Volume 70 Received 11 March 2013  Double antisymmetry and the rotationreversal space groups^{a}Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania, 16803, USA, and ^{b}Department of Physics, The Eberly College of Science, The Pennsylvania State University, Penn State Berks, PO Box 7009, Reading, Pennsylvania, 19610, USA Rotationreversal symmetry was recently introduced to generalize the symmetry classification of rigid static rotations in crystals such as tilted octahedra in perovskite structures and tilted tetrahedra in silica structures. This operation has important implications for crystallographic group theory, namely that new symmetry groups are necessary to properly describe observations of rotationreversal symmetry in crystals. When both rotationreversal symmetry and timereversal symmetry are considered in conjunction with spacegroup symmetry, it is found that there are 17 803 types of symmetry which a crystal structure can exhibit. These symmetry groups have the potential to advance understanding of polyhedral rotations in crystals, the magnetic structure of crystals and the coupling thereof. The full listing of the double antisymmetry space groups can be found in the supplementary materials of the present work and at http://sites.psu.edu/gopalan/research/symmetry/ . 
Rotationreversal symmetry (Gopalan & Litvin, 2011) was introduced to generalize the symmetry classification of tilted octahedra perovskite structures (Glazer, 2011). The rotationreversal operation, represented by 1* (previously represented by 1^{}) was compared by analogy to the well known timereversal operation, represented by 1' (see Fig. 1). Although reversing time in a crystal is not something that can be performed experimentally, it is nonetheless useful for describing the magnetic symmetry of a crystal structure and this magnetic symmetry description has important consequences which can be observed by experiment (Opechowski, 1986). Likewise, rotationreversal symmetry is useful in describing the symmetry of a crystal structure composed of molecules or polyhedral units. For example, the structure conventionally described in Glazer notation (Glazer, 1972) as a_{o}^{+} a_{o}^{+} c_{o}^{+} and classified with the group Immm1' is now classified with the rotationreversal group I4*/mmm*1' (No. 4206). An a_{o}^{+} a_{o}^{+} c_{o}^{+} structure with a spin along the z direction in each octahedron was formerly classified with the group Im'm'm and is now classified with the group I4*/mm'm'* (No. 16490). The rotationreversal space groups used in this new classification of tilted octahedra perovskites are isomorphic to, i.e. have the same abstract mathematical structure, as double antisymmetry space groups (Zamorzaev & Sokolov, 1957a,b; Zamorzaev, 1976). As rotationreversal space groups are isomorphic to double antisymmetry space groups, in the present work we will use the terminology and notation associated with double antisymmetry space groups.
 Figure 1 Identity (1) and antiidentities (1', 1* and 1'*) of the rotationreversal and timereversal space groups. 
Double antisymmetry space groups are among the generalizations of the crystallographic groups which began with Heesch (1930) and Shubnikov (1951) and continued to include a myriad of generalizations under various names as antisymmetry groups, cryptosymmetry groups, quasisymmetry groups, color groups and metacrystallographic groups (see reviews by Koptsik, 1967, 1968; Zamorzaev & Palistrant, 1980; Opechowski, 1986; Zamorzaev, 1988). Only some of these groups have been explicitly listed, e.g. the blackandwhite space groups (Belov et al., 1955, 1957a,b) and various multiple antisymmetry (Zamorzaev, 1976) and color groups (Zamorzaev et al., 1978). While no explicit listing of the double antisymmetry space groups has been given, the number of these groups, and other generalizations of the crystallographic groups, have been calculated (see Zamorzaev, 1976, 1988; Zamorzaev & Palistrant, 1980; Jablan, 1987, 1990, 1992, 1993a,b, 2002; Palistrant & Jablan, 1991; Radovic & Jablan, 2005).
In §2, we shall define double antisymmetry space groups and specify which of these groups we shall explicitly tabulate. This is followed by the details of the procedure used in their tabulation. In §3, we set out the format of the tables listing these groups. §4 describes example diagrams of double antisymmetry space groups (Fig. 4a: No. 8543 C2'/m*, Fig. 4b: No. 16490 I4*/mm'm'*, Fig. 4c: No. 13461 Ib'*c'a'). §5 gives the computational details of how the types were derived.
Zamorzaev & Palistrant (1980) have calculated not only the total number of types of double antisymmetry space groups, but they have also specified the number in subcategories. We have found errors in these numbers. Consequently, the total number of types of groups is different than that calculated by Zamorzaev & Palistrant (1980). This is discussed in §6.
Space groups, in the present work, will be limited to the conventional threedimensional crystallographic space groups as defined in Volume A of International Tables for Crystallography (Hahn, 2006). An antisymmetry space group is similar to a space group, but some of the symmetry elements may also `flip' space between two possible states, e.g. (r, t) and (r, t). A double antisymmetry space group extends this concept to allow symmetry elements to flip space in two independent ways between four possible states, e.g. (r, t, ), (r, t, ), (r, t, ) and (r, t, ). A more precise definition will be given below.
To precisely define antisymmetry space groups, we will start by defining an antiidentity. An operation, e.g. 1', is an antiidentity if it has the following properties:
(a) Selfinverse: 1'·1' = 1 where 1 is identity.
(b) Commutivity: 1'·g = g·1' for all elements g of E(3).
(c) 1' is not an element of E(3).
E(3) is the threedimensional Euclidean group, i.e. the group of all distancepreserving transformations of threedimensional Euclidean space. A space group can be defined as a group such that the subgroup composed of all translations in is minimally generated by a set of three translations with linearly independent translation vectors. We can extend this to define antisymmetry space groups as follows:
Let an ntuple antisymmetry space group be defined as a group such that the subgroup composed of all translations in G is minimally generated by a set of three translations with linearly independent translation vectors and P is minimally generated by a set of n antiidentities and is isomorphic to .
Thus, for single antisymmetry space groups, P is minimally generated by just one antiidentity, for double, two, for triple, three, and so forth. It should be noted that the above definition could be generalized to arbitrary spaces and coloring schemes by changing and P, respectively, but that is beyond the scope of the present work.
