Dichromatic symmetry

Two-colour symmetry (examples, history and dimensional counts)

Dichromatic triangle illustrating colour symmetry

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.

## Examples

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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.

Anti-identity operation and Cayley table

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.

The six operations making up the dichromatic D3 (3m') point groupD3 Cayley table with colour change representing the antisymmetry group 3m'

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 (e)
* rotation by 2π/3 (r)
* rotation by 4π/3 (r2)
* anti-mirror reflection (m')
* combination of m' with r (m'r)
* combination of m' with r2 (m'r2).

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

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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.

* 1951 Landau and Lifshitz reinterpret black and white colours to correspond to time reversal symmetry[16]
* 1953 Zamorzaev derives the 1651 3D antisymmetric space groups for the first time[17][18]
* 1956 Tavger and Zaitsev use the concept of vector reversal of magnetic moments to derive point groups for magnetic crystals[19]
* 1957 Belov and his colleagues independently derive the 2D and 3D antisymmetric groups[20]
* 1957 Zamorzaev and Sokolov begin the generalization of antisymmetry by introducing the concept of more than one kind of two-valued antisymmetry operation[11][21][22][23]
* 1957 Mackay publishes the first review of the Russian work in English.[9] Subsequent reviews were published by Holser (1961),[24] Koptsik (1968),[25] Schwarzenberger (1984),[26] in Grünbaum and Shephard's Tilings and patterns (1987),[27] and Brückler and Stilinović (2024)[28]
* Late 1950s M.C. Escher's artworks based on dichromatic and polychromatic patterns popularise colour symmetry amongst scientists[29][30]
* 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[31]
* 1964 First publication of Shubnikov and Belov's Colored Symmetry in English translation[3]
* 1965 Wladyslaw Opechowski and Rosalia Guccione provide a complete derivation and enumeration of the dichromatic 3D space groups[32]
* 1966 Publication by Koptsik of the complete atlas of dichromatic 3D space groups[33] (in Russian)
* 1971 Derivation by Loeb of 2D colour symmetry configurations using rotocenters[1]
* 1974 Publication of Symmetry in Science and Art by Shubnikov and Koptsik with extensive coverage of dichromatic symmetry in 1D, 2D and 3D[34]
* 1988 Washburn and Crowe apply colour symmetry analysis to cultural patterns and objects[35]
* 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[36]

## Dimensional counts

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The table below gives the number of ordinary and dichromatic groups by dimension. The Bohm[37] symbol  G o l a {\displaystyle G_{ol}^{a}} is used to denote the number of groups where  o {\displaystyle o} = overall dimension,  l {\displaystyle l} = lattice dimension and  a {\displaystyle a} = number of antisymmetry operation types.  a = 1 {\displaystyle a=1} 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  |  G 0 {\displaystyle G_{0}} | 1  |  |  G 0 1 {\displaystyle G_{0}^{1}} | 2  |
1  | 0  | One-dimensional point groups |  G 10 {\displaystyle G_{10}} | 2  |  |  G 10 1 {\displaystyle G_{10}^{1}} | 5  |
1  | One-dimensional discrete symmetry groups |  G 1 {\displaystyle G_{1}} | 2  |  |  G 1 1 {\displaystyle G_{1}^{1}} | 7  |
2  | 0  | Two-dimensional point groups (rosettes) |  G 20 {\displaystyle G_{20}} | 10  |  |  G 20 1 {\displaystyle G_{20}^{1}} | 31  |
1  | Frieze (strip) groups |  G 21 {\displaystyle G_{21}} | 7  | [38] |  G 21 1 {\displaystyle G_{21}^{1}} | 31  | [2][34]
2  | Wallpaper (plane) groups |  G 2 {\displaystyle G_{2}} | 17  | [39][40] |  G 2 1 {\displaystyle G_{2}^{1}} | 80  | [14][34][41]
3  | 0  | Three-dimensional point groups |  G 30 {\displaystyle G_{30}} | 32  | [42] |  G 30 1 {\displaystyle G_{30}^{1}} | 122  | [2][13]
1  | Rod (cylinder) groups |  G 31 {\displaystyle G_{31}} | 75  | [38] |  G 31 1 {\displaystyle G_{31}^{1}} | 394  | [43]
2  | Layer (sheet) groups |  G 32 {\displaystyle G_{32}} | 80  | [38] |  G 32 1 {\displaystyle G_{32}^{1}} | 528  | [44]
3  | Three-dimensional space groups |  G 3 {\displaystyle G_{3}} | 230  | [45] |  G 3 1 {\displaystyle G_{3}^{1}} | 1651  | [17][20]
4  | 0  | Four-dimensional point groups |  G 40 {\displaystyle G_{40}} | 271  | [46] |  G 40 1 {\displaystyle G_{40}^{1}} | 1202  | [47]
1  |  |  G 41 {\displaystyle G_{41}} | 343  | [48] |  |  |
2  |  |  G 42 {\displaystyle G_{42}} | 1091  | [49] |  |  |
3  |  |  G 43 {\displaystyle G_{43}} | 1594  | [50] |  |  |
4  | Four-dimensional discrete symmetry groups  |  G 4 {\displaystyle G_{4}} | 4894  | [46] |  G 4 1 {\displaystyle G_{4}^{1}} | 62227  | [47]

