Dichromatic Symmetry

Dichromatic symmetry,Loeb, A.L. (1971). ''Color and Symmetry'', Wiley, New York, also referred to as antisymmetry,Shubnikov, A.V. (1951). ''Symmetry and antisymmetry of finite figures'', Izv. Akad. Nauk SSSR, Moscow black-and-white symmetry, magnetic symmetry, counterchange symmetry or dichroic symmetry, is a symmetry operation which reverses an object to its opposite. A more precise definition is "operations of antisymmetry transform objects possessing two possible values of a given property from one value to the other."Mackay, A.L. (1957).
Extensions of space-group theory
', Acta Crystallogr. 10, 543-548, 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. 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,Padmanabhan, H., Munro, J.M., Dabo, I and Gopalan, V. (2020). ''Antisymmetry: fundamentals and applications'', Annual Review of Materials Research, 50, 255-281, such as time reversal, which require two-valued symmetry operations.

Examples

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:
* identity ()
* rotation by ()
* rotation by ()
* anti-mirror reflection ()
* combination of with ()
* combination of with ().

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.

History

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.Heesch, H. (1930). ''Über die vierdimensionalen Gruppen des dreidimensionalen Raumes'', Z. Krist., 73, 325-345, Heesch's work was influenced by Weber's 1929 paper on black-and-white colouring of 2D bands.Weber, L. (1929). ''Die Symmetrie homogener ebener Punktsysteme'', Z. Krist., 70, 309-327,

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 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.

* 1951 Landau and Lifshitz reinterpret black and white colours to correspond to time reversal symmetry
* 1953 Zamorzaev derives the 1651 3D antisymmetric space groups for the first timeZamorzaev, A.M. (1953). ''Generalization of the space groups'', Dissertation, Leningrad University
* 1956 Tavger and Zaitsev use the concept of vector reversal of magnetic moments to derive point groups for magnetic crystals
* 1957 Belov and his colleagues independently derive the 2D and 3D antisymmetric groupsBelov, N.V., Neronova, N.N. and Smirnova, T.S. (1957).
Shubnikov groups
', Sov. Phys. Cryst., 2(3), 311-322
* 1957 Zamorzaev and Sokolov begin the generalization of antisymmetry by introducing the concept of more than one kind of two-valued antisymmetry operation
* 1957 Mackay publishes the first review of the Russian work in English. Subsequent reviews were published by Holser (1961), Koptsik (1968), Schwarzenberger (1984), in Grünbaum and Shephard's ''Tilings and patterns'' (1987), and Brückler and Stilinović (2024)
* Late 1950s M.C. Escher's artworks based on dichromatic and polychromatic patterns popularise colour symmetry amongst scientists
* 1961 Clear definition by van der Waerden and Burckhardt of colour symmetry in terms of group theory, regardless of the number of colours or dimensions involved
* 1964 First publication of Shubnikov and Belov's '' Colored Symmetry'' in English translationShubnikov, A.V., Belov, N.V. et. al. (1964). ''Colored symmetry'', ed. W.T. Holser, Pergamon, New York
* 1965 Wladyslaw Opechowski and Rosalia Guccione provide a complete derivation and enumeration of the dichromatic 3D space groups
* 1966 Publication by Koptsik of the complete atlas of dichromatic 3D space groups (in Russian)
* 1971 Derivation by Loeb of 2D colour symmetry configurations using rotocenters
* 1974 Publication of '' Symmetry in Science and Art'' by Shubnikov and Koptsik with extensive coverage of dichromatic symmetry in 1D, 2D and 3DShubnikov, A.V. and Koptsik, V.A. (1974). ''Symmetry in science and art'', Plenum Press, New York, (original in Russian published by Nauka, Moscow, 1972.)
* 1988 Washburn and Crowe apply colour symmetry analysis to cultural patterns and objects
* 2008 Conway, Burgiel and Goodman-Strauss publish '' The Symmetries of Things'' which describes the colour-preserving symmetries of coloured objects using a new notation based on Orbifolds

Dimensional counts

The table below gives the number of ordinary and dichromatic groups by dimension. The BohmBohm, J. and Dornberger-Schiff, K. (1966). ''The nomenclature of crystallographic symmetry groups'', Acta Crystallogr., 21, 1000-1007, symbol $G_^a$ is used to denote the number of groups where $o$ = overall dimension, $l$ = lattice dimension and $a$ = number of antisymmetry operation types. $a = 1$ for dichromatic groups with a single antisymmetry operation .

References

External links

* Crowe, D.W. (1986).
The mosaic patterns of H.J. Woods
', Comput. Math. Applic., 12B(1/2), 407-411
* Schattschneider, D. (1986).
In black and white: how to create perfectly colored symmetric patterns
', Comput. Math. Applic., 12B(1/2), 673-695,
* Senechal, M. (1988).
Color symmetry
', Comput. Math. Applic., 16(5-8), 545-553, {{doi, 10.1016/0898-1221(88)90244-1
* Radovic, L. and Jablan, S. (2001).
Antisymmetry and modularity in ornamental art
', Proceedings of Bridges: Mathematical Connections in Art, Music, and Science, Kansas, 55–66

Symmetry