Let the two antiidentities which generate P for double antisymmetry space groups be labeled as 1' and 1*. The product of 1' and 1* is also an antiidentity which will be labeled 1'*. The coloring of 1', 1* and 1'* is intended to assist the reader and has no special meaning beyond that. Double antisymmetry has a total of three antiidentities: 1', 1* and 1'*. Note that these three antiidentities are not independent because each can be generated from the product of the other two. So although we have three antiidentities, only two are independent and thus we call it `double antisymmetry' (more generally nantisymmetry has 2^{n}  1 antiidentities). 1' generates the group 1' = {1,1'}, 1* generates the group 1* = {1,1*}, and together 1' and 1* generate the group 1'1* = {1,1',1*,1'*}. For double antisymmetry space groups, P = 1'1*.
Fig. 2 shows how the elements of 1'1* multiply. To evaluate the product of two elements of 1'1* with the multiplication table given in Fig. 2(a), we find the row associated with the first element and the column associated with the second element, e.g. for 1'·1*, go to the second row, third column to find 1'*. To evaluate the product of two elements of 1'1* with the Cayley graph given in Fig. 2(b), we start from the circle representing the first element and follow the arrow representing the second, e.g. for 1'·1*, we start on the red circle (1') and take the blue path (1*) to the green circle (1'*).
 Figure 2 (a) Multiplication table of 1'1*. To evaluate the product of two elements, we find the row associated with the first element and the column associated with the second element, e.g. for 1'·1*, go to the second row, third column to find 1'*. (b) Cayley graph generated by 1', 1* and 1'*. To evaluate the product of two elements, we start from the circle representing the first element and follow the arrow representing the second, e.g. for 1'·1*, we start on the red circle (1') and take the blue path (1*) to the green circle (1'*). 
When a spatial transformation is coupled with an antiidentity, we shall say it is colored with that antiidentity. This is represented by adding ', * or '* to the end of the symbol representing the spatial transformation, e.g. a fourfold rotation coupled with timereversal (i.e. the product of 4 and 1') is 4'.
We shall say that all double antisymmetry groups can be constructed by coloring the elements of a colorblind parent group. In the case of a double antisymmetry space group, the colorblind parent group, Q, is one of the crystallographic space groups. There are four different ways of coloring an element of Q, namely coloring with 1, 1', 1* or 1'* which we shall then refer to as being colorless, primed, starred or primestarred, respectively. Let Q1'1* be the group formed by including all possible colorings of the elements of Q, i.e. the direct product of Q and 1'1*. Since Q1'1* contains all possible colorings of the elements of Q, every double antisymmetry space group whose colorblind parent is Q must be a subgroup of Q1'1*.
Every subgroup of Q1'1* whose colorblind parent is Q is of the form of one of the 12 categories of double antisymmetry groups listed in Table 1. The formulae of Table 1 are represented visually in Appendix A using Venn diagrams.

As an example, consider applying the formulae in Table 1 to point group 222. 222 has four elements: {1,2_{x},2_{y},2_{z}}. 222 has three index2 subgroups: {1,2_{x}}, {1,2_{y}} and {1,2_{z}} which will be denoted by 2_{x}, 2_{y} and 2_{z}, respectively. The subscripts indicate the axes of rotation. Applying the formulae in Table 1 yields the following:
Category (1): Q
1. Q = 222 Q = {1,2_{x},2_{y},2_{z}}
Category (2): Q + Q1'
2. Q = 222 Q1' = {1,2_{x},2_{y},2_{z},1',2_{x}',2_{y}',2_{z}'}
Category (3): H + (Q  H)1'
