Polychromatic symmetry

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Three-colour symmetry operation of colour group p3[3]1 Threecolouranimation.gif
Three-colour symmetry operation of colour group p3[3]1

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]

Contents

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 3-coloured p3 patterns [4] : 415 
Uncoloured pattern p33-colour pattern p3[3]13-colour pattern p3[3]2
P3 pattern.svg Three-colour group p3(3)1.svg Three-colour group p3(3)2.svg

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 and it was also used as the cover art of Mott the Hoople’s debut album.

4-, 6-, 7-, 9- and 12-coloured p3 patterns
4 colours p3[4] [6] : 287 4.03.01 6 colours p3[6]7 colours p3[7]9 colours p3[9]112 colours p3[12]1
Four-colour group p3(4).svg Six-colour group p3(6).svg Seven-colour group p3(7).svg Nine-colour group p3(9)1.svg Twelve-colour group p3(12)1.svg

Group theory

Initial research by Wittke and Garrido (1959) [7] and by Niggli and Wondratschek (1960) [8] 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 [9] 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. [10] When each symmetry operation in G is associated with a unique colour permutation the pattern is said to be perfectly coloured. [11] [12]

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. [13] If the subgroup H is a normal subgroup then the quotient group G/H permutes all the colours. [14]

History

Number of colour groups

Number of strip (frieze) k-colour groups for k ≤ 12 [4] [6]
 Number of colours (k)
Underlying
group		23456789101112
p11111111111111
p1a111111111111
p1m131313131313
pm1121212121212
p11221212121212
pma231313131313
pmm251715171517
      Total strip
groups		17   719   717   719   717   719
Numbers of periodic (plane) k-colour groups for k ≤ 12 [4] [6] [37]
 Number of colours (k)
Underlying
group		23456789101112
p1 11211122112
pg 224252736211
pm 5210211216312223
cm 3272721338217
p2 21312142213
pgg 21414172519
pmg 5211211219312226
pmm 511319121210125
cmm 51111812129122
p3 -21-11-3--4
p31m 121-5-13--7
p3m1 121-4-13--7
p4 2-512-914-9
p4g 3-7-2-1313-10
p4m 5-13-2-2813-16
p6 121-5113--8
p6m 322-11-33--20
Total periodic
groups		462396149015166407513219

Both of the 3-colour p3 patterns, the unique 4-, 6-, 7-colour p3 patterns, one of the three 9-colour p3 patterns, and one of the four 12-colour p3 patterns are illustrated in the Example section above.

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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. 1 2 3 4 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. 1 2 3 Wieting, T.W. (1982). The mathematical theory of chromatic plane ornaments , Marcel Dekker, New York, ISBN   9780824715175
  7. ↑ 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
  8. ↑ 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
  9. 1 2 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
  10. 1 2 Senechal, M. (1990). Geometrical crystallography in Historical atlas of crystallography ed. Lima-de-Faria, J., Kluwer, Dordrecht, 52-53, ISBN   9780792306498
  11. ↑ Senechal, M. (1988). Color symmetry , Comput. Math. Applic., 16(5-8), 545-553, doi : 10.1016/0898-1221(88)90244-1
  12. ↑ Senechal, M. (1990). Crystalline symmetries: an informal mathematical introduction, Adam Hilger, Bristol, 74-87, ISBN   9780750300414
  13. 1 2 Senechal, M. (1983). Color symmetry and colored polyhedra , Acta Crystallogr., A39, 505-511, doi : 10.1107/s0108767383000987
  14. ↑ Coxeter, H.S.M. (1987). A simple introduction to colored symmetry, Int. J. Quantum Chemistry, 31, 455-461, doi : 10.1002/qua.560310317
  15. ↑ Belov, N.V. and Tarkhova, T.N. (1956). Colour symmetry groups, Sov. Phys. Cryst., 1, 5-11
  16. ↑ Belov, N.V. and Tarkhova, T.N. (1956). Colour symmetry groups, Sov. Phys. Cryst., 1, 487-488
  17. ↑ Belov, N.V. (1956). Moorish patterns of the Middle Ages and the symmetry groups, Sov. Phys. Cryst., 1, 482-483
  18. ↑ Belov, N.V. (1956). Three-dimensional mosaics with colored symmetry, Sov. Phys. Cryst., 1, 489-492
  19. ↑ 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
  20. ↑ Belov, N.V., Belova, E.N. and Tarkhova, T.N. (1959). More about the colour symmetry groups, Sov. Phys. Cryst., 3, 625-626
  21. ↑ 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
  22. ↑ Mackay, A.L. (1957). Extensions of space-group theory , Acta Crystallogr. 10, 543-548, doi : 10.1107/s0365110x57001966
  23. ↑ 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
  24. ↑ 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
  25. ↑ 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
  26. ↑ MacGillavry, C.H. (1976). Symmetry aspects of M. C. Escher's periodic drawings , International Union of Crystallography, Utrecht, ISBN   9789031301843
  27. ↑ Schnattschneider, D. (2004). M. C. Escher: Visions of Symmetry , Harry. N. Abrams, New York, ISBN   9780810943087
  28. ↑ Shubnikov, A.V., Belov, N.V. et. al. (1964). Colored symmetry, ed. W.T. Holser, Pergamon, New York
  29. ↑ Loeb, A.L. (1971). Color and Symmetry , Wiley, New York, ISBN   9780471543350
  30. ↑ 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)
  31. ↑ Senechal, M. (1983). Coloring symmetrical objects symmetrically, Math. Magazine, 56(1), 3-16, doi : 10.2307/2690259
  32. ↑ Cromwell, P.R. (1997). Polyhedra, Cambridge University Press, 327-348, ISBN   9780521554329
  33. ↑ Washburn, D.K. and Crowe, D.W. (1988). Symmetries of Culture: Theory and Practice of Plane Pattern Analysis , Washington University Press, Seattle, ISBN   9780295970844
  34. ↑ Makovicky, E. (2016). Symmetry through the eyes of old masters, de Gruyter, Berlin, 133-147, ISBN   9783110417050
  35. ↑ 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
  36. ↑ Conway, J.H., Burgeil, H. and Goodman-Strauss, C. (2008). The symmetries of things, A.K. Peters, Wellesley, MA, ISBN   9781568812205
  37. ↑ Jarratt, J.D. and Schwarzenberger, R.L.E. (1980). Coloured plane groups, Acta Crystallogr., A36, 884-888, doi : 10.1107/S0567739480001866

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