Abstract
As crystallography merges with materials science and engineering, mathematical crystallography is growing in new directions, including: Characterizing new materials with unusual properties; Imaging, including but not limited to diffraction; Exploring and exploiting superspaces; Mapping the aperiodic landscape, from chaos to classical periodicity and beyond; Re-modeling the structures of real crystals, both periodic and aperiodic; Modeling self-assembly and self-reorganization on the nanoscale. In short, it’s not (just) about space groups and tilings anymore.
References
[1] D. Hilbert, Mathematical problems. B. Am. Math. Soc.2000, 37, 407.Search in Google Scholar
[2] H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space, John Wiley & Sons, New York, 1978.Search in Google Scholar
[3] For a current summary, see https://en.wikipedia.org/wiki/Space_groups. Accessed September 7, 2015.Search in Google Scholar
[4] B. Yandell, ed., The Honors Class, A.K. Peters Natick, 2001.10.1201/9781439864227Search in Google Scholar
[5] M. Senechal, What is . . . a quasicrystal? Notices Amer. Math. Soc.2006, 53, 886; http://www.ams.org/notices/200608/whatis–senechal.pdf. Accessed September 7, 2015.Search in Google Scholar
[6] M. Senechal, J. E. Taylor, Quasicrystals: The view from Stockholm. Math. Intell.2013, 35, 1.Search in Google Scholar
[7] M. Senechal, Delaunay sets and condensed matter: The dialogue continues. Proc. Steklov I. Math.2015, 288, 259.Search in Google Scholar
[8] M. Senechal, Structures beyond superspace. Acta Cryst.2015, B71, 250.Search in Google Scholar
[9] P. M. De Wolff, The pseudo-symmetry of modulated crystal structures. Acta Cryst.1974, A30, 777.Search in Google Scholar
[10] A. Janner, T. Janssen, Symmetry of periodically distorted crystals. Phys. Rev. B1977, 15, 643.10.1103/PhysRevB.15.643Search in Google Scholar
[11] B. Grünbaum, G. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987.Search in Google Scholar
[12] A. Mackay, Crystallography and the penrose pattern. Physica A1982, 114, 609.10.1016/0378-4371(82)90359-4Search in Google Scholar
[13] N. G. De Bruijn, Algebraic theory of Penrose’s non-periodic tilings of the plane. I, II. Indag. Math.1981, 43, 39.Search in Google Scholar
[14] M. Senechal, Quasicrystals and Geometry, Cambridge University Press, Cambridge, 1995; corrected paperback edition 1996; reprinted, and ebook, 2007.Search in Google Scholar
[15] B. Delone, A. Alexandrov, N. Padurov, Mathematical Foundations of the Structural Analysis of Crystals, Leningrad, ONTI, 1934; [in Russian].Search in Google Scholar
[16] B. N. Delone, N. P. Dolbilin, M. I. Shtogrin, R. V. Galiulin, A local criterion for regularity of a system of points. Dokl. Akad. Nauk SSSR1976, 227, 19, Sov. Math. Dokl.1976, 17, 319.Search in Google Scholar
[17] N. P. Dolbilin, J. C. Lagarias, M. Senechal, Multiregular point systems. Discrete Comput. Geom.1998, 20, 477.Search in Google Scholar
[18] N. P. Dolbilin, E. Schulte, The local theorem for monotypic tilings. Electron. J. Comb. 2004, 11, R7.Search in Google Scholar
[19] D. Shechtman, I. Blech, D. Gratias, J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett.1984, 53, 1951.Search in Google Scholar
[20] International Union of Crystallography, Commission on Aperiodic Crystals, http://www.iucr.org/iucr/commissions/aperiodic-crystals/. Accessed September 7, 2015.Search in Google Scholar
[21] U. Grimm, Aperiodic crystals and beyond. Acta Cryst.2015, B71, 258.Search in Google Scholar
[22] M. Baake, R. Moody, P. Pleasants, Diffraction from visible lattice points and kth power free integers. Discrete Math.1999, 221, 3.Search in Google Scholar
[23] M. Höffe, M. Baake, Surprises in diffuse scattering. Z. Krist.2000, 215, 441.Search in Google Scholar
[24] M. Baake, U. Grimm, Theory of Aperiodic Order: A Mathematical Invitation, Cambridge University Press, Cambridge, 2013.10.1017/CBO9781139025256Search in Google Scholar
[25] J. Lagarias, Mathematical quasicrystals and the problem of diffraction. Directions in Mathematical Quasicrystals, (Eds. M. Baake, R. Moody) Providence RI, CRM Monograph series, AMS, 2000, 13, 61.10.1090/crmm/013/03Search in Google Scholar