3. Q = 222, H = 2_{x} Q(H) = {1,2_{x},2_{y}',2_{z}'}
4. Q = 222, H = 2_{y} Q(H) = {1,2_{x}',2_{y},2_{z}'}
5. Q = 222, H = 2_{z} Q(H) = {1,2_{x}',2_{y}',2_{z}}
Category (4): Q + Q1*
6. Q = 222 Q1* = {1,2_{x},2_{y},2_{z},1*,2_{x}*,2_{y}*,2_{z}*}
Category (5): Q + Q1' + Q1* + Q1'*
7. Q = 222 Q1'1* = {1,2_{x},2_{y},2_{z},1',2_{x}',2_{y}',2_{z}',1*,2_{x}*,2_{y}*,2_{z}*, 1'*,2_{x}'*,2_{y}'*,2_{z}'*}
Category (6): H + (Q  H)1' + H1* + (Q  H)1'*
8. Q = 222, H = 2_{x} Q(H)1* = {1,2_{x},2_{y}',2_{z}',1*,2_{x}*,2_{y}'*,2_{z}'*}
9. Q = 222, H = 2_{y} Q(H)1* = {1,2_{x}',2_{y},2_{z}',1*,2_{x}'*,2_{y}*,2_{z}'*}
10. Q = 222, H = 2_{z} Q(H)1* = {1,2_{x}',2_{y}',2_{z},1*,2_{x}'*,2_{y}'*,2_{z}*}
Category (7): H + (Q  H)1*
11. Q = 222, H = 2_{x} Q{H} = {1,2_{x},2_{y}*,2_{z}*}
12. Q = 222, H = 2_{y} Q{H} = {1,2_{x}*,2_{y},2_{z}*}
13. Q = 222, H = 2_{z} Q{H} = {1,2_{x}*,2_{y}*,2_{z}}
Category (8): Q + Q1'*
14. Q = 222 Q1'* = {1,2_{x},2_{y},2_{z},1'*,2_{x}'*,2_{y}'*,2_{z}'*}
Category (9): H + (Q  H)1* + H1' + (Q  H)1'*
15. Q = 222, H = 2_{x} Q{H}1' = {1,2_{x},2_{y}*,2_{z}*,1',2_{x}',2_{y}'*,2_{z}'*}
16. Q = 222, H = 2_{y} Q{H}1' = {1,2_{x}*,2_{y},2_{z}*,1',2_{x}'*,2_{y}',2_{z}'*}
17. Q = 222, H = 2_{z} Q{H}1' = {1,2_{x}*,2_{y}*,2_{z},1',2_{x}'*,2_{y}'*,2_{z}'}
Category (10): H + (Q  H)1' + H1'* + (Q  H)1*
18. Q = 222, H = 2_{x} Q(H)1'* = {1,2_{x},2_{y}',2_{z}',1'*,2_{x}'*,2_{y}*,2_{z}*}
19. Q = 222, H = 2_{y} Q(H)1'* = {1,2_{x}',2_{y},2_{z}',1'*,2_{x}*,2_{y}'*,2_{z}*}
20. Q = 222, H = 2_{z} Q(H)1'* = {1,2_{x}',2_{y}',2_{z},1'*,2_{x}*,2_{y}*,2_{z}'*}
Category (11): H + (Q  H)1'*
21. Q = 222, H = 2_{x} Q(H){H} = {1,2_{x},2_{y}'*,2_{z}'*}
22. Q = 222, H = 2_{y} Q(H){H} = {1,2_{x}'*,2_{y},2_{z}'*}
23. Q = 222, H = 2_{z} Q(H){H} = {1,2_{x}'*,2_{y}'*,2_{z}}
Category (12): H K + (H  K)1* + (K  H)1' + (Q  (H + K))1'*
24. Q = 222, H = 2_{y}, K = 2_{z} Q(H){K} = {1,2_{x}'*,2_{y}*,2_{z}'}
25. Q = 222, H = 2_{x}, K = 2_{z} Q(H){K} = {1,2_{x}*,2_{y}'*,2_{z}'}
26. Q = 222, H = 2_{x}, K = 2_{y} Q(H){K} = {1,2_{x}*,2_{y}',2_{z}'*}
27. Q = 222, H = 2_{z}, K = 2_{y} Q(H){K} = {1,2_{x}'*,2_{y}',2_{z}*}
28. Q = 222, H = 2_{z}, K = 2_{x} Q(H){K} = {1,2_{x}',2_{y}'*,2_{z}*}
29. Q = 222, H = 2_{y}, K = 2_{x} Q(H){K} = {1,2_{x}',2_{y}*,2_{z}'*}.
Note that although 29 double antisymmetry point groups are generated from using 222 as a colorblind parent group, they are not all of distinct antisymmetry pointgroup types, as is explained in §2.3. In the above example, there are only 12 unique types of groups, as discussed further on. Two additional examples, using point group 2/m and space group Cc, are given in Appendix B.
Since the index2 subgroups of the crystallographic space groups are already known and available in International Tables for Crystallography Volume A, applying this set of formulae is straightforward. If applied to a representative group of each of the 230 crystallographic spacegroup types, 38 290 double antisymmetry space groups are generated. These 38 290 generated groups can be sorted into 17 803 equivalence classes, i.e. double antisymmetry spacegroup types, by applying an equivalence relation.
The well known 230 crystallographic spacegroup types given in International Tables for Crystallography Volume A (Hahn, 2006) are the proper affine classes of space groups (`types' is used instead of `classes' to avoid confusion with `crystal classes'). The equivalence relation of proper affine classes is as follows: two space groups are equivalent if and only if they can be bijectively mapped by a proper affine transformation (proper means chiralitypreserving) (Opechowski, 1986). In the literature, the `spacegroup types' are often referred to as simply `space groups' when the distinction is unnecessary.
For the present work, we will use `double antisymmetry spacegroup types' to refer to the proper affine classes of double antisymmetry space groups. This is consistent with Zamorzaev's works on generalized antisymmetry (Zamorzaev & Sokolov, 1957a,b; Zamorzaev, 1976, 1988; Zamorzaev et al., 1978; Zamorzaev & Palistrant, 1980).
As an example, we consider the proper affine equivalence classes of the 29 Q = 222 double antisymmetry groups generated using the formulae given in Table 1. Only 12 such classes exist, one in each category. For categories (1), (2), (4), (5) and (8), the reason for this is that there is only one group generated in each to begin with. For categories (3), (6), (7), (9), (10) and (11), there are three groups generated which are related to each other by 120° rotations (e.g. {1,2_{x},2_{y}',2_{z}'} = 3_{xyz}·{1,2_{x}',2_{y},2_{z}'}·3_{xyz}^{1} = 3_{xyz}^{1}·{1,2_{x}',2_{y}',2_{z}}·3_{xyz}) and therefore they are members of the same equivalence class. For category (12), the six generated groups are all in the same equivalence class because {1,2_{x}'*,2_{y}*,2_{z}'} = 3_{xyz}·{1,2_{x}*,2_{y}',2_{z}'*}·3_{xyz}^{1} = 3_{xyz}^{1}·{1,2_{x}',2_{y}'*,2_{z}*}·3_{xyz} = 4_{x}·{1,2_{x}'*,2_{y}',2_{z}*}·4_{x}^{1} = 4_{x}·3_{xyz}·{1,2_{x}',2_{y}*,2_{z}'*}·3_{xyz}^{1}·4_{x}^{1} = 4_{x}·3_{xyz}^{1}·{1,2_{x}*,2_{y}'*,2_{z}'}·3_{xyz}·4_{x}^{1}. This is demonstrated with pointgroup diagrams in Fig. 3.
 Figure 3 Demonstration of proper affine equivalence of Q(H){K} groups generated for Q = 222 using pointgroup diagrams (stereographic projections). 
Proper affine equivalence and other definitions of equivalence are discussed in Appendix C.
Double antisymmetry spacegroup types of categories (1) through (11) of Table 1 are already known or easily derived. The group types of category (1) are the well known 230 conventional spacegroup types. The groups of categories (2), (4), (5) and (8) are effectively just products of the groups of category (1) with 1', 1*, 1'1* and 1'*, respectively. The groups of category (3) are the well known blackandwhite space groups (Belov et al., 1955, 1957a,b) [also known as type M magnetic space groups (Opechowski, 1986)]. The groups of categories (7) and (11) are derived by substituting starred operations and primestarred operations, respectively, for the primed operations of category (3) groups. And finally, the groups of categories (6), (9) and (10) are products of the groups of categories (3), (7) and (11), respectively, with 1*, 1' and 1'*, respectively.
For the groups of category (12) we have used the following fourstep procedure:
(i) For one representative group Q from each of the 230 types of crystallographic space groups, we list all subgroups of index 2 (Aroyo, Kirov et al., 2006; Aroyo, PerezMato et al., 2006; Aroyo et al., 2011).
(ii) We construct and list all double antisymmetry space groups Q(H){K} for each representative group Q and pairs of distinct subgroups, H and K, of index 2. This step results in 26 052 Q(H){K} groups.
(iii) For every pair of groups Q_{1}(H_{1}){K_{1}} and Q_{2}(H_{2}){K_{2}} where Q_{1} and Q_{2}, H_{1} and H_{2} and K_{1} and K_{2} are pairwise of the same spacegroup type, we evaluate the proper affine equivalence relation to determine if Q_{1}(H_{1}){K_{1}} and Q_{2}(H_{2}){K_{2}} are of the same double antisymmetry spacegroup type.
