions of the irreducible components. This scheme can be refined as follows. Suppose that the holohedral K can be written as an external product . If one gives the reduction scheme for each of the components by putting it in parentheses, the reduction scheme gives more information. For example, in three dimensions, the holohedral point groups of the triclinic, monoclinic and orthorhombic families are all 1+1+1reducible. Addition of the new information gives the reduction scheme (1+1+1), (1+1)+1 or 1+1+1, respectively.
In the opinion of the Subcommittee, both schemes for ordering the families have advantages and disadvantages but there is a preference for the scheme chosen, primarily because the number of free parameters in the et al. (1998) have recently introduced, in a survey of algorithms used in the computer program CARAT, an alternative grouping of crystal families in n dimensions based on the decomposition pattern, group–subgroup relations and metric tensors in their Section 2.4. The main issue in this section of the recommendations, however, is the choice of the standard basis.
plays an important rôle. Neither of the schemes allows for a natural grouping of families in all cases. OpgenorthA recommendation for the choice of the standard basis is given in Table 4 for four dimensions, in Table 7 for five dimensions, and in Table 9 for six dimensions. In general, the chosen standard basis for a family is that of a lattice with the highest symmetry. Sometimes this is not the most convenient choice. In the 4D family No. 23, there are two lattice systems, one is the hypercubic system with of order 384, the other one with of order 1152. The transformations of the groups of the first system have a very simple form when described on a basis of the hypercubic lattice. In each row and column, there is exactly one nonzero element. If one chooses a basis in the other system this is not the case. However, point groups that do not leave the hypercubic lattice invariant are represented by noninteger matrices if one uses the hypercubic basis. Therefore, we propose using the hypercubic basis for all point groups belonging to pointgroup system 23_1, and the second basis (that of the lattice with holohedral of order 1152) for the point groups in system 23_2 only. The alternatives are given in Table 5.
The full list of 4D families was given for the first time by Brown et al. (1978). Comparison of the numbering in that publication with the numbering in Table 4 is given in Table 6.
Recommendation 6. Recommended symbols for families are the same as those for corresponding holohedral point groups.
Recommendation 7. The family order is first according to the number of free parameters in the (in decreasing order), and second according to the order of the holohedral (in increasing order).
Recommendation 8. The recommended standard basis for each family is chosen such that the reducibility becomes maximally evident. Orthogonal invariant subspaces are ordered according to decreasing orders of the corresponding point groups. The for a is the cell of the corresponding family. As an alternative, a special basis for each in the family may be used. The recommended bases and metric tensors for four, five and sixdimensional families are given in Tables 4, 7 and 9.
6. Relation with symmetries of aperiodic crystals
nD crystallography as such has become a topic for research. On the other hand, nD space groups are widely used for the description of aperiodic crystals. In this section, we discuss the relation between the general problem and the applications to aperiodic structures. It should be noted that the use of nD crystallographic groups is not the only method for describing aperiodic structures. Colour symmetry or Fourier transforms in 3D space are also used. However, these are strongly related to a description in terms of higherdimensional crystallography. Here we only note one of the applications of the groups discussed in this Report. We do not specifically recommend this method, but simply draw attention to the fact that the symbols used by many crystallographers for quasiperiodic systems are closely related to those that arise naturally for nD groups with a reducible point group.
