Dichromatic symmetry,[1] also referred to as antisymmetry,[2][3] black-and-white symmetry,[4] magnetic symmetry,[5] counterchange symmetry[6] or dichroic symmetry,[7] is a symmetry operation which reverses an object to its opposite.[8] A more precise definition is "operations of antisymmetry transform objects possessing two possible values of a given property from one value to the other."[9] Dichromatic symmetry refers specifically to two-coloured symmetry; this can be extended to three or more colours in which case it is termed polychromatic symmetry.[10] A general term for dichromatic and polychromatic symmetry is simply colour symmetry. Dichromatic symmetry is used to describe magnetic crystals and in other areas of physics,[11] such as time reversal,[12] which require two-valued symmetry operations.
A simple example is to take a white object, such as a triangle, and apply a colour change resulting in a black triangle. Applying the colour change once more yields the original white triangle.
The colour change, here termed an anti-identity operation (1'), yields the identity operation (1) if performed twice.
Another example is to construct an anti-mirror reflection (m') from a mirror reflection (m) and an anti-identity operation (1') executed in either order.
The m' operation can then be used to construct the antisymmetry point group 3m' of a dichromatic triangle.
There are no mirror reflection (m) operations for the dichromatic triangle, as there would be if all the smaller component triangles were coloured white. However, by introducing the anti-mirror reflection (m') operation the full dihedral D3 symmetry is restored. The six operations making up the dichromatic D3 (3m') point group are:
Note that the vertex numbers do not form part of the triangle being operated on - they are shown to keep track of where the vertices end up after each operation.
In 1930 Heinrich Heesch was the first person to formally postulate an antisymmetry operation in the context of examining the 3D space groups in 4D.[13] Heesch's work was influenced by Weber's 1929 paper on black-and-white colouring of 2D bands.[14]
In 1935-1936 H.J. Woods published a series of four papers with the title The geometrical basis of pattern design. The last of these[15] was devoted to counterchange symmetry and in which was derived for the first time the 46 dichromatic 2D point groups.
The work of Heesch and Woods were not influential at the time, and the subject of dichromatic symmetry did not start to become important until the publication of A.V. Shubnikov's book Symmetry and antisymmetry of finite figures in 1951. Thereafter the subject developed rapidly, initially in Russia but subsequently in many other countries, because of its importance in magnetic structures and other physical fields.
The table below gives the number of ordinary and dichromatic groups by dimension. The Bohm[36] symbol is used to denote the number of groups where = overall dimension, = lattice dimension and = number of antisymmetry operation types. for dichromatic groups with a single antisymmetry operation .
Overall dimension |
Lattice dimension |
Ordinary groups | Dichromatic groups | |||||
---|---|---|---|---|---|---|---|---|
Name | Symbol | Count | Refs | Symbol | Count | Refs | ||
0 | 0 | Zero-dimensional symmetry group | 1 | 2 | ||||
1 | 0 | One-dimensional point groups | 2 | 5 | ||||
1 | One-dimensional discrete symmetry groups | 2 | 7 | |||||
2 | 0 | Two-dimensional point groups (rosettes) | 10 | 31 | ||||
1 | Frieze (strip) groups | 7 | [37] | 31 | ||||
2 | Wallpaper (plane) groups | 17 | [38][39] | 80 | [14][33][40] | |||
3 | 0 | Three-dimensional point groups | 32 | [41] | 122 | [2][13] | ||
1 | Rod (cylinder) groups | 75 | [37] | 394 | [42] | |||
2 | Layer (sheet) groups | 80 | [37] | 528 | [43] | |||
3 | Three-dimensional space groups | 230 | [44] | 1651 | [17][20] | |||
4 | 0 | Four-dimensional point groups | 271 | [45] | 1202 | [46] | ||
1 | 343 | [47] | ||||||
2 | 1091 | [48] | ||||||
3 | 1594 | [49] | ||||||
4 | Four-dimensional discrete symmetry groups | 4894 | [45] | 62227 | [46] |