(iv) From each set of groups belonging to the same type, we list one representative double antisymmetry space group.
The serial number, symbol and symmetry operations of a representative group of each of the 17 803 double antisymmetry spacegroup types are given in the supplementary material `Double Antisymmetry Space Groups.pdf'.^{1} The double antisymmetry spacegroup symbols are based on the HermannMauguin symbol of the colorblind parent space group, e.g. C2'/m* is based on C2/m.
The first part of the symbol gives the lattice centering (or more precisely the translational subgroup). If there are no colored translations in the group, then this part of the symbol is given as P (primitive), C (Cface centered), A (Aface centered), I (body centered), F (allface centered) or R (rhombohedrally centered). If there are colored translations, then P, C, A, I, F or R is followed by three color operations in parentheses, e.g. C(1,1'*,1')2/m'*. These three color operations denote the coloring of a minimal set of generating lattice translations indicated in Table 2. For example, consider C(1,1'*,1')2/m'*: 1 is in the first position, 1'* is in the second position and 1' is in the third position. Looking up the lattice symbol `C' in the first column of Table 2, we find that the first, second and third positions correspond to t_{[100]}, t_{[001]} and t_{[0]}, respectively. The translations of C(1,1'*,1')2/m'* are therefore generated by t_{[100]}, t_{[001]}'* and t_{[0]}'.

The second part of the symbol gives the remaining generators for the double antisymmetry space group. This is also based on the corresponding part of the HermannMauguin symbol of the colorblind parent space group. The Seitz notation of each character is given in `secondPartOfSymbolGenerators.pdf' in the supplementary material.
Finally, if the group is a member of category (2), (4), (5), (6), (8), (9) or (10), then 1', 1*, 1'1*, 1*, 1'*, 1', 1'* or 1'*, respectively, is appended to the end of the symbol.
The Computable Document Format (CDF) file `Double antisymmetry space groups.cdf' (supplementary material) provides an interactive way to find the symbols and operations of double antisymmetry space groups. The file is opened with the Wolfram CDF Player which can be downloaded from http://www.wolfram.com/cdfplayer/ . After opening the file, click `Enable Dynamics' if prompted. Provide the necessary input with the dropdown menus.
A tutorial with screenshots is given in `Double antisymmetry space groups CDF tutorial.pdf (supplementary material).
In the PDF file `Double Antisymmetry Space Groups.pdf' (supplementary material), the 17 803 double antisymmetry space groups are listed sequentially. The first portion of the file contains links to each group entry. These links are sorted by the colorblind parent group, e.g. C2'/m* is listed under the spacegroup number of C2/m (i.e. `SG. 12').
The first line of each entry gives the sequential serial number (1 through 17 803), the double antisymmetry spacegroup symbol and the Xray diffraction symmetry group (i.e. the symmetry group obtained by removing all of the starred and primestarred operations and changing all of the primed operations to colorless operations). The second line gives the number of the colorblind parent group, the double antisymmetry point group and the crystal system. The remaining lines give the symmetry operations of the group: a set of coset representatives of the group with respect to the translational subgroup generated by translations of the conventional unit cell. These symmetry operations are given in International Tables Volume A notation and Seitz notation. Three examples from the listings of double antisymmetry space groups are given in Table 3.

The `machinereadable' file `DASGMachineReadable.txt' (supplementary material) is intended to provide a simple way to use the double antisymmetry space groups in code or software such as MatLab or Mathematica. The structure of the file is given in the supplementary material file called `Using the Machine Readable File.pdf'. The file `Import DASGMachineReadable.nb' (supplementary material) has been provided to facilitate loading into Mathematica.
Symmetry diagrams have been made for the example double antisymmetry space groups listed in Table 3. These diagrams are intended to extend the conventional spacegroup diagrams such as those in International Tables for Crystallography Volume A.
In Fig. 4(a), double antisymmetry space group No. 8543, C2'/m*, is projected along the b axis. In Fig. 4(b), double antisymmetry space group No. 16490, I4*/mm'm'*, is projected along the c axis. In Fig. 4(c), double antisymmetry space group No. 13461, Ib'*c'a', is projected along the c axis. As with Fig. 4(a), the symbols in these diagrams are naturally extended from those used for conventional space groups in International Tables for Crystallography Volume A.
 Figure 4 Example double antisymmetry spacegroup diagrams. (a) No. 8543 C2'/m*, (b) No. 16490 I4*/mm'm'*, (c) No. 13461, Ib'*c'a'. The legend relates to part (a). 
The majority of the computation was performed in Mathematica using 4 × 4 augmented matrices to represent spacegroup operations and unitcell transformations (more generally, affine transformations). The use of augmented matrices to represent spacegroup operations and unitcell transformations is described in International Tables for Crystallography Volume A, Chapters 5.1 and 8.1. These matrices were downloaded for each spacegroup type from the GENPOS tool on the Bilbao Crystallographic Server, http://www.cryst.ehu.es/cryst/get_gen.html (Aroyo, Kirov et al., 2006; Aroyo, PerezMato et al., 2006; Aroyo et al., 2011). The standard setting was used for each of the 230 spacegroup types as given in International Tables for Crystallography Volume A.
Every spacegroup operation can be broken up into a linear transformation R and a translation t which transform the coordinates (r_{1}, r_{2}, r_{3}) into (r_{1}', r_{2}', r_{3}'):
It is convenient to condense this linear transformation and translation into a single square matrix called an augmented matrix:
Note that the final row is necessary to make a square 4 × 4 matrix but contains no information specific to the transformation. When using these augmented matrices to represent spacegroup operations, the product of two spacegroup operations is evaluated by matrix multiplication, i.e. g_{1}g_{2} is performed by multiplying the matrices which represent g_{1} and g_{2} to get a product matrix which is also an augmented matrix that represents a spacegroup operation. The inverse of a spacegroup operation is represented by the matrix inverse of the operation's augmented matrix. The sign of the determinant of an augmented matrix representing a spacegroup operation determines if it is proper (orientationpreserving). A positive determinant means that it is proper.
For a set of 4 × 4 matrices S and a 4 × 4 matrix A, we will denote the set formed by the similarity transformation of each element of S by A as , i.e. .
