Abstract
In the previous chapter we gave a characterization of the standard realization by means of a minimal principle. The existence was also confirmed. What remains to be done is to establish its explicit construction fitting in with the enumeration of topological crystals. The idea is quite simple. We make up a candidate for the building block \(\{{\mathbf{v}}_{0}{(e)\}}_{e\in {E}_{0}}\) by the method suggested in Notes (III) in Chap. 7. To this end, we shall equip the vector space \({C}_{1}({X}_{0}, \mathbb{R})\) of 1-chains with the canonical Euclidean structure. Thus, homology theory plays again a key role in our discussion.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
One can prove this claim by using the adjoint operator \({\partial }^{{_\ast}} : {C}_{0}({X}_{0}, \mathbb{Z})\rightarrow {C}_{1}({X}_{0}, \mathbb{R})\) of the boundary operator \(\partial \) which will be introduced in Sect. 10.2.
- 3.
The only problem we have when we want to describe standard realizations is that to say nothing of the higher dimensional case, it is not easy to imagine the complete geometric shape of a 3D crystal net when we display it on the plane (a computer screen). The best way to understand the shape is to use a crystalline compound kit (but such a kit is not always available for the crytal model we want to get).
- 4.
The three-letter names for crystal structures are proposed by O’Keeffe et al. [77].
- 5.
It is given its name because several streets in Cairo are paved in this design (strictly speaking, it is a bit distorted). This is also called Macmahon’s net, mcm, and a fourfold pentille (Conway).
- 6.
This figure is placed in my book [93] published in 2006, and is called a Pythagorian lattice because it is related to rational solutions of the equation \({x}^{2} + {y}^{2} = 1\). This is also an example of coincidence-site lattices [see Notes (I) in the present chapter].
- 7.
See Notes in this chapter and Ebeling [38] for general root lattices.
- 8.
Richard Buckminster Fuller (1895–1983) is an American engineer, architectural designer, and inventor, and is best known for his geodesic dome. The carbon molecule “Fullerene” (C60) was named after him for its resemblance to the geodesic dome.
- 9.
However Lonsdaleite is not isotropic.
- 10.
A grain boundary is represented as a two-dimensional section of the CSL.
- 11.
A triple of positive integers (x, y, z) is called primitive Pythagorean if they are pair-wise coprime, and \({x}^{2} + {y}^{2} = {z}^{2}\).
- 12.
Note that the symbol D d used here does not mean the graph introduced in Sect. 8.3 (II).
- 13.
This pamphlet was dedicated on the occasion of the New Year of 1611 to his friend and patron, the scholar and imperial privy councilor Johannes Matthäus Wackher von Wackenfels.
- 14.
The atomism was advocated by Leucippus and Democritus in the fifth century BC.
- 15.
In 1998 Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he proved the Kepler conjecture.
- 16.
In connection with the hexagonal arrangement, Kepler noticed that the honeycomb structure maximizes the number of wax walls each bee shares with his neighbor, thereby allowing bees to collaborate in constructing the shared walls of a cell. The hexagon also turns out to be most efficient in terms of exploiting the maximum space using the minimal amount of wax. Pappus (290 AD–350 AD) of Alexandria had already noticed that bees have foresight at their disposal in allowing them to understand that a hexagonal partition is more effective than a triangular or square partition.
- 17.
Needless to say, Kepler is renowned as an astronomer who discovered the three laws of planetary motion. His essay demonstrates that he had been thinking of not only the law of the vast universe but also the smallest aspects of nature. It should be emphasized, however, that he was an astrologer and theologian as well, and always sought an interpretation of the harmony of forms as God’s choice [106]. That is, he asks the snowflakes the same question as he did the planets: which form follows God’s order? Kepler’s achievement tells us, though his study of geometric figures cannot be separated from his theological dogmatism, that he was one of the most outstanding mathematicians of his day.
- 18.
