Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Topological Crystallography

Part of the book series: Surveys and Tutorials in the Applied Mathematical Sciences ((STAMS,volume 6))

  • 2291 Accesses

Abstract

Applications of mathematics to crystallography have a long history. The theory of crystallographic groups (space groups in jargon) is a traditional field dating back to the first half of the nineteenth century, which, needless to say, has been playing a significant role in the classification of crystals in view of the symmetry. Graph theory is another powerful area for the obvious reason that it is used to study the microscopic structure of a crystal (and any molecule) as a 3D (three-dimensional) network, in which each atom (or each cluster of atoms) is represented by a vertex of the net, and each edge of the net represents a bond (or a polymeric ligand) in the crystal structure.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    To be precise, the notion of crystallographic groups has its origin in the study of morphology (the observed shapes) of crystals. In this sense, the link between mathematics and crystallography is ancient. See the beginning of Chap. 6.

  2. 2.

    In this book, crystals mean solids composed of atoms arranged in an orderly repetitive array. In crystallography, more general materials such as quasicrystals are counted as crystals.

  3. 3.

    Wells [109] initiated a systematic study of crystal structures as 3D networks. Applications of graph theory to chemistry could be traced back to 1864 when the Edinburgh chemist Crum Brown proposed representing chemical compounds by graphs.

  4. 4.

    By a crystal structure we mean an abstract graph associated with a crystal net.

  5. 5.

    The notion of period lattices is a generalization of Bravais lattices in crystallography. Precisely speaking, a Bravais lattice is a representative of period lattices when classified by symmetry.

  6. 6.

    Diamond is an allotrope of carbon which is formed and synthesized at high-pressure and high-temperature conditions, and is known to be less stable than graphite though the conversion rate from diamond to graphite is negligible at ambient conditions. Silicon and germanium adopt similar types of crystal structure.

  7. 7.

    See Sect. 8.3 and Notes (IV) in Chap. 8 for the detailed account where we explain the reason why the K 4 crystal deserves to be called “diamond twin”. The picture of the K 4 crystal is given in Fig. 1.4.

  8. 8.

    This carbon allotrope, formed when meteorites containing graphite strike the earth, is named in honour of crystallographer Kathleen Lonsdale, also referred to as the hexagonal diamond.

  9. 9.

    Source of the figure: WebElements (http://www.webelements.com/).

  10. 10.

    They also use the term “minimal nets” for maximal abelian covering graphs [9] and the term “cycle spaces” for homology groups [42]. See Notes (V) in Chap. 8.

  11. 11.

    Even for the description of crystallographic groups there are three main systems of notations; one used in mathematics, and other two (the Schoenflies system and International system) used by chemists and crystallographers.

  12. 12.

    In chemistry, the term “topological crystal(lography)” is used sometimes in a different context.

  13. 13.

    See the book [44] by J-G. Eon, W.E. Klee, B. Souvignier and J.S. Rutherford as a reference in crystallography.

  14. 14.

    In [58], we have used the term “crystal lattice” instead.

  15. 15.

    This is the so-called Platonic view; that is, we mathematicians insist that mathematical entities are abstract in not being spatiotemporally located, and hence lie outside of the real world.

  16. 16.

    Algorithm means a step-by-step procedure involving a precise set of instructions for what to do next. With access to a computer and with some work in computer graphics based on our algorithm, 3D crystal structures may be displayed on a screen.

  17. 17.

    In [96], I used the term “canonical placement”.

  18. 18.

    For a closed curve in the plane, if its perimeter is L and the area that it encloses is A, then 4πA ≤ L 2, where equality holds if and only if the curve is a circle.

  19. 19.

    A minimizer of a given function (or functional) is a point (or function) at which the minimum value is attained.

  20. 20.

    Fixing a period lattice is equivalent to imposing a periodic boundary condition.

  21. 21.

    The program is available at http://www.gavrog.org/.

  22. 22.