To evaluate the formulae in Table 1 for any given colorblind parent space group Q, the index2 subgroups of Q are needed. The index2 subgroups of a space group are also space groups themselves. As such, every index2 subgroup of space group Q must be one of the 230 types of space groups. Using this, we can specify an index2 subgroup of Q by specifying the type of the subgroup (1 to 230) and a transformation from a standard representative group of that type. This is to say that H, a subgroup of Q, can be specified by a standard representative group H_{0} and the transformation T such that . Note, this is just the usual linear algebra changeofbasis formula; we are simply using T to transform from the standard conventional basis to the basis which makes H a subgroup of Q.
Q, H and H_{0} are each represented by a set of 4 × 4 real matrices. T is represented by a single 4 × 4 real matrix. For every element h in H there is an element h_{0} in H_{0} such that h = Th_{0}T^{  1}. Thus, we make the set of matrices representing H by computing Th_{0}T^{  1} for each matrix in the set representing H_{0}.
The index2 subgroup data were downloaded from the Bilbao Crystallographic Server. Altogether there are 1848 index2 subgroups among the 230 representative space groups. Each entry (out of 1848) consisted of: a number between 1 and 230 for the spacegroup type of Q, a number between 1 and 230 for the spacegroup type of H_{0} and a 4 × 4 matrix for T.
The proper affine equivalence relation can be defined as: two groups, G_{1} and G_{2}, are equivalent if and only if G_{1} can be bijectively mapped to G_{2} by a proper affine transformation, a. Expressed in mathematical shorthand, this is
where is the proper affine equivalence relational operator, is logical equivalence (`is logically equivalent to'), is the existential quantification of a (`there exists a'), means `an element of', : means `such that' and is the group of proper affine transformations. An affine transformation is the combination of a linear transformation and a translation. A proper affine transformation is an affine transformation that preserves chirality.
For evaluating the proper affine equivalence of a pair of Q(H){K} groups, if the spacegroup types of Q, H and K of one of the Q(H){K} groups are not the same as those of the other, then the proper affine equivalence relation fails and no further work is necessary. For a pair of Q(H){K} groups where they are the same, we have derived a method to evaluate proper affine equivalence based on affine normalizer groups. To begin this derivation, we can expand the proper affine equivalence for Q(H){K} groups to
where denotes logical conjunction (`and'). The subscripts have been omitted for Q_{1} and Q_{2} because we are testing the equivalence of Q(H){K} groups where . We can use the definition of the proper affine normalizer group of Q, i.e. ^{2} (Opechowski, 1986), to get
H_{1} and H_{2} are mapped by proper affine transformations, T_{H1} and T_{H2}, respectively, from a standard representative group, H_{0}, as follows: and . Likewise, K_{1} and K_{2} are mapped by proper affine transformations, T_{K1} and T_{K2}, respectively, from a standard representative group, K_{0}, as follows: and . Thus, by substitution, is equivalent to , which can be rearranged to . By applying the definition of a normalizer group again, we find that and , which rearrange to and , respectively. Therefore, the proper affine equivalence of Q(H_{1}){K_{1}} and Q(H_{2}){K_{2}} is logically equivalent to the existence of a nonempty intersection of , and :
This simplifies the problem of evaluating the equivalence relation to either proving that the intersection has at least one member or proving that it does not. To do this, we applied Mathematica's builtin `FindInstance' function. As with the subgroup data, the normalizer group data were downloaded from the Bilbao Crystallographic Server. As previously discussed, the formulae in Table 1 generate 38 290 double antisymmetry space groups when applied to all 230 representative space groups. With the aid of Mathematica, these 38 290 double antisymmetry space groups were partitioned by the equivalence relation given in equation (3) into 17 803 proper affine equivalence classes, i.e. 17 803 double antisymmetry spacegroup types.
This method can be easily generalized to other types of antisymmetry and color symmetry. For example, for antisymmetry groups formed from one index2 subgroup, such as Q(H) groups, the condition simply reduces to the following:
For finding the double antisymmetry spacegroup types, only conditions for Q(H) and Q(H){K} groups are necessary. This is because it is trivial to map these results to all the other categories of double antisymmetry space groups. This normalizer method is demonstrated in Appendix B to derive all double antisymmetry spacegroup types where Q = Cc.
The total number of types of double antisymmetry space groups listed by the present work is 17 803. The total number of Q(H){K} types listed by the present work is 9507. These values differ from those given by Zamorzaev & Palistrant (1980). We have found four fewer Q(H){K} types where Q = Ibca (No. 73 in International Tables for Crystallography Volume A). Since there are only a small number of discrepancies between our listing and the numbers calculated by Zamorzaev & Palistrant, each will be addressed explicitly.
Zamorzaev & Palistrant (1980) gave a list of double antisymmetry spacegroup generators in noncoordinate notation (Koptsik & Shubnikov, 1974). For Ibca [21a in Zamorzaev & Palistrant (1980)] the following generators are used:
a, b, , , and can be interpreted, in Seitz notation, as (1100), (1010), (1), (2_{z}0), (m_{x}0) and (2_{x}0), respectively.
Zamorzaev & Palistrant (1980) couple these generators with antiidentities to give generating sets for double antisymmetry space groups. However, unlike the more explicit listing given in the present work, Zamorzaev & Palistrant give only generating sets and only those that are unique under the permutations of the elements of 1'1* that preserve the group structure, i.e. the automorphisms of 1'1*. Because of this concise method of listing generating sets (only 1846 Q(H){K} generating sets are necessary), a single generating set given by Zamorzaev & Palistrant (1980) can represent up to six types under the proper affine equivalence relation. The six possible types correspond to the six automorphisms of 1'1*.
The automorphisms of 1'1* correspond to the possible permutations of the three antiidentities of double antisymmetry: (1', 1*, 1'*), (1', 1'*, 1*), (1*, 1', 1'*), (1*, 1'*, 1'), (1'*, 1', 1*) and (1'*, 1*, 1'), i.e. . Zamorzaev & Palistrant give the number of types represented by each line, but not which automorphisms must be applied to get them. This is discussed in the supplementary material file called `Color Automorphisms of Double Antisymmetry.pdf'. There are only two lines of generators from Zamorzaev & Palistrant for which their resulting number of Q(H){K} types differs from this work. These lines are given in Table 4.