Pappus testifies in his Collection that Archimedes (287 BC–212 BC) proposed, in a now-lost work, the notion of semi-regular polyhedron as a generalization of regular polyhedron, and found 13 such solids. Here a polyhedron is said to be semi-regular if it has regular faces and a symmetry group that is transitive on its vertices. Precisely speaking, there are two infinite series of convex prisms and convex antiprisms satisfying this condition.
- 19.
- 20.
The term “isotropic” is used in a different context in crystallography. That is, an isotropic crystal is a crystal which has the same optical properties in all directions.
- 21.
See Sect. 9.6.
- 22.
The two-dimensional crystal net in Fig. 8.3 is such an example.
- 23.
- 24.
Leopold Kronecker (1823–1891). His dream was described in a letter to Dedekind in 1880.
- 25.
This problem was officially posed by Hilbert in his famous 12th problem. The exponential function can be used to describe abelian extensions of the rational number field \(\mathbb{Q}\) (the Kronecker–Weber theorem). There are partial results in the cases of CM-fields or real quadratic fields, but the problem is largely still open.
References
Artamkin IV (2006) Discrete Torelli theorem. Sbornik: Math 197:1109–1120
Auslander L, Kuranishi M (1957) On the holonomy group of locally Euclidean spaces. Ann Math 65:411–415
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
Bader M, Klee WE, Thimm G (1997) The 3-regular nets with four and six vertices per unit cell. Z Kristallogr 212:553–558
Baker M, Norine S (2007) Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv Math 215:766–788
Baker M, Norine S (2009) Harmonic morphisms and hyperelliptic graphs. Int Math Res Notices (15):2914–2955
Baker M, Faber X (2011) Metric properties of the tropical Abel-Jacobi map. J Algebraic Combin 33:349–381
Bass H (1992) The Ihara-Selberg zeta function of a tree lattice. Int J Math 3:717–797
Beukemann A, Klee WE (1992) Minimal nets. Z Kristallogr 201:37–51
Biggs NL, Lloyd EK, Wilson RJ (1999) Graph theory 1736–1936. Oxford University Press, Oxford
Biggs NL (1993) Algebraic graph theory. Cambridge University Press, Cambridge
Biggs NL (1997) Algebraic potential theory on graphs. Bull Lond Math Soc 29:641–682
Blatov V (2000) Search for isotypism in crystal structures by means of the graph theory. Acta Crystallogr A 56:178–188
Bollmann W (1972) The basic concepts of the O-lattice theory. Surf Sci 31:1–11
Bollobas B (1998) Modern graph theory. Springer, New York
Brown KS (1972) Cohomology of groups. Springer, New York
Bryant PR (1967) Graph theory applied to electrical networks. In: Harary F (ed) Graph theory and theoretical physics. Academic, New York, pp 111–137
Caporaso L, Viviani F (2010) Torelli theorem for graphs and tropical curves. Duke Math J 153:129–171
Charlap LS (1986) Bieberbach groups and flat manifolds. Springer, New York
Chung SJ, Hahn T, Klee WE (1984) Nomenclature and generation of three-periodic nets: the vector method. Acta Crystallogr A 40:42–50
Conway JH, Burgiel H, Goodman-Strauss C (2008) The symmetries of things. A K Peters Ltd, Wellesley
Coxeter HSM (1955) On Laves’ graph of girth ten. Can J Math 7:18–23
Coxeter HSM (1973) Regular polytopes. Dover, New York
Coxeter HSM (1974) Regular complex polytopes. Cambridge University Press, Cambridge
Cromwell P (1999) Polyhedra. Cambridge University Press, Cambridge
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
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
Danzer L. Something about [10, 3] a . Unpublished
Delgado-Friedrichs O, Dress A, Huson D, Klinowski J, Mackay A (1999) Systematic enumeration of crystalline networks. Nature 400:644–647
Delgado-Friedrichs O, O’Keeffe M (2003) Identification of and symmetry computation for crystal nets. Acta Crystallogr A 59:351–360