## References

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1. ^ a b Loeb, A.L. (1971). Color and Symmetry, Wiley, New York, ISBN 9780471543350
2. ^ a b c Shubnikov, A.V. (1951). Symmetry and antisymmetry of finite figures, Izv. Akad. Nauk SSSR, Moscow
3. ^ a b Shubnikov, A.V., Belov, N.V. et. al. (1964). Colored symmetry, ed. W.T. Holser, Pergamon, New York
4. ^ Gévay, G. (2000). Black-and-white symmetry, magnetic symmetry, self-duality and antiprismatic symmetry: the common mathematical background, Forma, 15, 57–60
5. ^ Tavger, B.A. (1958). The symmetry of ferromagnetics and antiferromagnetics, Sov. Phys. Cryst., 3, 341-343
6. ^ 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
7. ^ Makovicky, E. (2016). Symmetry through the eyes of old masters, de Gruyter, Berlin, ISBN 9783110417050
8. ^ Atoji, A. (1965). Graphical representations of magnetic space groups, American Journal of Physics, 33(3), 212–219, doi:10.1119/1.1971375
9. ^ a b Mackay, A.L. (1957). Extensions of space-group theory, Acta Crystallogr. 10, 543-548, doi:10.1107/S0365110X57001966
10. ^ Lockwood, E.H. and Macmillan, R.H. (1978). Geometric symmetry , Cambridge University Press, Cambridge, 67-70 & 206-208 ISBN 9780521216852
11. ^ a b Padmanabhan, H., Munro, J.M., Dabo, I and Gopalan, V. (2020). Antisymmetry: fundamentals and applications, Annual Review of Materials Research, 50, 255-281, doi:10.1146/annurev-matsci-100219-101404
12. ^ Shubnikov, A.V. (1960). Time reversal as an operation of antisymmetry, Sov. Phys. Cryst., 5, 309-314
13. ^ a b Heesch, H. (1930). Über die vierdimensionalen Gruppen des dreidimensionalen Raumes, Z. Krist., 73, 325-345, doi:10.1524/zkri.1930.73.1.325
14. ^ a b Weber, L. (1929). Die Symmetrie homogener ebener Punktsysteme, Z. Krist., 70, 309-327, doi:10.1524/zkri.1929.70.1.309
15. ^ Woods, 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
16. ^ Landau, L.D. and Lifshitz E.M. (1951). Course of theoretical physics, vol. 5. Statistical physics, 1st edition, Nauka, Moscow
17. ^ a b Zamorzaev, A.M. (1953). Generalization of the space groups, Dissertation, Leningrad University
18. ^ Zamorzaev, A.M. (1957). Generalization of Fedorov groups, Sov. Phys. Cryst., 2, 10-15
19. ^ Tavger, B.A. and Zaitsev, V.M. (1956). Magnetic symmetry of crystals, Soviet Physics JETP, 3(3), 430-436
20. ^ a b Belov, N.V., Neronova, N.N. and Smirnova, T.S. (1957). Shubnikov groups, Sov. Phys. Cryst., 2(3), 311-322
21. ^ Zamorzaev, A.M. and Sokolov, E.I. (1957). Symmetry and various kinds of antisymmetry of finite bodies, Sov. Phys. Cryst., 2(1), 5-9
22. ^ Zamorzaev, A.M. and Palistrant, A.F. (1980). Antisymmetry, its generalizations and geometrical applications, Z. Krist., 151, 231-248, doi:10.1524/zkri.1980.151.3-4.231
23. ^ Zamorzaev, A.M. (1988). Generalized antisymmetry, Comput. Math. Applic., 16(5-8), 555-562, doi:10.1016/0898-1221(88)90245-3
24. ^ Holser, W.T. (1961). Classification of symmetry groups, Acta Crystallogr., 14, 1236-1242, doi:10.1107/S0365110X61003612
25. ^ 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
26. ^ Schwarzenberger, R.L.E. (1984). Colour symmetry, Bull. London Math. Soc., 16, 209-240, doi:10.1112/blms/16.3.209
27. ^ Grünbaum, B. and Shephard, G.V. (1987). Tilings and patterns, W.H. Freeman, New York, ISBN 9780716711933
28. ^ Brückler, F.M. and Stilinović, V. (2024) From friezes to quasicrystals: a history of symmetry groups, 1823-1864, doi:10.1007/978-3-030-19071-2_132-2, in Sriraman, B. (ed.) Handbook of the history and philosophy of mathematical practice, Springer, pp 3200, ISBN 978-3-030-19071-2