Applying the six automorphisms of 1'1* to Ibc'a'* from the first line of Table 4, we get Ibc'a'*, Ibc'a*, Ibc*a'*, Ibc*a', Ibc'*a* and Ibc'*a'. According to Zamorzaev & Palistrant, these are six distinct types whereas there are actually only three distinct types. Ibc'a'* and Ibc'*a' are of type No. 13460. Ibc'a* and Ibc*a' are of type No. 13462. Ibc*a'* and Ibc'*a* are of type No. 13450. Applying the six automorphisms of 1'1* to Ib*c'*a' from the second line of Table 4, we get Ib*c'*a', Ib'*c*a', Ib'c'*a*, Ib'*c'a*, Ib'c*a'* and Ib*c'a'*. According to Zamorzaev & Palistrant, these correspond to two distinct types whereas they are actually all the same type, No. 13447.
Consequently, the generators in the first line of Table 4 map to three types and the generators in the second line map to one type, not six and two, respectively. Thus, there are four fewer Q(H){K} types than the number given by Zamorzaev & Palistrant (1980), i.e. 9507 rather than 9511. This error likely affects the number of higherorder multiple antisymmetry groups calculated by Zamorzaev & Palistrant as well. We conjecture that there are 24 fewer nontrivial triple antisymmetry space groups than calculated by Zamorzaev & Palistrant but that the numbers for other multiple antisymmetries are correct. If we are correct, this would mean that the numbers in the final column of Table 1 of `Generalized Antisymmetry' by Zamorzaev (1988) should read 230, 1191, 9507, 109115, 1640955, 28331520 and 419973120, rather than 230, 1191, 9511, 109139, 1640955, 28331520 and 419973120 (the numbers which differ are underlined). Similarly, if we are correct, the final column of Table 3 of the same work should read 230, 1651, 17803, 287574, 6879260, 240768842 and 12209789596, rather than 230, 1651, 17807, 287658, 6880800, 240800462 (mistyped as 240900462) and 12210589024.
Our results also confirm that there are 5005 types of Q(H){K} Mackay groups (Jablan, 1993a, 2002; Radovic & Jablan, 2005).
It was found that there are 17 803 types of double antisymmetry space groups. This is four fewer than previously stated by Zamorzaev (1988). When rotationreversal symmetry and timereversal symmetry are considered together with the periodic spatial symmetry of a threedimensional crystal, our results show that there are 17 803 distinct types of symmetry that a crystal may exhibit.
The set structure of each category is visually represented in Fig. 5. These category symbols, e.g. Q(H){K}, were introduced by Litvin et al. (1994, 1995). They are intended as a generalization of the widely used Q(H) notation for magnetic groups. The `()' brackets enclose a subset which is not primed. The `{ }' brackets enclose a subset which is not starred. The 1', 1*, 1'* or 1'1* at the end of a symbol indicates that 1', 1*, 1'*, or 1' and 1*, respectively, are elements of the group.
 Figure 5 Representation of the 12 categories of double antisymmetry groups. 
2/m has four elements: {1,2,m,1}. 2/m has three index2 subgroups: {1,2}, {1,m} and {1,1}, which will be referred to as 2, m and 1, respectively.
Category (1): Q
1. Q = 2/m Q = {1,2,m,1}
Category (2): Q + Q1'
2. Q = 2/m Q1' = {1,2,m,1,1',2',m',1'}
Category (3): H + (Q  H)1'
3. Q = 2/m, H = 2 Q(H) = {1,2,m',1'}
4. Q = 2/m, H = m Q(H) = {1,2',m,1'}
5. Q = 2/m, H = 1 Q(H) = {1,2',m',1}
Category (4): Q + Q1*
6. Q = 2/m Q1* = {1,2,m,1,1*,2*,m*,1*}
Category (5): Q + Q1' + Q1* + Q1'*
7. Q = 2/m Q1'1* = {1,2,m,1,1',2',m',1',1*,2*,m*,1*,1'*,2'*,m'*,1'*}
Category (6): H + (Q  H)1' + H1* + (Q  H)1'*
8. Q = 2/m, H = 2 Q(H)1* = {1,2,m',1',1*,2*,m'*,1'*}
9. Q = 2/m, H = m Q(H)1* = {1,2',m,1',1*,2'*,m*,1'*}
10. Q = 2/m, H = 1 Q(H)1* = {1,2',m',1,1*,2'*,m'*,1*}
Category (7): H + (Q  H)1*
11. Q = 2/m, H = 2 Q{H} = {1,2,m*,1*}
12. Q = 2/m, H = m Q{H} = {1,2*,m,1*}
13. Q = 2/m, H = 1 Q{H} = {1,2*,m*,1}
Category (8): Q + Q1'*
14. Q = 2/m Q1'* = {1,2,m,1,1'*,2'*,m'*,1'*}
Category (9): H + (Q  H)1* + H1' + (Q  H)1'*
15. Q = 2/m, H = 2 Q{H}1' = {1,2,m*,1*,1',2',m'*,1'*}
16. Q = 2/m, H = m Q{H}1' = {1,2*,m,1*,1',2'*,m',1'*}
17. Q = 2/m, H = 1 Q{H}1' = {1,2*,m*,1,1',2'*,m'*,1'}
Category (10): H + (Q  H)1' + H1'* + (Q  H)1*
18. Q = 2/m, H = 2 Q(H)1'* = {1,2,m',1',1'*,2'*,m*,1*}
19. Q = 2/m, H = m Q(H)1'* = {1,2',m,1',1'*,2*,m'*,1*}
20. Q = 2/m, H = 1 Q(H)1'* = {1,2',m',1,1'*,2*,m*,1'*}
Category (11): H + (Q  H)1'*
21. Q = 2/m, H = 2 Q(H){H} = {1,2,m'*,1'*}
22. Q = 2/m, H = m Q(H){H} = {1,2'*,m,1'*}
23. Q = 2/m, H = 1 Q(H){H} = {1,2'*,m'*,1}
Category (12): H K + (H  K)1* + (K  H)1' + (Q  (H + K))1'*
24. Q = 2/m, H = m, K = 1 Q(H){K} = {1,2'*,m*,1'}
25. Q = 2/m, H = 2, K = 1 Q(H){K} = {1,2*,m'*,1'}
26. Q = 2/m, H = 2, K = m Q(H){K} = {1,2*,m',1'*}
27. Q = 2/m, H = 1, K = m Q(H){K} = {1,2'*,m',1*}
28. Q = 2/m, H = 1, K = 2 Q(H){K} = {1,2',m'*,1*}
29. Q = 2/m, H = m, K = 2 Q(H){K} = {1,2',m*,1'*}.
Both 2/m and 222 (given as an example in §2) have three index2 subgroups. Consequently, they generate the same number of double antisymmetry groups: 29. However, unlike with 222, none of the 29 groups formed from 2/m are in the same equivalence class. This may seem surprising given that 2/m is isomorphic to 222. This can be thought of as being a consequence of the fact that none of the elements of 2/m can be rotated into one another, whereas the three twofold axes of 222 can. Another way to look at it is to consider that 2/m's proper affine normalizer group (2) does not contain nontrivial automorphisms whereas 222's proper affine normalizer group (432) does.
