Dichromatic symmetry

Dichromatic symmetryLoeb, A.L. (1971). Color and Symmetry, Wiley, New York, also called antisymmetry,Shubnikov, A.V. (1951). Symmetry and antisymmetry of finite figures, Izv. Akad. Nauk SSSR, Moscow black-and-white symmetry,Gévay, G. (2000). Black-and-white symmetry, magnetic symmetry, self-duality and antiprismatic symmetry: the common mathematical background, Forma, 15, 57–60 magnetic symmetry,Tavger, B.A. (1958). The symmetry of ferromagnetics and antiferromagnetics, Sov. Phys. Cryst., 3, 341-343 counterchange symmetry,Woods, H.J. (1935). The geometric basis of pattern design part I: point and line symmetry in simple figures and borders, Journal of the Textile Institute, Transactions, 26, T197-T210 or dichroic symmetryMakovicky, E. (2016). Symmetry through the eyes of old masters, de Gruyter, Berlin, is a symmetry operation which reverses an object to its opposite.Atoji, A. (1965). Graphical representations of magnetic space groups, American Journal of Physics, 33(3), 212–219, 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.Lockwood, E.H. and Macmillan, R.H. (1978). Geometric symmetry , Cambridge University Press, Cambridge, 67-70 & 206-208 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,Shubnikov, A.V. (1960). Time reversal as an operation of antisymmetry, Sov. Phys. Cryst., 5, 309-314 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 theseWoods, H.J. (1936). The geometric basis of pattern design part IV: counterchange symmetry in plane patterns, Journal of the Textile Institute, Transactions, 27, T305-320 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 symmetryLandau, L.D. and Lifshitz E.M. (1951). Course of theoretical physics, vol. 5. Statistical physics, 1st edition, Nauka, Moscow 1952 W. Cochran re-derives the 46 dichromatic 2D point groups quoting the previous work of Alexander and Herrmann (1928–9) and Woods 1935 1953 Zamorzaev derives the 1651 3D antisymmetric space groups for the first timeZamorzaev, A.M. (1953). Generalization of the space groups, Dissertation, Leningrad UniversityZamorzaev, A.M. (1957). Generalization of Fedorov groups, Sov. Phys. Cryst., 2, 10-15 1956 Tavger and Zaitsev use the concept of vector reversal of magnetic moments to derive point groups for magnetic crystalsTavger, B.A. and Zaitsev, V.M. (1956). Magnetic symmetry of crystals, Soviet Physics JETP, 3(3), 430-436 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 operationZamorzaev, A.M. and Sokolov, E.I. (1957). Symmetry and various kinds of antisymmetry of finite bodies, Sov. Phys. Cryst., 2(1), 5-9Zamorzaev, A.M. and Palistrant, A.F. (1980). Antisymmetry, its generalizations and geometrical applications, Z. Krist., 151, 231-248, Zamorzaev, A.M. (1988). Generalized antisymmetry, Comput. Math. Applic., 16(5-8), 555-562, 1957 Mackay publishes the first review of the Russian work in English. Subsequent reviews were published by Holser (1961),Holser, W.T. (1961). Classification of symmetry groups, Acta Crystallogr., 14, 1236-1242, Koptsik (1968),Koptsik, V.A. (1968). A general sketch of the development of the theory of symmetry and its applications in physical crystallography over the last 50 years, Sov. Phys. Cryst., 12(5), 667-683 Schwarzenberger (1984),Schwarzenberger, R.L.E. (1984). Colour symmetry, Bull. London Math. Soc., 16, 209-240, in Grünbaum and Shephard's Tilings and patterns (1987),Grünbaum, B. and Shephard, G.V. (1987). Tilings and patterns, W.H. Freeman, New York, and Brückler and Stilinović (2024)Brückler, F.M. and Stilinović, V. (2024) From friezes to quasicrystals: a history of symmetry groups, 1823-1864, , in Sriraman, B. (ed.) Handbook of the history and philosophy of mathematical practice, Springer, pp 3200, Late 1950s M.C. Escher's artworks based on dichromatic and polychromatic patterns popularise colour symmetry amongst scientistsMacGillavry, C.H. (1976). Symmetry aspects of M. C. Escher's periodic drawings, International Union of Crystallography, Utrecht, Schnattschneider, D. (2004). M. C. Escher: Visions of Symmetry, Harry. N. Abrams, New York, 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 involvedvan der Waerden, B.L. and Burkhardt, J.J. (1961). Farbgruppen, Z. Krist, 115, 231-234, 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 groupsOpechowski, W. and Guccione, R. (1965). Magnetic symmetry in Magnetism, vol. IIA ed. Rado, G.T. and Suhl, H., Academic Press, New York, pp 105-165 1966 Publication by Koptsik of the complete atlas of dichromatic 3D space groupsKoptsik, V.A. (1966). Shubnikov groups: Handbook on the symmetry and physical properties of crystal structures, Moscow University, Moscow (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 objectsWashburn, D.K. and Crowe, D.W. (1988). Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, Washington University Press, Seattle, 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 OrbifoldsConway, J.H., Burgeil, H. and Goodman-Strauss, C. (2008). The symmetries of things, A.K. Peters, Wellesley, MA,

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 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 Symbol Count 00Zero-dimensional symmetry group121 0One-dimensional point groups251One-dimensional discrete symmetry groups2720Two-dimensional point groups (rosettes)10311Frieze (strip) groups7Kopský, V. and Litvin, D.B. (eds.) (2010). International Tables for Crystallography Volume E: Subperiodic groups, Second online edition , 312Wallpaper (plane) groups17Pólya, G. (1924). Über die Analogie der Kristallsymmetrie in der Ebene (On the analog of crystal symmetry in the plane), Z. Krist., 60, 278–282, 80Alexander, E. and Herrman, K. (1929). Die 80 zweidimensionalen Raumgruppen, Z. Krist. 70, 328-345, 30Three-dimensional point groups32Frankenheim, M.L. (1826). Crystallonomische Aufsätze, Isis (Jena) 19, 497-515, 542-5651221Rod (cylinder) groups75394Neronova, N.N. and Belov, N.V. (1961). A single scheme for the classical and black-and-white crystallographic symmetry groups, Sov. Phys. Cryst., 6, 3-122Layer (sheet) groups80528Litvin, D.B. (1999). Magnetic subperiodic groups, Acta Cryst. A, 55(5), 963–964, 3Three-dimensional space groups230Burckhardt, J.J. (1967). Zur Geschichte der Entdeckung der 230 Raumgruppen [On the history of the discovery of the 230 space groups], Archive for History of Exact Sciences, 4(3), 235-246, 165140Four-dimensional point groups271Brown, H., Bulow, R., Neubuser, J. et al. (1978). Crystallographic groups of four-dimensional space, Wiley, New York, 1202Souvignier, B. (2006). The four-dimensional magnetic point and space groups, Z. Krist., 221, 77-82, 1343Palistrant, A.F. and Zamorzaev, A.M. (1992). Symmetry space groups: on the 100th anniversary of their discovery, ed. Vainshtein, B.K., Nauka, Moscow (in Russian)21091Zamorzaev, A.M., Karpova, Yu.S., Lungu, A.P. and Palistrant, A.F. (1986). P-symmetry and its further development, Shtiintsa, Chisinau (in Russian)31594Palistrant, A.F. (2012). Complete scheme of four-dimensional crystallographic symmetry groups, Crystallography Reports, 57(4), 471-477. 4Four-dimensional discrete symmetry groups489462227

Dichromatic symmetry

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