    Once we find a hypothetical crystal, a systematic prediction of its physical properties for appropriate atoms can be carried out by first principles calculations used in chemistry. The prediction appealing to the computer power encourages (or discourages) material scientists to synthesize the hypothetical crystals [26].

  23. 23.

    The usage of the term “lattice” for crystal structures may give rise to confusion because customarily a lattice means a discrete subgroup of \({\mathbb{R}}^{d}\) (or more generally a discrete subgroup of a Lie group). But we will follow the convention to use “lattice” for some crystal structures (see Remark in Sect. 2.2).

  24. 24.

    A “proof” is a logical procedure to derive what we anticipate to be true from what we have already known to be true.

References

  1. Artamkin IV (2006) Discrete Torelli theorem. Sbornik: Math 197:1109–1120

    Article  MathSciNet  MATH  Google Scholar 

  2. Auslander L, Kuranishi M (1957) On the holonomy group of locally Euclidean spaces. Ann Math 65:411–415

    Article  MathSciNet  MATH  Google Scholar 

  3. Bacher R, De La Harpe P, Nagnibeda T (1997) The lattice of integral flows and the lattice of integral cuts on a finite graph. Bull Soc Math Fr 125:167–198

    MATH  Google Scholar 

  4. Bader M, Klee WE, Thimm G (1997) The 3-regular nets with four and six vertices per unit cell. Z Kristallogr 212:553–558

    Article  Google Scholar 

  5. Baker M, Norine S (2007) Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv Math 215:766–788

    Article  MathSciNet  MATH  Google Scholar 

  6. Baker M, Norine S (2009) Harmonic morphisms and hyperelliptic graphs. Int Math Res Notices (15):2914–2955

    MathSciNet  Google Scholar 

  7. Baker M, Faber X (2011) Metric properties of the tropical Abel-Jacobi map. J Algebraic Combin 33:349–381

    Article  MathSciNet  MATH  Google Scholar 

  8. Bass H (1992) The Ihara-Selberg zeta function of a tree lattice. Int J Math 3:717–797

    Article  MathSciNet  MATH  Google Scholar 

  9. Beukemann A, Klee WE (1992) Minimal nets. Z Kristallogr 201:37–51

    Article  MathSciNet  MATH  Google Scholar 

  10. Biggs NL, Lloyd EK, Wilson RJ (1999) Graph theory 1736–1936. Oxford University Press, Oxford

    Google Scholar 

  11. Biggs NL (1993) Algebraic graph theory. Cambridge University Press, Cambridge

    Google Scholar 

  12. Biggs NL (1997) Algebraic potential theory on graphs. Bull Lond Math Soc 29:641–682

    Article  MathSciNet  Google Scholar 

  13. Blatov V (2000) Search for isotypism in crystal structures by means of the graph theory. Acta Crystallogr A 56:178–188

    Article  Google Scholar 

  14. Bollmann W (1972) The basic concepts of the O-lattice theory. Surf Sci 31:1–11

    Article  Google Scholar 

  15. Bollobas B (1998) Modern graph theory. Springer, New York

    Book  MATH  Google Scholar 

  16. Brown KS (1972) Cohomology of groups. Springer, New York

    Google Scholar 

  17. Bryant PR (1967) Graph theory applied to electrical networks. In: Harary F (ed) Graph theory and theoretical physics. Academic, New York, pp 111–137

    Google Scholar 

  18. Caporaso L, Viviani F (2010) Torelli theorem for graphs and tropical curves. Duke Math J 153:129–171

    Article  MathSciNet  MATH  Google Scholar 

  19. Charlap LS (1986) Bieberbach groups and flat manifolds. Springer, New York

    Book  MATH  Google Scholar 

  20. Chung SJ, Hahn T, Klee WE (1984) Nomenclature and generation of three-periodic nets: the vector method. Acta Crystallogr A 40:42–50