As with all crystallographic space groups, Cc has an infinite number of elements due to the infinite translational subgroup. Cc's elements will be represented as (t_{[100]},t_{[001]},t_{[0]}){1,c} where (t_{[100]},t_{[001]},t_{[0]}) represent the generators of the translation subgroup and {1,c} are coset representatives of the corresponding decomposition. Cc has three index2 subgroups: (t_{[100]},t_{[001]},t_{[0]}){1}, (t_{[100]},t_{[010]},t_{[001]}){1,c} and (t_{[100]},t_{[010]},t_{[001]}){1, t_{[0]}·c}.
Category (1): Q
1. Q = Cc Q = (t_{[100]},t_{[001]},t_{[0]}){1,c}
Category (2): Q + Q1'
2. Q = Cc Q1' = (t_{[100]},t_{[001]},t_{[0]}){1,c,1',c'}
Category (3): H + (Q  H)1'
3. Q = Cc, H = (t_{[100]},t_{[001]},t_{[0]}){1} Q(H) = (t_{[100]},t_{[001]},t_{[0]}){1,c'}
4. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,c} Q(H) = (t_{[100]},t_{[001]},t_{[0]}'){1,c}
5. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c} Q(H) = (t_{[100]},t_{[001]},t_{[0]}'){1,c'}
Category (4): Q + Q1*
6. Q = Cc Q1* = (t_{[100]},t_{[001]},t_{[0]}){1,c,1*,c*}
Category (5): Q + Q1' + Q1* + Q1'*
7. Q = Cc Q1'1* = (t_{[100]},t_{[001]},t_{[0]}){1,c,1',c',1*,c*,1'*,c'*}
Category (6): H + (Q  H)1' + H1* + (Q  H)1'*
8. Q = Cc, H = (t_{[100]},t_{[001]},t_{[0]}){1} Q(H)1* = (t_{[100]},t_{[001]},t_{[0]}){1,c',1*,c'*}
9. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,c} Q(H)1* = (t_{[100]},t_{[001]},t_{[0]}'){1,c,1*,c*}
10. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c} Q(H)1* = (t_{[100]},t_{[001]},t_{[0]}'){1,c',1*,c'*}
Category (7): H + (Q  H)1*
11. Q = Cc, H = (t_{[100]},t_{[001]},t_{[0]}){1} Q{H} = (t_{[100]},t_{[001]},t_{[0]}){1,c*}
12. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,c} Q{H} = (t_{[100]},t_{[001]},t_{[0]}*){1,c}
13. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c} Q{H} = (t_{[100]},t_{[001]},t_{[0]}*){1,c*}
Category (8): Q + Q1'*
14. Q = Cc Q1'* = (t_{[100]},t_{[001]},t_{[0]}){1,c,1'*,c'*}
Category (9): H + (Q  H)1* + H1' + (Q  H)1'*
15. Q = Cc, H = (t_{[100]},t_{[001]},t_{[0]}){1} Q{H}1' = (t_{[100]},t_{[001]},t_{[0]}){1,c*,1',c'*}
16. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,c} Q{H}1' = (t_{[100]},t_{[001]},t_{[0]}*){1,c,1',c'}
17. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c} Q{H}1' = (t_{[100]},t_{[001]},t_{[0]}*){1,c*,1',c'*}
Category (10): H + (Q  H)1' + H1'* + (Q  H)1*
18. Q = Cc, H = (t_{[100]},t_{[001]},t_{[0]}){1} Q(H)1'* = (t_{[100]},t_{[001]},t_{[0]}){1,c',1'*,c*}
19. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,c} Q(H)1'* = (t_{[100]},t_{[001]},t_{[0]}'){1,c,1'*,c'*}
20. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c} Q(H)1'* = (t_{[100]},t_{[001]},t_{[0]}'){1,c',1'*,c*}
Category (11): H + (Q  H)1'*
21. Q = Cc, H = (t_{[100]},t_{[001]},t_{[0]}){1} Q(H){H} = (t_{[100]},t_{[001]},t_{[0]}){1,c'*}
22. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,c} Q(H){H} = (t_{[100]},t_{[001]},t_{[0]}'*){1,c}
23. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c} Q(H){H} = (t_{[100]},t_{[001]},t_{[0]}'*){1,c'*}
Category (12): H K + (H  K)1* + (K  H)1' + (Q  (H + K))1'*
24. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,c}, K = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c} Q(H){K} = (t_{[100]},t_{[001]},t_{[0]}'*){1,c*}
25. Q = Cc, H = (t_{[100]},t_{[001]},t_{[0]}){1}, K = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c} Q(H){K} = (t_{[100]},t_{[001]},t_{[0]}*){1,c'*}
26. Q = Cc, H = (t_{[100]},t_{[001]},t_{[0]}){1}, K = (t_{[100]},t_{[010]},t_{[001]}){1,c} Q(H){K} = (t_{[100]},t_{[001]},t_{[0]}*){1,c'}
27. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c}, K = (t_{[100]},t_{[010]},t_{[001]}){1,c} Q(H){K} = (t_{[100]},t_{[001]},t_{[0]}'*){1,c'}
28. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c}, K = (t_{[100]},t_{[001]},t_{[0]}){1} Q(H){K} = (t_{[100]},t_{[001]},t_{[0]}'){1,c'*}
29. Q = Cc, H = (t_{[100]},t_{[010]},t_{[001]}){1,c}, K = (t_{[100]},t_{[001]},t_{[0]}){1} Q(H){K} = (t_{[100]},t_{[001]},t_{[0]}'){1,c*}.
Note that although 29 double antisymmetry space groups are generated from using Cc as a colorblind parent group, they are not all of unique types. This is because there exist proper affine transformations which map some of these into each other. We show this by applying the results of §5.3.
