Polychromatic symmetry

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Short description: Symmetry with three or more colours

Polychromatic symmetry is a colour symmetry which interchanges three or more colours in a symmetrical pattern. It is a natural extension of dichromatic symmetry. The coloured symmetry groups are derived by adding to the position coordinates (x and y in two dimensions, x, y and z in three dimensions) an extra coordinate, k, which takes three or more possible values (colours).[1]

An example of an application of polychromatic symmetry is crystals of substances containing molecules or ions in triplet states, that is with an electronic spin of magnitude 1, should sometimes have structures in which the spins of these groups have projections of + 1, 0 and -1 onto local magnetic fields. If these three cases are present with equal frequency in an orderly array, then the magnetic space group of such a crystal should be three-coloured.[2][3]

Example

The group p3 has three different rotation centres of order three (120°), but no reflections or glide reflections.

Uncoloured and three-colored p3 patterns[4]
Uncoloured pattern p3 Three-colour pattern p3[3]1 Three-colour pattern p3[3]2
P3 pattern.svg 307px Three-colour group p3(3)2.svg
Three-colour symmetry operation of colour group p3[3]1

There are two distinct ways of colouring the p3 pattern with three colours: p3[3]1 and p3[3]2 where the figure in square brackets indicates the number of colours, and the subscript distinguishes between multiple cases of coloured patterns.[5]

Taking a single motif in the pattern p3[3]1 it has a symmetry operation 3', consisting of a rotation by 120° and a cyclical permutation of the three colours white, green and red as shown in the animation.

This pattern p3[3]1 has the same colour symmetry as M. C. Escher's Hexagonal tessellation with animals: study of regular division of the plane with reptiles (1939). Escher reused the design in his 1943 lithograph Reptiles.

Group theory

Initial research by Wittke and Garrido (1959)[6] and by Niggli and Wondratschek (1960)[7] identified the relation between the colour groups of an object and the subgroups of the object's geometric symmetry group. In 1961 van der Waerden and Burckhardt[8] built on the earlier work by showing that colour groups can be defined as follows: in a colour group of a pattern (or object) each of its geometric symmetry operations s is associated with a permutation σ of the k colours in such a way that all the pairs (s,σ) form a group. Senechal showed that the permutations are determined by the subgroups of the geometric symmetry group G of the uncoloured pattern.[9] When each symmetry operation in G is associated with a unique colour permutation the pattern is said to be perfectly coloured.[10][11]

The Waerden-Burckhardt theory defines a k-colour group G(H) as being determined by a subgroup H of index k in the symmetry group G.[12] If the subgroup H is a normal subgroup then the quotient group G/H permutes all the colours.[13]

History

  • 1956 First papers on polychromatic, as opposed to dichromatic, symmetry groups are published by Belov and his co-workers.[14][15][16][17][18][19] Vainshtein and Koptsik (1994) summarise the Russian work.[20]
  • 1957 Mackay publishes the first review of the Russian work in English.[21] Subsequent reviews were published by Koptsik (1968),[22] Schwarzenberger (1984),[23] in Grünbaum and Shephard's Tilings and patterns (1987),[4] by Senechal (1990)[9] and by Thomas (2012).[24]
  • Late 1950s M.C. Escher's artworks based on dichromatic and polychromatic patterns popularise colour symmetry amongst scientists.[25][26]
  • 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.[8]
  • 1964 First publication of Shubnikov and Belov's Colored Symmetry in English translation[27]
  • 1971 Derivation by Loeb of 2D colour symmetry configurations using rotocenters.[28]
  • 1974 Publication of Symmetry in science and art by Shubnikov and Koptsik with extensive coverage of polychromatic symmetry.[29]
  • 1983 Senechal examines the problem of colouring polyhedra symmetrically using group theory.[12][30] Cromwell later uses an algorithmic counting approach (1997).[31]
  • 1988 Washburn and Crowe apply colour symmetry analysis to cultural patterns and objects.[32] Washburn and Crowe inspired further work, for example by Makovicky.[33]
  • 1997 Lifshitz extends the theory of color symmetry from periodic to quasiperiodic crystals.[34]
  • 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.[35]