    Article  MathSciNet  MATH  Google Scholar 

  21. Conway JH, Burgiel H, Goodman-Strauss C (2008) The symmetries of things. A K Peters Ltd, Wellesley

    MATH  Google Scholar 

  22. Coxeter HSM (1955) On Laves’ graph of girth ten. Can J Math 7:18–23

    Article  MathSciNet  MATH  Google Scholar 

  23. Coxeter HSM (1973) Regular polytopes. Dover, New York

    Google Scholar 

  24. Coxeter HSM (1974) Regular complex polytopes. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  25. Cromwell P (1999) Polyhedra. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  26. Curtarolo S, Morgan D, Persson K, Rodgers J, Ceder G (2003) Predicting crystal structures with data mining of quantum calculations. Phys Rev Lett 91:135503

    Article  Google Scholar 

  27. Dai J, Li Z, Yang J (2010) Boron K 4 crystal: a stable chiral three-dimensional sp2 network. Phys Chem Chem Phys 12:12420–12422

    Article  Google Scholar 

  28. Danzer L. Something about [10, 3] a . Unpublished

    Google Scholar 

  29. Delgado-Friedrichs O, Dress A, Huson D, Klinowski J, Mackay A (1999) Systematic enumeration of crystalline networks. Nature 400:644–647

    Article  Google Scholar 

  30. Delgado-Friedrichs O, O’Keeffe M (2003) Identification of and symmetry computation for crystal nets. Acta Crystallogr A 59:351–360

    Article  MathSciNet  Google Scholar 

  31. Delgado-Friedrichs O, O’Keeffe M, Yaghin OM (2003) Three-periodic nets and tilings: regular and quasiregular nets. Acta Crystallogr A 59:22–27

    Article  Google Scholar 

  32. Delgado-Friedrichs O (2004) Barycentric drawings of periodic graphs. Lect Notes Comput Sci 2912:178–189

    Article  MathSciNet  Google Scholar 

  33. Delgado-Friedrichs O, Foster MD, O’Keeffe M, Proserpio DM, Treacy MMJ, Yaghi OM (2005) What do we know about three-periodic nets? J Solid State Chem 178:2533–2554

    Article  Google Scholar 

  34. Delgado-Friedrichs O, O’Keeffe M (2007) Three-periodic tilings and nets: face-transitive tilings and edge-transitive nets revisited. Acta Crystallogr A 63:344–347

    Article  MathSciNet  Google Scholar 

  35. Delgado-Friedrichs O, O’Keeffe M (2009) Edge-transitive lattice nets. Acta Crystallogr A 65:360–363

    Article  MathSciNet  Google Scholar 

  36. Diestel R (2006) Graph theory. Springer, New York

    Google Scholar 

  37. Dixmier J (1981) Von Neumann algebras. North-Holland, Amsterdam

    MATH  Google Scholar 

  38. Ebeling W (1994) Lattices and codes. Vieweg, Wiesbaden

    Book  MATH  Google Scholar 

  39. Eells J, Sampson JH (1964) Harmonic mappings of Riemannian manifolds. Am J Math 86:109–160

    Article  MathSciNet  MATH  Google Scholar 

  40. Eells J, Fuglede B (2001) Harmonic maps between Riemannian polyhedra. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  41. Eon J-G (1998) Geometrical relationships between nets mapped on isomorphic quotient graphs: examples. J Solid State Chem 138:55–65

    Article  Google Scholar 

  42. Eon J-G (1999) Archetypes and other embeddings of periodic nets generated by orthogonal projection. J Solid State Chem 147:429–437

    Article  Google Scholar 

  43. Eon J-G (2011) Euclidean embeddings of periodic nets: definition of a topologically induced complete set of geometric descriptors for crystal structures. Acta Crystallogr A 67:68–86

    Article  MathSciNet  Google Scholar 

  44. Eon J-G, Klee WE, Souvignier B, Rutherford JS (2012) Graph-theory in crystallography and crystal chemistry. Oxford University Press with IUCr (to be published)