[26] J. Lagarias, P. Pleasants, Repetitive Delone sets and perfect quasicrystals. arXiv:math/9909033 v.3.Search in Google Scholar
[27] J. Lagarias, Geometric models for quasicrystals. I: Delone sets of finite type. Discrete. Comput. Geom.1999, 21, 161.Search in Google Scholar
[28] N. Strungaru, Almost periodic measures and long-range order in Meyer sets. Discrete Comput. Geom.2005, 33, 483.Search in Google Scholar
[29] H. Takakura, C. P. Gómez, A. Yamamoto, M. De Boissieu, A. P. Tsai, Atomic structure of the binaryicosahedral Yb–Cd quasicrystal. Nature Mater.2007, 6, 58.Search in Google Scholar
[30] D. Gratias, F. Puyraimond, M. Quiquandon, A. Katz, Atomic clusters in icosahedral F–type quasicrystals. Phys. Rev. B2000, 63, 024202.10.1103/PhysRevB.63.024202Search in Google Scholar
[31] T. Kuhn, The Structure of Scientific Revolutions, Univ. Chicago Press, Chicago, 1962.Search in Google Scholar
[32] A. Pankova, T. Akhmetshina, V. Blatov, D. Proserpio, A collection of topological types of nanoclusters and its application to icosahedron – based intermetallics. http://pubs.acs.org/doi/pdf/10.1021/acs.inorgchem.5b00960. Accessed September 7, 2015.Search in Google Scholar
[33] http://www.nobelprize.org/nobelprizes/chemistry/laureates/2011/press.html. Accessed September 7, 2015.Search in Google Scholar
[34] C. Radin, A revolutional material. Notices Amer. Math. Soc.2013, 60, 310; http://www.ams.org/notices/201303/rnoti-p310.pdf/. Accessed September 7, 2015.Search in Google Scholar
[35] J. Lagarias, (ed.) The Kepler Conjecture: The Hales-Ferguson Proof, Springer, New York, 2011.10.1007/978-1-4614-1129-1Search in Google Scholar
[36] J. M. Mortejano-Carrizales, J. L. Rodríguez, C. Gutierrez-Wing, M. Miki, M. Jose-Yacaman. Crystallography and Shape of Nanoparticles and Clusters. In Encyclopedia of Nanoscience and Nanotechnology, Vol. 2, (Eds. H. S. Nalwa) American Scientific Publishers, Valencia, California, 2004, 238.Search in Google Scholar
[37] F. C. Frank, Supercooling of liquids. Proc. Roy. Soc. A.1952, 215, 43.Search in Google Scholar
[38] J. D. Bernal, The structure of liquids. Proc. Roy. Soc. A1964, 280, 299.10.1098/rspa.1964.0147Search in Google Scholar
[39] C. Day, Experiments vindicate a 50 year old explanation of how liquid metals resist solidification. Phys. Today2003, 56, 24.10.1063/1.1603067Search in Google Scholar
[40] K. Kelton, The influence of icosahedral ordering in metallic liquids. http://www.youtube.com/watch?v=5m4cyP0YTk8/. Accessed September 7, 2015.Search in Google Scholar
[41] M. V. Jaríc, D. Gratias, (eds). Extended Icosahedral Structures, Boston, Academic, 1989.Search in Google Scholar
[42] B. Dume, Glass arrested on the road to crystallization. http://www.Physicsworld.com, Jun 26, 2008. Accessed September 7, 2015.Search in Google Scholar
[43] A. Mackay, A dense non-crystallographic packing of equal spheres. Acta Cryst.1962, 15, 916.Search in Google Scholar
[44] Y. Wang, S. Teitel, C. Dellago, Melting of icosahedral gold nanoclusters from molecular dynamics simulations. J. Chem. Phys.2005, 122, 214722.Search in Google Scholar
[45] http://glotzerlab.engin.umich.edu/home/. Accessed September 7, 2015.Search in Google Scholar
[46] M. Engel, A. Haji-Akbari, S. C. Glotzer, A dense quasicrystal phase of hard tetrahedra. Acta Cryst.2011, A67, C144. For a history of tetrahedral packing, see J. Lagarias, C. M. Zong, Mysteries in packing regular tetrahedra. Notices Amer. Math. Soc.2012, 59, 1540. http://www.ams.org/notices/201211/rtx121101540p.pdf/. Accessed September 7, 2015.Search in Google Scholar
[47] A. Keys, S. C. Glotzer How do quasicrystals grow? Phys. Rev. Lett.2007, 99, 235503.Search in Google Scholar
[48] M. Engel, P. F. Damasceno, C. L. Phillips, S. C. Glotzer, Computational self-assembly of a one-component icosahedral quasicrystal. Nature Mater.2015, 14, 109.Search in Google Scholar
[49] B. Mann, 23 Mathematical Challenges. DARPA Washington, D. C., Defense Sciences Offices, 2007.Search in Google Scholar
©2015 by De Gruyter
