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Zeitschrift für Kristallographie - Crystalline Materials

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Volume 230, Issue 12


Coincidence indices of sublattices and coincidences of colorings

Manuel Joseph C. Loquias
  • Institute of Mathematics, University of the Philippines Diliman, C.P. Garcia Avenue, University of the Philippines Campus, Diliman, Quezon City, 1101, Philippines
  • Lehrstuhl für Mathematik und Statistik, Montanuniversität Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Peter Zeiner
Published Online: 2015-12-01 | DOI: https://doi.org/10.1515/zkri-2015-1864


Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice Λ1 and to a sublattice Λ2 may differ. Here, we examine the coloring of Λ1 induced by Λ2 to identify how the coincidence indices with respect to Λ1 and to Λ2 are related. This leads to a generalization of the notion of color symmetries of lattices to what we call color coincidences of lattices. Examples involving the cubic and hypercubic lattices are given to illustrate these ideas.

Keywords: coincidence index; coincidence site lattice; color coincidences; color symmetry


  • [1]

    G. Friedel, Leçons de Cristallographie, Hermann, Paris, 1911.Google Scholar

  • [2]

    S. Ranganathan, On the geometry of coincidence-site lattices, Acta Cryst. 1966, 21, 197.Google Scholar

  • [3]

    W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer, Berlin, 1970.Google Scholar

  • [4]

    D. H. Warrington, R. Lück, The use of the Wieringa roof to examine coincidence site quasilattices in icosahedral quasicrystals, in Aperiodic ’94 (Singapore), (Eds. G. Chapuis and W. Paciorek) World Scientific, p. 30, 1995.Google Scholar

  • [5]

    D. H. Warrington, R. Lück, Healing of slip planes and interfaces in quasiperiodic patterns, Ferroelectrics 2001, 250, 357.Google Scholar

  • [6]

    O. Radulescu, D. H. Warrington, R. Lück, Phason flips and reconstruction of grain boundaries in quasicrystals, in Aperiodic ’97 (Singapore), (Eds. M. de Boissieu, J.-L. Verger-Gaugry and R. Currat) World Scientific, p. 783, 1998.Google Scholar

  • [7]

    D. H. Warrington, Coincidence site lattices in quasicrystal tilings, Materials Science Forum, vol. 126, Trans Tech Publications, Dürnten, Switzerland, p. 57, 1993.Google Scholar

  • [8]

    H. Grimmer, W. Bollmann, D. H. Warrington, Coincidence-site lattices and complete pattern-shift lattices in cubic crystals, Acta Cryst. 1974, A30, 197.Google Scholar

  • [9]

    R. L. E. Schwarzenberger, N-dimensional crystallography, Res. Notes in Math., vol. 41, Pitman, London, 1980.Google Scholar

  • [10]

    P. A. B. Pleasants, M. Baake, J. Roth, Planar coincidences for N-fold symmetry, J. Math. Phys. 1996, 37, 1029.Google Scholar

  • [11]

    M. Baake, Combinatorial aspects of colour symmetries, J. Phys. A 1997, 30, 2687.Google Scholar

  • [12]

    R. Lück, Colour symmetry of 25 colours in quasiperiodic patterns, Phil. Mag. 2008, 88, 2049.Google Scholar

  • [13]

    R. Lück, D. Frettlöh, Ten colours in quasiperiodic and regular hyperbolic tilings, Z. Krist. 2008, 223, 782.Web of ScienceGoogle Scholar

  • [14]

    A. V. Shubnikov, N. V. Belov, Colored Symmetry, Pergamon, Oxford, 1964.Google Scholar

  • [15]

    D. B. Litvin, Magnetic Group Tables, IUCr, 2013.Google Scholar

  • [16]

    D. Harker, Colored lattices, Proc. Natl. Acad. Sci. USA 1978, 75, 5264.Google Scholar

  • [17]

    R. L. E. Schwarzenberger, Colour symmetry, Bull. Lond. Math. Soc. 1984, 16, 209.CrossrefGoogle Scholar

  • [18]

    M.L.A.N. De Las Peñas, R.P. Felix, Color groups associated with square and hexagonal lattices, Z. Krist. 2007, 222, 505.Google Scholar

  • [19]

    H. S. M. Coxeter, Coloured symmetry, in M.C. Escher: Art and Science, (Eds. H. S. M. Coxeter, M. Emmer, R. Penrose and M. Teuber) North-Holland, Amsterdam, p. 15, 1986.Google Scholar