Number of colour groups

Number of strip (frieze) k-colour groups for k ≤ 12[4][36]
  Number of colours (k)
Underlying
group		 2 3 4 5 6 7 8 9 10 11 12
p111 1 1 1 1 1 1 1 1 1 1 1
p1a1 1 1 1 1 1 1 1 1 1 1 1
p1m1 3 1 3 1 3 1 3 1 3 1 3
pm11 2 1 2 1 2 1 2 1 2 1 2
p112 2 1 2 1 2 1 2 1 2 1 2
pma2 3 1 3 1 3 1 3 1 3 1 3
pmm2 5 1 7 1 5 1 7 1 5 1 7
      Total strip
groups		 17    7 19    7 17    7 19    7 17    7 19
Numbers of periodic (plane) k-colour groups for k ≤ 12[4][36][37]
  Number of colours (k)
Underlying
group		 2 3 4 5 6 7 8 9 10 11 12
p1 1 1 2 1 1 1 2 2 1 1 2
pg 2 2 4 2 5 2 7 3 6 2 11
pm 5 2 10 2 11 2 16 3 12 2 23
cm 3 2 7 2 7 2 13 3 8 2 17
p2 2 1 3 1 2 1 4 2 2 1 3
pgg 2 1 4 1 4 1 7 2 5 1 9
pmg 5 2 11 2 11 2 19 3 12 2 26
pmm 5 1 13 1 9 1 21 2 10 1 25
cmm 5 1 11 1 8 1 21 2 9 1 22
p3 - 2 1 - 1 1 - 3 - - 4
p31m 1 2 1 - 5 - 1 3 - - 7
p3m1 1 2 1 - 4 - 1 3 - - 7
p4 2 - 5 1 2 - 9 1 4 - 9
p4g 3 - 7 - 2 - 13 1 3 - 10
p4m 5 - 13 - 2 - 28 1 3 - 16
p6 1 2 1 - 5 1 1 3 - - 8
p6m 3 2 2 - 11 - 3 3 - - 20
Total periodic
groups		 46 23 96 14 90 15 166 40 75 13 219

The two three-colour p3 patterns are illustrated in the Example section above.