To do this, we need to know the transformation matrices mapping the standard representative groups to the actual subgroups, and the proper affine normalizer groups of Cc and its index2 subgroups. For Cc's three index2 subgroups:
(t_{[100]},t_{[001]},t_{[0]}){1} is type P1 and can be mapped from the standard P1 by
(t_{[100]},t_{[010]},t_{[001]}){1,c} is type Pc and can be mapped from the standard Pc by
(t_{[100]},t_{[010]},t_{[001]}){1,t_{[0]}·c} is type Pc and can be mapped from the standard Pc by
The proper affine normalizers of Cc are
The proper affine normalizers of the standard representative groups of the two types of subgroups (P1 and Pc) are:
P1 normalizers
Pc normalizers
Having collected all this information, we can now evaluate the proper affine equivalence of the 29 double antisymmetry groups generated from Q = Cc.
We know that groups from different categories can never be equivalent; therefore categories (1), (2), (4), (5) and (8) must contain only one type as only one group has been generated.
For category (3), we have three generated groups. Thus there are three pairs for which we can test for equivalence, , and . For group 3, H is P1 type whereas for 4 and 5 H is Pc type. Therefore, and are false. For , we can evaluate:
In this case,
and
From substituting these in and simplifying, we can show that is logically equivalent to the existence of a solution with a positive determinant to the following:
There are clearly many solutions, e.g. one solution is where n_{2} = 1, n_{i2} = r = t = 0 and p = 1. Thus, 4 and 5 are equivalent and therefore for category (3) there are only two types of groups where Q = Cc. It is trivial to extend these results to show that categories (6), (7), (9), (10) and (11) similarly have two types.
For category (12), we have six generated groups. Thus there are 15 pairs for which we can test for equivalence. Only three of the 15 have the same H and K types (, and ) and therefore only these need to be evaluated using
For ,
and
From substituting these in and simplifying, we can show that is logically equivalent to the existence of a solution with a positive determinant to the following:
There are clearly many solutions, e.g. one solution is where n_{2} = 1, n_{i2} = r = t = s = 0, p = 1, n_{i=j} = 1 and n_{ij} = 0. Thus 25 and 26 are equivalent. Since 24, 27, 28 and 29 can be related to 25 and 26 by automorphisms of 1'1*, implies and . Therefore there are only three types of category (12) groups and a total of 20 double antisymmetry spacegroup types for Q = Cc.
An equivalence relation can be used to partition a set of groups into equivalence classes. For example, an equivalence relation can be applied to partition the set of crystallographic space groups (which is uncountably infinite) into a finite number of classes. The proper affine equivalence relation is used to classify space groups into 230 proper affine classes or `types'. The proper affine equivalence relation can be defined as: two groups, G_{1} and G_{2}, are equivalent if and only if G_{1} can be bijectively mapped to G_{2} by a proper affine transformation, a.
If it is known that G_{1} and G_{2} have the same colorblind parent group Q, then, instead of using the entire proper affine group , it is sufficient to use the proper affine normalizer group of Q, denoted (see footnote 2).
The proper affine equivalence relation does not allow for any permutations of antiidentities. Other works give another set of equivalence classes of antisymmetry groups called Mackay groups (Jablan, 1993a, 2002; Radovic & Jablan, 2005). The equivalence relation of Mackay groups allows some color permutations in addition to proper affine transformation. For double antisymmetry, the Mackay equivalence relation allows for 1' and 1* to be permuted, i.e. all the primed operations become starred and vice versa:
The Mackay equivalence relation does not allow for 1'* to be permuted (Radovic & Jablan, 2005). Note that Radovic & Jablan do give the Mackay equivalence relation as permuting `antiidentities' but 1'* is not considered an antiidentity in their work (it is simply the product of 1' and 1*). They also conclude that Mackay groups are the minimal representation of `Zamorzaev groups'. This seems potentially inconsistent with the aforementioned restriction on color permutation. If we instead allow for all possible color permutations that preserve the group structure of 1'1*, i.e. the automorphisms of 1'1*, then we can clearly further reduce the representation beyond that of the Mackay groups, contrary to what has been claimed. This is demonstrated by Zamorzaev & Palistrant's listing of double antisymmetry spacegroupgenerating sets. In their listing, they gave only those sets which were unique up to the automorphisms of 1'1* (Zamorzaev & Palistrant, 1980). Such a listing only needs to contain 1846 Q(H){K} generating sets, far fewer than the 5005 Mackay equivalence classes of Q(H){K} groups.
If all possible color permutations that preserve the group structure of 1'1*, i.e. the automorphisms of 1'1*, are allowed, then the equivalence relation can be expressed as
This proper affine color equivalence relation results in 1846 classes for category (12) Q(H){K} groups. The equivalence classes of this kind of relation are similar to what Koptsik & Shubnikov (1974) refer to as `Belov groups'.
Generalized to an arbitrary coloring scheme, P, the color equivalence relation, can be defined as
The advantage of using the color equivalence relation to reduce the number of equivalence classes becomes greater as the number of colors (the order of P) increases. For example, for nontrivial double antisymmetry space groups (where and thus ), there are 9507 proper affine equivalence classes [equation (12)], 5005 Mackay equivalence classes [equation (13)] and 1846 color equivalence classes [equation (14)]. Whereas for nontrivial sextuple antisymmetry space groups (where and thus ), there are 419 973 120 proper affine equivalence classes, 598 752 Mackay equivalence classes and just one color equivalence class. Although the number of colors only increased from four to 64 by going from nontrivial double antisymmetry space groups to nontrivial sextuple antisymmetry space groups, the number of proper affine classes [equation (12)] increased from 9507 to 419 973 120 whereas the number of color equivalence classes [equation (15)] actually decreased from 1846 to one.
Although these colorpermuting equivalence relations reduce the number of equivalence classes significantly, they are not suitable when the differences between the colors are important. With timereversal as 1' and rotationreversal as 1*, the differences are clearly very important. However, there may be applications where the color equivalence relation is suitable, for example, in making patterns for aesthetic purposes.
We acknowledge support from the Penn State Center for Nanoscale Science through the NSFMRSEC DMR No. 0820404. We also acknowledge NSF DMR0908718 and DMR1210588. DBL would like to acknowledge informative correspondence with S. V. Jablan. BKV would like to acknowledge useful feedback and corrections from Mantao Huang. BKV also acknowledges Mantao Huang for compiling and formatting the supplementary material `Double Antisymmetry Space Groups.pdf'.
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