  • [20]

    M. Senechal, Color symmetry, Comput. Math. Appl. 1988, 16, 545.CrossrefGoogle Scholar

  • [21]

    R. V. Moody, J. Patera, Colourings of quasicrystals, Can. J. Phys. 1994, 72, 442.Google Scholar

  • [22]

    R. Lifshitz, Theory of color symmetry for periodic and quasiperiodic crystals, Rev. Modern Phys. 1997, 69, 1181.Google Scholar

  • [23]

    E. P. C. Bugarin, M. L. A. N. De Las Peñas, D. Frettlöh, Perfect colourings of cyclotomic integers, Geom. Dedicata 2013, 162, 271.Google Scholar

  • [24]

    J. M. Howe, Interfaces in Materials, John Wiley & Sons, New York, 1997.Google Scholar

  • [25]

    A. P. Sutton, R. W. Balluffi, Interfaces in Crystalline Materials, Oxford University Press, Oxford, 2006.Google Scholar

  • [26]

    M. J. C. Loquias, P. Zeiner, The coincidence problem for shifted lattices and crystallographic point packings, Acta Cryst. 2014, A70, 656.Web of ScienceGoogle Scholar

  • [27]

    M. J. C. Loquias, P. Zeiner, Colourings of lattices and coincidence site lattices, Phil. Mag. 2011, 91, 2680.Google Scholar

  • [28]

    M. Baake, U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, Cambridge University Press, Cambridge, 2013.Google Scholar

  • [29]

    H. Grimmer, Disorientations and coincidence rotations for cubic lattices, Acta Cryst. 1974, A30, 685.Google Scholar

  • [30]

    M. Baake, Solution of the coincidence problem in dimensions d≤4, in The Mathematics of Long-Range Aperiodic Order, (Ed. R. V. Moody), Kluwer, Dordrecht, p. 9, 1997.Google Scholar

  • [31]

    P. Zeiner, Coincidences of hypercubic lattices in 4 dimensions, Z. Krist. 2006, 221, 105.Google Scholar

  • [32]

    B. Grünbaum, G. C. Shephard, Tilings and Patterns, W.H. Freeman, New York, 1987.Google Scholar

  • [33]

    P. Zeiner, Multiplicativity in the theory of coincidence site lattices, J. Phys.: Conf. Ser. 2010, 226, 012025.Google Scholar

  • [34]

    M. Koecher, R. Remmert, Hamilton’s quaternions, in Numbers, (Eds. H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, A. Prestel and R. Remmert) Graduate Texts in Mathematics, vol. 123, Springer, New York, p. 189, 1991.Google Scholar

  • [35]

    J. H. Conway, D. A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A.K. Peters, Ltd., Massachusetts, 2003.Google Scholar

  • [36]

    A. Hurwitz, Vorlesungen über die Zahlentheorie der Quaternionen, Springer, Berlin, 1919.Google Scholar

  • [37]

    G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford, 1979.Google Scholar

  • [38]

    H. Grimmer, The generating function for coincidence site lattices in the cubic system, Acta Cryst. 1984, A40, 108.Google Scholar

  • [39]

    J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, Fundamental Principles of Mathematical Sciences, vol. 290, Springer, New York, 1999.Google Scholar

  • [40]

    H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, H. Zassenhaus, Crystallographic groups of four-dimensional space, Wiley-Interscience, New York-Chichester-Brisbane, 1978.Google Scholar

  • [41]

    M. Baake, P. Zeiner, Coincidences in four dimensions, Phil. Mag. 2008, 88, 2025.Google Scholar

  • [42]

    S. Glied, M. Baake, Similarity versus coincidence rotations of lattices, Z. Krist. 2008, 223, 770.Web of ScienceGoogle Scholar

About the article

Corresponding author: Peter Zeiner, Fakultät für Mathematik, Universität Bielefeld, Universitätsstraße 25, 33501, Germany, Tel.: +49-521-1064791, Fax: +49-521-1066481, E-mail:

Received: 2015-05-29

Accepted: 2015-11-12

Published Online: 2015-12-01

Published in Print: 2015-11-01

Citation Information: Zeitschrift für Kristallographie - Crystalline Materials, Volume 230, Issue 12, Pages 749–759, ISSN (Online) 2196-7105, ISSN (Print) 2194-4946, DOI: https://doi.org/10.1515/zkri-2015-1864.

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