29. ^ MacGillavry, C.H. (1976). Symmetry aspects of M. C. Escher's periodic drawings, International Union of Crystallography, Utrecht, ISBN 9789031301843
30. ^ Schnattschneider, D. (2004). M. C. Escher: Visions of Symmetry, Harry. N. Abrams, New York, ISBN 9780810943087
31. ^ van der Waerden, B.L. and Burkhardt, J.J. (1961). Farbgruppen, Z. Krist, 115, 231-234, doi:10.1524/zkri.1961.115.3-4.231
32. ^ Opechowski, 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
33. ^ Koptsik, V.A. (1966). Shubnikov groups: Handbook on the symmetry and physical properties of crystal structures, Moscow University, Moscow
34. ^ a b c Shubnikov, A.V. and Koptsik, V.A. (1974). Symmetry in science and art, Plenum Press, New York, ISBN 9780306307591 (original in Russian published by Nauka, Moscow, 1972.)
35. ^ Washburn, D.K. and Crowe, D.W. (1988). Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, Washington University Press, Seattle, ISBN 9780295970844
36. ^ Conway, J.H., Burgeil, H. and Goodman-Strauss, C. (2008).  The symmetries of things, A.K. Peters, Wellesley, MA, ISBN 9781568812205
37. ^ Bohm, J. and Dornberger-Schiff, K. (1966). The nomenclature of crystallographic symmetry groups, Acta Crystallogr., 21, 1000-1007, doi:10.1107/S0365110X66004389
38. ^ a b c Kopský, V. and Litvin, D.B. (eds.) (2010). International Tables for Crystallography Volume E: Subperiodic groups, Second online edition ISBN 978-0-470-68672-0, doi:10.1107/97809553602060000109
39. ^ Fedorov, E.S. (1891). "Симметрія на плоскости" [Simmetriya na ploskosti, Symmetry in the plane]. Записки Императорского С.-Петербургского Минералогического Общества (Zapiski Imperatorskova Sankt-Petersburgskova Mineralogicheskova Obshchestva, Proceedings of the Imperial St. Petersburg Mineralogical Society). 2nd series (in Russian). 28: 345–390.
40. ^ Pólya, G. (1924). Über die Analogie der Kristallsymmetrie in der Ebene (On the analog of crystal symmetry in the plane), Z. Krist., 60, 278–282, doi:10.1524/zkri.1924.60.1.278
41. ^ Alexander, E. and Herrman, K. (1929). Die 80 zweidimensionalen Raumgruppen, Z. Krist. 70, 328-345, doi:10.1524/zkri.1929.70.1.328
42. ^ Frankenheim, M.L. (1826). Crystallonomische Aufsätze, Isis (Jena) 19, 497-515, 542-565
43. ^ Neronova, N.N. and Belov, N.V. (1961). A single scheme for the classical and black-and-white crystallographic symmetry groups, Sov. Phys. Cryst., 6, 3-12
44. ^ Litvin, D.B. (1999). Magnetic subperiodic groups, Acta Cryst. A, 55(5), 963–964, doi:10.1107/S0108767399003487
45. ^ Burckhardt, 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, doi:10.1007/BF00412962
46. ^ a b Brown, H., Bulow, R., Neubuser, J. et. al. (1978). Crystallographic groups of four-dimensional space, Wiley, New York, ISBN 9780471030959
47. ^ a b Souvignier, B. (2006). The four-dimensional magnetic point and space groups, Z. Krist., 221, 77-82, doi:10.1524/zkri.2006.221.1.77
48. ^ Palistrant, 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)
49. ^ Zamorzaev, A.M., Karpova, Yu.S., Lungu, A.P. and Palistrant, A.F. (1986). P-symmetry and its further development, Shtiintsa, Chisinau (in Russian)
50. ^ Palistrant, A.F. (2012). Complete scheme of four-dimensional crystallographic symmetry groups, Crystallography Reports, 57(4), 471-477. doi:10.1134/S1063774512040104

## External links

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* 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, doi:10.1016/0898-1221(86)90418-9
* 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