    Google Scholar 

  45. Greenberg M (1971) Lectures on algebraic topology. Benjamin, Menlo Park

    Google Scholar 

  46. Gromov M (1999) Metric structures for Riemannian and non-Riemannian spaces. Birkhäuser, Basel

    MATH  Google Scholar 

  47. Harper PG (1955) Single band motion of conduction electrons in a uniform magnetic field. Proc Phys Soc Lond A 68:874–878

    Article  MATH  Google Scholar 

  48. Hashimoto K (1990) On zeta and L-functions of finite graphs. Int J Math 1:381–396

    Article  MATH  Google Scholar 

  49. Hörmander L (1983) The analysis of linear partial differential operators I. Springer, New York

    Google Scholar 

  50. Hyde ST, O’Keeffe M, Proserpio DM (2008) A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics. Angew Chem Int Ed 47:7996–8000

    Article  Google Scholar 

  51. Ihara Y (1966) On discrete subgroups of the two by two projective linear group over p-adic fields. J Math Soc Jpn 18:219–235

    Article  MathSciNet  MATH  Google Scholar 

  52. Itoh M, Kotani M, Naito H, Kawazoe Y, Adschiri T (2009) New metallic carbon crystal. Phys Rev Lett 102:055703

    Article  Google Scholar 

  53. Jost J (1996) Generalized harmonic maps between metric spaces. In: Geometric analysis and calculus of variations. International Press, Cambridge, pp 143–174

    Google Scholar 

  54. Katsuda A, Sunada T (1990) Closed orbits in homology classes. Publ Math IHES 71:5–32

    MathSciNet  MATH  Google Scholar 

  55. Klein H-J (1996) Systematic generation of models for crystal structures. Math Model Sci Comput 6:325–330

    Google Scholar 

  56. Koch E, Fischer W (1995) Sphere packings with three contacts per sphere and the problem of the least dense sphere packing. Z Kristallogr 210:407–414

    Article  Google Scholar 

  57. Kotani M, Sunada T (2000) Zeta functions of finite graphs. J Math Sci Univ Tokyo 7:7–25

    MathSciNet  MATH  Google Scholar 

  58. Kotani M, Sunada T (2000) Standard realizations of crystal lattices via harmonic maps. Trans Am Math Soc 353:1–20

    Article  MathSciNet  Google Scholar 

  59. Kotani M, Sunada T (2000) Jacobian tori associated with a finite graph and its abelian covering graphs. Adv Appl Math 24:89–110

    Article  MathSciNet  MATH  Google Scholar 

  60. Kotani M, Sunada T (2000) Albanese maps and off diagonal long time asymptotics for the heat kernel. Commun Math Phys 209:633–670

    Article  MathSciNet  MATH  Google Scholar 

  61. Kotani M, Sunada T (2003) Spectral geometry of crystal lattices. Contemporary Math 338:271–305

    Article  MathSciNet  Google Scholar 

  62. Kotani M, Sunada T (2006) Large deviation and the tangent cone at infinity of a crystal lattice. Math Z 254:837–870

    Article  MathSciNet  MATH  Google Scholar 

  63. Krámli A, Szász D (1983) Random walks with internal degree of freedom, I. Local limit theorem. Z. Wahrscheinlichkeittheorie 63:85–95

    Article  MATH  Google Scholar 

  64. Kuchment P (1993) Floquet theory for partial differential operators. Birkhäuser, Basel

    Book  Google Scholar 

  65. Lang S (1987) Linear algebra. Springer, Berlin

    MATH  Google Scholar 

  66. Magnus W, Karrass A, Solitar D (1976) Combinatorial group theory. Dover, New York

    MATH  Google Scholar 

  67. Mikhalkin G, Zharkov I (2008) Tropical curves, their Jacobians and theta functions. In: Alexeev V et al (eds) Curves and abelian varieties. International conference, 2007. Contemporary Math 465:203–230