Delgado-Friedrichs O, O’Keeffe M, Yaghin OM (2003) Three-periodic nets and tilings: regular and quasiregular nets. Acta Crystallogr A 59:22–27
Delgado-Friedrichs O (2004) Barycentric drawings of periodic graphs. Lect Notes Comput Sci 2912:178–189
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
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
Delgado-Friedrichs O, O’Keeffe M (2009) Edge-transitive lattice nets. Acta Crystallogr A 65:360–363
Diestel R (2006) Graph theory. Springer, New York
Dixmier J (1981) Von Neumann algebras. North-Holland, Amsterdam
Ebeling W (1994) Lattices and codes. Vieweg, Wiesbaden
Eells J, Sampson JH (1964) Harmonic mappings of Riemannian manifolds. Am J Math 86:109–160
Eells J, Fuglede B (2001) Harmonic maps between Riemannian polyhedra. Cambridge University Press, Cambridge
Eon J-G (1998) Geometrical relationships between nets mapped on isomorphic quotient graphs: examples. J Solid State Chem 138:55–65
Eon J-G (1999) Archetypes and other embeddings of periodic nets generated by orthogonal projection. J Solid State Chem 147:429–437
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
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)
Greenberg M (1971) Lectures on algebraic topology. Benjamin, Menlo Park
Gromov M (1999) Metric structures for Riemannian and non-Riemannian spaces. Birkhäuser, Basel
Harper PG (1955) Single band motion of conduction electrons in a uniform magnetic field. Proc Phys Soc Lond A 68:874–878
Hashimoto K (1990) On zeta and L-functions of finite graphs. Int J Math 1:381–396
Hörmander L (1983) The analysis of linear partial differential operators I. Springer, New York
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
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
Itoh M, Kotani M, Naito H, Kawazoe Y, Adschiri T (2009) New metallic carbon crystal. Phys Rev Lett 102:055703
Jost J (1996) Generalized harmonic maps between metric spaces. In: Geometric analysis and calculus of variations. International Press, Cambridge, pp 143–174
Katsuda A, Sunada T (1990) Closed orbits in homology classes. Publ Math IHES 71:5–32
Klein H-J (1996) Systematic generation of models for crystal structures. Math Model Sci Comput 6:325–330
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
Kotani M, Sunada T (2000) Zeta functions of finite graphs. J Math Sci Univ Tokyo 7:7–25
Kotani M, Sunada T (2000) Standard realizations of crystal lattices via harmonic maps. Trans Am Math Soc 353:1–20
Kotani M, Sunada T (2000) Jacobian tori associated with a finite graph and its abelian covering graphs. Adv Appl Math 24:89–110
Kotani M, Sunada T (2000) Albanese maps and off diagonal long time asymptotics for the heat kernel. Commun Math Phys 209:633–670
Kotani M, Sunada T (2003) Spectral geometry of crystal lattices. Contemporary Math 338:271–305
Kotani M, Sunada T (2006) Large deviation and the tangent cone at infinity of a crystal lattice. Math Z 254:837–870
Krámli A, Szász D (1983) Random walks with internal degree of freedom, I. Local limit theorem. Z. Wahrscheinlichkeittheorie 63:85–95
Kuchment P (1993) Floquet theory for partial differential operators. Birkhäuser, Basel
Lang S (1987) Linear algebra. Springer, Berlin
Magnus W, Karrass A, Solitar D (1976) Combinatorial group theory. Dover, New York
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
Milnor J (1969) Morse theory. Princeton University Press, Princeton
Nagano T, Smith B (1975) Minimal varieties and harmonic maps in tori. Commun Math Helv 50:249–265
Nagnibeda T (1997) The Jacobian of a finite graph. Contemporary Math 206:149–151
Neukirch J (1999) Algebraic number theory. Springer, Berlin
Newman P, Stoy G, Thompson E (1994) Groups and geometry. Oxford University Press, Oxford
Oganov A (ed) (2010) Modern methods of crystal structure prediction. Wiley-VCH, Berlin
Oda T, Seshadri CS (1979) Compactifications of the generalized Jacobian variety. Trans Am Math Soc 253:1–90