References

  1. ↑ Bradley, C.J. and Cracknell, A.P. (2010). The mathematical theory of symmetry in solids: representation theory for point groups and space groups, Clarendon Press, Oxford, 677–681, ISBN:9780199582587
  2. ↑ Harker, D. (1981). The three-colored three-dimensional space groups, Acta Crystallogr., A37, 286-292, doi:10.1107/s0567739481000697
  3. ↑ Mainzer, K. (1996). Symmetries of nature: a handbook for philosophy of nature and science, de Gruyter, Berlin, 162-168, ISBN:9783110129908
  4. ↑ 4.0 4.1 4.2 4.3 Grünbaum, B. and Shephard, G.C. (1987). Tilings and patterns, W.H. Freeman, New York, ISBN:9780716711933
  5. ↑ Hann, M.A. and Thomas, B.G. (2007). Beyond black and white: a note concerning three-colour-counterchange patterns, J. Textile Inst., 98(6), 539-547, doi:10.1080/00405000701502446
  6. ↑ Wittke O. and Garrido J. (1959). Symétrie des polyèdres polychromatiques, Bull. Soc. française de Minéral. et de Crist., 82(7-9), 223-230; doi:10.3406/bulmi.1959.5332
  7. ↑ Niggli, A. and Wondratschek, H. (1960). Eine Verallgemeinerung der Punktgruppen. I. Die einfachen Kryptosymmetrien, Z. Krist., 114(1-6), 215-231 doi:10.1524/zkri.1960.114.16.215
  8. ↑ 8.0 8.1 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
  9. ↑ 9.0 9.1 Senechal, M. (1990). Geometrical crystallography in Historical atlas of crystallography ed. Lima-de-Faria, J., Kluwer, Dordrecht, 52-53, ISBN:9780792306498
  10. ↑ Senechal, M. (1988). Color symmetry, Comput. Math. Applic., 16(5-8), 545-553, doi:10.1016/0898-1221(88)90244-1
  11. ↑ Senechal, M. (1990). Crystalline symmetries: an informal mathematical introduction, Adam Hilger, Bristol, 74-87, ISBN:9780750300414
  12. ↑ 12.0 12.1 Senechal, M. (1983). Color symmetry and colored polyhedra, Acta Crystallogr., A39, 505-511,doi:10.1107/s0108767383000987
  13. ↑ Coxeter, H.S.M. (1987). A simple introduction to colored symmetry, Int. J. Quantum Chemistry, 31, 455-461, doi:10.1002/qua.560310317
  14. ↑ Belov, N.V. and Tarkhova, T.N. (1956). Colour symmetry groups, Sov. Phys. Cryst., 1, 5-11
  15. ↑ Belov, N.V. and Tarkhova, T.N. (1956). Colour symmetry groups, Sov. Phys. Cryst., 1, 487-488
  16. ↑ Belov, N.V. (1956). Moorish patterns of the Middle Ages and the symmetry groups, Sov. Phys. Cryst., 1, 482-483
  17. ↑ Belov, N.V. (1956). Three-dimensional mosaics with colored symmetry, Sov. Phys. Cryst., 1, 489-492
  18. ↑ Belov, N.V. and Belova, E.N. (1956). Mosaics for the 46 plane (Shubnikov) groups of anti-symmetry and for the 15 (Fedorov) colour groups, Sov. Phys. Cryst., 2, 16-18
  19. ↑ Belov, N.V., Belova, E.N. and Tarkhova, T.N. (1959). More about the colour symmetry groups, Sov. Phys. Cryst., 3, 625-626
  20. ↑ Vainshtein, B.K. and Koptsik, V.A. (1994). Modern crystallography. Volume 1. Fundamentals of crystals: symmetry, and methods of structural crystallography, Springer, Berlin, 158-179, ISBN:9783540565581
  21. ↑ Mackay, A.L. (1957). Extensions of space-group theory, Acta Crystallogr. 10, 543-548, doi:10.1107/s0365110x57001966
  22. ↑ 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
  23. ↑ Schwarzenberger, R.L.E. (1984). Colour symmetry, Bull. London Math. Soc., 16, 209-240, doi:10.1112/blms/16.3.209, doi:10.1112/blms/16.3.216, doi:10.1112/blms/16.3.229
  24. ↑ Thomas, B.G. (2012). Colour symmetry: the systematic coloration of patterns and tilings in Colour Design, ed. Best, J., Woodhead Publishing, 381-432, ISBN:9780081016480
  25. ↑ MacGillavry, C.H. (1976). Symmetry aspects of M.C. Escher's periodic drawings, International Union of Crystallography, Utrecht, ISBN:9789031301843
  26. ↑ Schnattschneider, D. (2004). M.C. Escher: Visions of symmetry, Harry. N. Abrams, New York, ISBN:9780810943087
  27. ↑ Shubnikov, A.V., Belov, N.V. et. al. (1964). Colored symmetry, ed. W.T. Holser, Pergamon, New York
  28. ↑ Loeb, A.L. (1971). Color and symmetry, Wiley, New York, ISBN:9780471543350
  29. ↑ 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)
  30. ↑ Senechal, M. (1983). Coloring symmetrical objects symmetrically, Math. Magazine, 56(1), 3-16, doi:10.2307/2690259
  31. ↑ Cromwell, P.R. (1997). Polyhedra, Cambridge University Press, 327-348, ISBN:9780521554329
  32. ↑ Washburn, D.K. and Crowe, D.W. (1988). Symmetries of culture: theory and practice of plane pattern analysis, Washington University Press, Seattle, ISBN:9780295970844
  33. ↑ Makovicky, E. (2016). Symmetry through the eyes of old masters, de Gruyter, Berlin, 133-147, ISBN:9783110417050
  34. ↑ Lifshitz, R. (1997). Theory of color symmetry for periodic and quasiperiodic crystals, Rev. Mod. Phys., 69(4), 1181–1218, doi:10.1103/RevModPhys.69.1181
  35. ↑ Conway, J.H., Burgeil, H. and Goodman-Strauss, C. (2008). The symmetries of things, A.K. Peters, Wellesley, MA, ISBN:9781568812205
  36. ↑ 36.0 36.1 Wieting, T.W. (1982). Mathematical theory of chromatic plane ornaments, Marcel Dekker, New York, ISBN:9780824715175
  37. ↑ Jarratt, J.D. and Schwarzenberger, R.L.E. (1980). Coloured plane groups, Acta Crystallogr., A36, 884-888, doi:10.1107/S0567739480001866

Further reading



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