    Article  MathSciNet  Google Scholar 

  68. Milnor J (1969) Morse theory. Princeton University Press, Princeton

    Google Scholar 

  69. Nagano T, Smith B (1975) Minimal varieties and harmonic maps in tori. Commun Math Helv 50:249–265

    Article  MATH  Google Scholar 

  70. Nagnibeda T (1997) The Jacobian of a finite graph. Contemporary Math 206:149–151

    Article  MathSciNet  Google Scholar 

  71. Neukirch J (1999) Algebraic number theory. Springer, Berlin

    Book  MATH  Google Scholar 

  72. Newman P, Stoy G, Thompson E (1994) Groups and geometry. Oxford University Press, Oxford

    Google Scholar 

  73. Oganov A (ed) (2010) Modern methods of crystal structure prediction. Wiley-VCH, Berlin

    Google Scholar 

  74. Oda T, Seshadri CS (1979) Compactifications of the generalized Jacobian variety. Trans Am Math Soc 253:1–90

    Article  MathSciNet  MATH  Google Scholar 

  75. Oda T (2011) Voronoi tilings hidden in crystals—the case of maximal abelian coverings arXiv:1204.6555 [math.CO]

    Google Scholar 

  76. O’Keeffe M (1991) N-dimensional diamond, sodalite and rare sphere packings. Acta Crystallogr A 47:748–753

    Article  Google Scholar 

  77. O’Keeffe M, Peskov MA, Ramsden SJ, Yaghi OM (2008) The reticular chemistry structure resource (RCSR) database of, and symbols for, crystal nets. Acc Chem Res 41:1782–1789

    Article  Google Scholar 

  78. Peresypkina E, Blatov V (2000) Molecular coordination numbers in crystal structures of organic compounds. Acta Crystallogr B 56:501–511

    Article  Google Scholar 

  79. Radin C (1987) Low temperature and the origin of crystalline symmetry. Int J Mod Phys B 1:1157–1191

    Article  MathSciNet  Google Scholar 

  80. Radin C (1991) Global order from local sources. Bull AMS 25:335–364

    Article  MathSciNet  MATH  Google Scholar 

  81. Rangnathan S (1966) On the geometry of coincidence-site lattices. Acta Crystallogr 21: 197–199

    Article  Google Scholar 

  82. Resnikoff H, Wells Jr R (1998) Wavelet analysis. Springer, Heidelberg

    Book  MATH  Google Scholar 

  83. Scott L (2012) A primer on ice (in preparation)

    Google Scholar 

  84. Serre JP (1980) Trees. Springer, Berlin

    Google Scholar 

  85. Shubin M, Sunada T (2006) Mathematical theory of lattice vibrations and specific heat. Pure Appl Math Q 2:745–777

    MathSciNet  MATH  Google Scholar 

  86. Strong R, Packard CJ (2004) Systematic prediction of crystal structures: an application to sp 3-hybridized carbon polymorphs. Phys Rev B 70:045101

    Article  Google Scholar 

  87. Sunada T (1984) Geodesic flows and geodesic random walks. In: Geometry of geodesics and related topics (Tokyo, 1982). Advanced Studies in Pure Mathematics, vol 3. North-Holland, Amsterdam, pp 47–85

    Google Scholar 

  88. Sunada T (1985) Riemannian coverings and isospectral manifolds. Ann Math 121:169–186

    Article  MathSciNet  MATH  Google Scholar 

  89. Sunada T (1986) L-functions in geometry and some applications. In: K. Shiohama, T Sakai, T. Sunada (ed) Proceedings of the 17th International Taniguchi symposium, 1985. Curvature and topology of Riemannian manifolds. Lecturer notes in mathematics, vol 1201. Springer, Berlin, pp 266–284

    Google Scholar 

  90. Sunada T (1988) Fundamental groups and Laplacians. In: T. Sunada (ed) Proceedings of the Taniguchi symposium, 1987. Geometry and analysis on manifolds. Lecture notes in mathematics, vol 1339. Springer, Berlin, pp 248–277