Oda T (2011) Voronoi tilings hidden in crystals—the case of maximal abelian coverings arXiv:1204.6555 [math.CO]
O’Keeffe M (1991) N-dimensional diamond, sodalite and rare sphere packings. Acta Crystallogr A 47:748–753
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
Peresypkina E, Blatov V (2000) Molecular coordination numbers in crystal structures of organic compounds. Acta Crystallogr B 56:501–511
Radin C (1987) Low temperature and the origin of crystalline symmetry. Int J Mod Phys B 1:1157–1191
Radin C (1991) Global order from local sources. Bull AMS 25:335–364
Rangnathan S (1966) On the geometry of coincidence-site lattices. Acta Crystallogr 21: 197–199
Resnikoff H, Wells Jr R (1998) Wavelet analysis. Springer, Heidelberg
Scott L (2012) A primer on ice (in preparation)
Serre JP (1980) Trees. Springer, Berlin
Shubin M, Sunada T (2006) Mathematical theory of lattice vibrations and specific heat. Pure Appl Math Q 2:745–777
Strong R, Packard CJ (2004) Systematic prediction of crystal structures: an application to sp 3-hybridized carbon polymorphs. Phys Rev B 70:045101
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
Sunada T (1985) Riemannian coverings and isospectral manifolds. Ann Math 121:169–186
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
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
Sunada T (1989) Unitary representations of fundamental groups and the spectrum of twisted Laplacians. Topology 28:125–132
Sunada T (1994) A discrete analogue of periodic magnetic Schrödinger operators. Contemporary Math 173:283–299
Sunada T (2006) Why do diamonds look so beautiful? Springer, Tokyo (in Japanese)
Sunada T (2008) Crystals that nature might miss creating. Notices Am Math Soc 55:208–215
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
Sunada T (2012) Lecture on topological crystallography. Jpn J Math 7:1–39
Sunada T (2012) Commensurable Euclidean lattices (in preparation)
Tanaka R (2011) Large deviation on a covering graph with group of polynomial growth. Math Z 267:803–833
Tanaka R (2011) Hydrodynamic limit for weakly asymmetric exclusion processes in crystal lattices. arXiv:1105.6220v1 [math.PR]
Tate T, Sunada T (2012) Asymptotic behavior of quantum walks on the line. J Funct Anal 262:2608–2645
Terras A (2010) Zeta functions of graphs: a stroll through the garden. Cambridge Studies in Advanced Mathematics, Cambridge
Tutte WT (1960) Convex representations of graphs. Proc Lond Math Soc 10:304–320
Tutte WT (1963) How to draw a graph. Proc Lond Math Soc 13:743–767
Uralawa H (2000) A discrete analogue of the harmonic morphism and Green kernel comparison theorems. Glasgow Math J 42:319–334
van Lint JH, Wilson RM (1992) A course in combinatorics. Cambridge University Press, Cambridge
van der Schoot A (2001) Kepler’s search for forms and proportion. Renaissance Stud 15: 59–78
Vick JW (1994) Homology theory, 2nd edn. Springer, New York
Wells AF (1954) The geometrical basis of crystal chemistry. Acta Crystallogr 7:535
Wells AF (1977) Three dimensional nets and polyhedra. Wiley, New York
Weyl H (1983) Symmetry. Princeton University Press, Princeton
Wolf JA (1967) Spaces of constant curvature. McGraw-Hill, New York
Woess W (2000) Random walks on infinite graphs and groups. Cambridge University Press, Cambridge
Wood EA (1977) Crystals and light, an introduction to optical crystallography, 2nd revised edn. Dover, New York
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Japan
About this chapter
Cite this chapter
Sunada, T. (2013). Explicit Construction. 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_8
Download citation
DOI: https://doi.org/10.1007/978-4-431-54177-6_8
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-54176-9
Online ISBN: 978-4-431-54177-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)