Polychromatic symmetry

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).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,

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.Harker, D. (1981). The three-colored three-dimensional space groups, Acta Crystallogr., A37, 286-292, Mainzer, K. (1996). Symmetries of nature: a handbook for philosophy of nature and science, de Gruyter, Berlin, 162-168,

Example

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

+ Uncoloured and 3-coloured p3 patternsGrünbaum, B. and Shephard, G.C. (1987). Tilings and patterns, W.H. Freeman, New York, Uncoloured pattern p3 3-colour pattern p31 3-colour pattern p32 There are two distinct ways of colouring the p3 pattern with three colours: p31 and p32 where the figure in square brackets indicates the number of colours, and the subscript distinguishes between multiple cases of coloured patterns.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,

Taking a single motif in the pattern p31 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 p31 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 p3Wieting, T.W. (1982). The mathematical theory of chromatic plane ornaments, Marcel Dekker, New York, 6 colours p3 7 colours p3 9 colours p31 12 colours p31

Group theory

Initial research by Wittke and Garrido (1959)Wittke O. and Garrido J. (1959). Symétrie des polyèdres polychromatiques, Bull. Soc. française de Minéral. et de Crist., 82(7-9), 223-230; and by Niggli and Wondratschek (1960)Niggli, A. and Wondratschek, H. (1960). Eine Verallgemeinerung der Punktgruppen. I. Die einfachen Kryptosymmetrien, Z. Krist., 114(1-6), 215-231 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 Burckhardtvan der Waerden, B.L. and Burkhardt, J.J. (1961). Farbgruppen, Z. Krist, 115, 231-234, 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.Senechal, M. (1990). Geometrical crystallography in Historical atlas of crystallography ed. Lima-de-Faria, J., Kluwer, Dordrecht, 52-53, When each symmetry operation in G is associated with a unique colour permutation the pattern is said to be perfectly coloured.Senechal, M. (1988). Color symmetry, Comput. Math. Applic., 16(5-8), 545-553, Senechal, M. (1990). Crystalline symmetries: an informal mathematical introduction, Adam Hilger, Bristol, 74-87,

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.Senechal, M. (1983). Color symmetry and colored polyhedra, Acta Crystallogr., A39, 505-511, If the subgroup H is a normal subgroup then the quotient group G/H permutes all the colours.Coxeter, H.S.M. (1987). A simple introduction to colored symmetry, Int. J. Quantum Chemistry, 31, 455-461,

History

1956 First papers on polychromatic, as opposed to dichromatic, symmetry groups are published by Belov and his co-workers.Belov, N.V. and Tarkhova, T.N. (1956). Colour symmetry groups, Sov. Phys. Cryst., 1(1), 5-11Belov, N.V. and Tarkhova, T.N. (1956). Colour symmetry groups, Sov. Phys. Cryst., 1(6), 487-488Belov, N.V. (1956). Moorish patterns of the Middle Ages and the symmetry groups, Sov. Phys. Cryst., 1(5), 482-483Belov, N.V. (1956). Three-dimensional mosaics with colored symmetry, Sov. Phys. Cryst., 1(6), 489-492Belov, 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(1), 16-18Belov, N.V., Belova, E.N. and Tarkhova, T.N. (1959). More about the colour symmetry groups, Sov. Phys. Cryst., 3, 625-626 Vainshtein and Koptsik (1994) summarise the Russian work.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, 1957 Mackay publishes the first review of the Russian work in English.Mackay, A.L. (1957). Extensions of space-group theory, Acta Crystallogr. 10, 543-548, Subsequent reviews were published by 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), by Senechal (1990) and by Thomas (2012).Thomas, B.G. (2012). Colour symmetry: the systematic coloration of patterns and tilings in Colour Design, ed. Best, J., Woodhead Publishing, 381-432, Late 1950s M.C. Escher's artworks based on dichromatic and polychromatic patterns popularise colour symmetry amongst scientists.MacGillavry, 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 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 1971 Derivation by Loeb in Color and Symmetry of 2D colour symmetry configurations using rotocenters.Loeb, A.L. (1971). Color and Symmetry, Wiley, New York, 1974 Publication of Symmetry in Science and Art by Shubnikov and Koptsik with extensive coverage of polychromatic symmetry.Shubnikov, A.V. and Koptsik, V.A. (1974). Symmetry in science and art, Plenum Press, New York, (original in Russian published by Nauka, Moscow, 1972) 1983 Senechal examines the problem of colouring polyhedra symmetrically using group theory.Senechal, M. (1983). Coloring symmetrical objects symmetrically, Math. Magazine, 56(1), 3-16, Cromwell later uses an algorithmic counting approach (1997).Cromwell, P.R. (1997). Polyhedra, Cambridge University Press, 327-348, 1988 Washburn and Crowe apply colour symmetry analysis to cultural patterns and objects.Washburn, D.K. and Crowe, D.W. (1988). Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, Washington University Press, Seattle, Washburn and Crowe inspired further work, for example by Makovicky.Makovicky, E. (2016). Symmetry through the eyes of old masters, de Gruyter, Berlin, 133-147, 1997 Lifshitz extends the theory of color symmetry from periodic to quasiperiodic crystals.Lifshitz, R. (1997). Theory of color symmetry for periodic and quasiperiodic crystals, Rev. Mod. Phys., 69(4), 1181–1218, 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.Conway, J.H., Burgeil, H. and Goodman-Strauss, C. (2008). The symmetries of things, A.K. Peters, Wellesley, MA,

Number of colour groups + Number of strip (frieze) k-colour groups for k ≤ 12 Number of colours (k) Underlyinggroup 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 ≤ 12Jarratt, J.D. and Schwarzenberger, R.L.E. (1980). Coloured plane groups, Acta Crystallogr., A36, 884-888, Number of colours (k) Underlyinggroup 2 3 4 5 6 7 8 9 10 11 12 1 1 2 1 1 1 2 2 1 1 2 2 2 4 2 5 2 7 3 6 2 11 5 2 10 2 11 2 16 3 12 2 23 3 2 7 2 7 2 13 3 8 2 17 2 1 3 1 2 1 4 2 2 1 3 2 1 4 1 4 1 7 2 5 1 9 5 2 11 2 11 2 19 3 12 2 26 5 1 13 1 9 1 21 2 10 1 25 5 1 11 1 8 1 21 2 9 1 22 - 2 1 - 1 1 - 3 - - 4 1 2 1 - 5 - 1 3 - - 7 1 2 1 - 4 - 1 3 - - 7 2 - 5 1 2 - 9 1 4 - 9 3 - 7 - 2 - 13 1 3 - 10 5 - 13 - 2 - 28 1 3 - 16 1 2 1 - 5 1 1 3 - - 8 3 2 2 - 11 - 3 3 - - 20 Total periodic groups 46 23 96 14 90 15 166 40 75 13 219 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.

References

Polychromatic symmetry

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