A latticeperiodic function in n variables gives an aperiodic structure if the space is intersected by an oblique subspace. Consider a function f(x_{1}, x_{2}) of two variables which is periodic in its two arguments: f(x_{1} +1,x_{2}) = f(x_{1}, x_{2} +1) = f(x_{1}, x_{2}). Take two real numbers and such that is irrational and define a function g(x) of one variable through . Its Fourier series follows from that of f(x_{1}, x_{2}):
This expression is a special case of a quasiperiodic function
where the set M^{*} is defined as
If n is larger than the dimension of the space, the function is not lattice periodic but is quasiperiodic. The set M^{*} is not a but is the projection of an nD . A vector [(k)\vec] in M^{*} can be written as the projection and the nD reciprocallattice vector [(k)\vec]_{s} is a linear combination of basis vectors: . On the other hand, if the diffraction pattern has intensities that are the square of the modulus of a function on a set M^{*}, then there is an nD periodic function that corresponds to the nD Fourier transform:
This function has lattice periodicity in n dimensions and the lattice has as its Moreover, as can be seen from the definition, the restriction of the nD function to the hyperplane on which the is projected is just the starting function . Consequently, a quasiperiodic function is the restriction of an nD lattice periodic function to physical space, and the number of dimensions is equal to the number of indices h_{i} necessary for indexing the diffraction pattern.
Because there is a onetoone correspondence between the function in physical space and the function in higherdimensional space, which is essentially given by its value in the nD with an associated nD Because the construction is based on a projection from higherdimensional space, it is intuitively clear that symmetry operators cannot mix the physical space and the space of the additional dimensions. This statement can be made rigorous. It implies that the point groups are necessarily Rreducible. A then consists of pairs () of an orthogonal transformation in physical space and another in the additional space. The elements R form a finite K_{E} and the elements a finite . Therefore, the point groups for aperiodic structures are subdirect products of K_{E} and .
the symmetry of the quasiperiodic structure can be identified with that of the higherdimensional structure. This symmetry group is anThe literature contains hundreds of examples of such aperiodic crystals. Two follow:
(i) K_{2}SeO_{4} has a modulated phase in a certain temperature region. The structure in this phase can be described as derived from a basic structure with Pcmn via a periodic displacement modulation with wavevector . The diffraction pattern needs four indices, because a diffraction vector can be written as
The lattice periodic function in four dimensions has a m_{x},1), (m_{y},1) and (m_{z},1). Such a is the subdirect product of the 3D mmm and the 1D m. In the recommended notation, it is the group . In the literature for modulated structures, it usually has the symbol .
generated by the pairs ((ii) The quasicrystalline phase of AlMnPd has a diffraction pattern with icosahedral symmetry, requiring six indices. The number of dimensions for the n = 6. The is the subdirect product of the icosahedral point groups in the physical and the 3D additional space. It is generated by pairs 5(5^{2}), and m(m). Therefore, the recommended notation for the is . In the literature, it is generally denoted by , which is shorter but is also ambiguous because it is the symbol for a 3D and it suppresses important information as the components in additional space are not indicated. The nomenclature for aperiodic systems can therefore benefit from the more general approach discussed in this Report in terms of nD crystallography. On the other hand, the notation schemes are not so different that it is difficult to change from one to the other.
is hence7. Conclusions and recommendations
Recommendations for the symbols of ndimensional orthogonal transformations, ndimensional the order of ndimensional families, and the standard bases in four, five and six dimensions are given at the end of §§2, 3 and 4. Specific recommendations for dimensions up to six are given in the tables.
The recommendations are summarized as follows.

Footnotes
†Established 23 February 1990 by the IUCr Commission on Crystallographic Nomenclature, with all members including A. J. C. Wilson (Chairman) and W. Opechowski appointed 7 May 1990, under general ground rules outlined in Acta Cryst. (1979), A35, 1072. A. J. C. Wilson resigned 1 July 1993, W. Opechowski 25 February 1992 and H. Wondratschek 10 August 1996. T. Janssen was appointed Chairman 1 July 1993. Professor Opechowski died 27 September 1993 and Professor Wilson 1 July 1995. Original version of the Report received by the Chair of the Commission as ex officio member of the Subcommittee on 5 January 1997, in revised form 16 March 1998, final version accepted by the Commission 16 November 1998.
‡Chairman.
Acknowledgements
We thank Subcommittee advisors G. Chapuis, N. D. Mermin, R. Veysseyre and E. J. W. Whittaker for useful comments. We thank, in particular, former Subcommittee member H. Wondratschek for many important contributions.
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