    Google Scholar 

  91. Sunada T (1989) Unitary representations of fundamental groups and the spectrum of twisted Laplacians. Topology 28:125–132

    Article  MathSciNet  MATH  Google Scholar 

  92. Sunada T (1994) A discrete analogue of periodic magnetic Schrödinger operators. Contemporary Math 173:283–299

    Article  MathSciNet  Google Scholar 

  93. Sunada T (2006) Why do diamonds look so beautiful? Springer, Tokyo (in Japanese)

    Google Scholar 

  94. Sunada T (2008) Crystals that nature might miss creating. Notices Am Math Soc 55:208–215

    MathSciNet  MATH  Google Scholar 

  95. Sunada T (2008) Discrete geometric analysis. In: Exner P, Keating JP, Kuchment P, Sunada T, Teplyaev A (eds) Geometry on Graphs and Its Applications, Proceedings of symposia in pure mathematics, vol 77, pp 51–86

    Google Scholar 

  96. Sunada T (2012) Lecture on topological crystallography. Jpn J Math 7:1–39

    Article  MathSciNet  MATH  Google Scholar 

  97. Sunada T (2012) Commensurable Euclidean lattices (in preparation)

    Google Scholar 

  98. Tanaka R (2011) Large deviation on a covering graph with group of polynomial growth. Math Z 267:803–833

    Article  MathSciNet  MATH  Google Scholar 

  99. Tanaka R (2011) Hydrodynamic limit for weakly asymmetric exclusion processes in crystal lattices. arXiv:1105.6220v1 [math.PR]

    Google Scholar 

  100. Tate T, Sunada T (2012) Asymptotic behavior of quantum walks on the line. J Funct Anal 262:2608–2645

    Article  MathSciNet  MATH  Google Scholar 

  101. Terras A (2010) Zeta functions of graphs: a stroll through the garden. Cambridge Studies in Advanced Mathematics, Cambridge

    Google Scholar 

  102. Tutte WT (1960) Convex representations of graphs. Proc Lond Math Soc 10:304–320

    Article  MathSciNet  MATH  Google Scholar 

  103. Tutte WT (1963) How to draw a graph. Proc Lond Math Soc 13:743–767

    Article  MathSciNet  MATH  Google Scholar 

  104. Uralawa H (2000) A discrete analogue of the harmonic morphism and Green kernel comparison theorems. Glasgow Math J 42:319–334

    Article  Google Scholar 

  105. van Lint JH, Wilson RM (1992) A course in combinatorics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  106. van der Schoot A (2001) Kepler’s search for forms and proportion. Renaissance Stud 15: 59–78

    Article  Google Scholar 

  107. Vick JW (1994) Homology theory, 2nd edn. Springer, New York

    Book  MATH  Google Scholar 

  108. Wells AF (1954) The geometrical basis of crystal chemistry. Acta Crystallogr 7:535

    Article  Google Scholar 

  109. Wells AF (1977) Three dimensional nets and polyhedra. Wiley, New York

    Google Scholar 

  110. Weyl H (1983) Symmetry. Princeton University Press, Princeton

    Google Scholar 

  111. Wolf JA (1967) Spaces of constant curvature. McGraw-Hill, New York

    MATH  Google Scholar 

  112. Woess W (2000) Random walks on infinite graphs and groups. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  113. Wood EA (1977) Crystals and light, an introduction to optical crystallography, 2nd revised edn. Dover, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita Tama-ku, Kawasaki-shi, Kanagawa, Japan

    Toshikazu Sunada (Professor)

Authors
  1. Toshikazu Sunada
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Japan

About this chapter

Cite this chapter

Sunada, T. (2013). Introduction. In: Topological Crystallography. Surveys and Tutorials in the Applied Mathematical Sciences, vol 6. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54177-6_1

Download citation

Publish with us

Policies and ethics

Search

Navigation