Advertisement

Lattices and packings in higher dimensions

  • Marcel BergerEmail author
Chapter

Abstract

A lattice in \(\mathbb{R}^3\) is a Λ that can be written as the set of integer combinations of three linearly independent vectors \(\{a,b,c\}\), say \(\Lambda= \mathbb{Z} \cdot a+\mathbb{Z} \cdot b+\mathbb{Z} \cdot c\). As in Sect. IX.4, two Euclidean invariants are immediately associated with a lattice; they are practically dictated when we seek to pack balls of like radius in the densest possible way, thus the most economical for practical life; see more in Sect. X.4.

Keywords

Modular Form Theta Function Minimal Norm Arbitrary Dimension Voronoi Cell 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [B] Berger, M. (1987, 2009) Geometry I, II. Berlin/Heidelberg/New York: SpringerGoogle Scholar
  2. [BG] Berger, M., & Gostiaux, B. (1987) Differential Geometry: Manifolds, Curves and Surfaces. Berlin/Heidelberg/New York: SpringerGoogle Scholar
  3. Aigner, M., & Ziegler, G.(1998). Proofs from THE BOOK. SpringerGoogle Scholar
  4. Ball, K. (1992). A lower boundfor the optimal density of lattice packings. International Mathematics Research Notices, 10, 217–221CrossRefGoogle Scholar
  5. Berger, M. (1999). Riemannian geometry during the second half of the twentieth century. Providence: American Mathematical SocietyGoogle Scholar
  6. Berger, M. (2001a). Peut-ondéfinir la géométrie aujourd’hui? In B. Engquist, W.Schmid (Eds.), Mathematics unlimited- 2001 and beyond.Berlin/Heidelberg/New York: SpringerGoogle Scholar
  7. Berger, M. (2003). Apanoramic introduction to Riemannian geometry. Berlin/Heidelberg/New York: SpringerGoogle Scholar
  8. Betke, U., Henk, M., & Wills,J. (1994). Finite and infinite packings. Journal f¸r dieReine und Angewandte Mathematik, 453, 165–191zbMATHMathSciNetGoogle Scholar
  9. Boerdijk, A. (1952). Someremarks concerning close-packing of equal spheres. Philips Research Reports, 7, 303–313zbMATHMathSciNetGoogle Scholar
  10. Christophe Lesapeur Camembert. Albin MichelGoogle Scholar
  11. Conway, J. (1995). Spherepackings, lattices, codes and greed. In Proceedings of theInternational Congress of Mathematicians (Zürich, 1994), Vol. 1. Birkhäuser, 45–55Google Scholar
  12. Conway, J. H., & Sloane, N.J. A. (1999). Sphere packings, Lattices and Groups (3rd ed.). Berlin/Heidelberg/New York: SpringerzbMATHGoogle Scholar
  13. Conway, J. H., & Sloane, N.J. A. (1993). Sphere packings, lattices and groups (2nd ed.). Berlin/Heidelberg/New York: SpringerzbMATHGoogle Scholar
  14. De la Harpe, P., & Venkov, B. (2001). Groupes engendrés par des réflexions, designssphériques et réseau de Leech. Comptes Rendus, AcadÈmie des sciences de Paris, 333, 745–750zbMATHCrossRefGoogle Scholar
  15. Dhombres, J., & Robert,J.-B. (1998). Fourier, créateur de la physique mathématique. Paris: BelinGoogle Scholar
  16. Elkies, N. (2000). Lattices,linear codes, and invariants, Part I. Notices of the American Mathematical Society, 47(10), 1238–1945zbMATHMathSciNetGoogle Scholar
  17. Erdös, P., Gruber, P., & Hammer, J. (1989). Lattice points. New York: Longman Scientific and Technical, John WileyGoogle Scholar
  18. Fejes Tóth, L. (1972). Lagerungen in der Ebene, auf der Kugel und im Raum (2nd ed.). Berlin/Heidelberg/New York: SpringerzbMATHGoogle Scholar
  19. Fournier, J.-C. (1977).Le théorème du coloriage des cartes (ex-conjecturedes quatre couleurs), Séminaire Bourbaki, 1977–78 Lecture Notes in Mathematics, 710, Springer, 41–64Google Scholar
  20. Gruber, P., & Lekerkerker, C. (1987). Geometry of numbers. Amsterdam: North-HollandGoogle Scholar
  21. Gruber, P., & Wills, J.(Eds.). (1993). Handbook of convex geometry. Amsterdam: North-HollandGoogle Scholar
  22. Hales, T. (1994). The status ofthe Kepler conjecture. The Mathematical Intelligencer, 16, 47–58zbMATHCrossRefMathSciNetGoogle Scholar
  23. Hales, T. (1997). Spherepackings. Discrete & Computational Geometry, 17, 1–51zbMATHCrossRefMathSciNetGoogle Scholar
  24. Hales, T. (1999). Anoverview of the Kepler conjecture, http://arxiv.org/abs/math/9811071
  25. Hales, T. (2000). Cannonballs and honeycombs. Notices of the American Mathematical Society, 47(4), 440–449zbMATHMathSciNetGoogle Scholar
  26. Hsiang, W.-Y. (1993). On thesphere packing problem and the proof of Kepler’s conjecture. International Journal of Mathematics, 4, 739–831zbMATHCrossRefMathSciNetGoogle Scholar
  27. Kabatianski,G., & Levenshtein, V. (1978). Bounds for packings on a sphere andin a space. Problems of Information Transmission, 14, 1–17Google Scholar
  28. Kac, M. (1996). Can one hear theshape of a drum? The American Mathematical Monthly, 73(4), part II, 1–23CrossRefGoogle Scholar
  29. Lee, R., & Szczarba, R. (1978). On the torsion in K4(Z) and K5(Z). Duke MathematicalJournal, 45, 101–129zbMATHCrossRefMathSciNetGoogle Scholar
  30. Martinet, J. (1996). Les réseaux parfaits des espaces euclidiens. Paris: MassonGoogle Scholar
  31. Mattila, P. (1995). Geometry of sets and measures in euclidean spaces.Cambridge: Cambridge University PressCrossRefGoogle Scholar
  32. Milnor, J. (1964). Eigenvalues of the Laplace operator on certain manifolds.Proceedings of the National Academy of Sciences of the USA, 51, 542zbMATHCrossRefMathSciNetGoogle Scholar
  33. Milnor, J. (1994). Hilbert’sproblem 18: on crystallographic groups, fundamental domains, and onsphere packing. In J. Milnor Collected papers, Publish or Perish,Houston, 173–187Google Scholar
  34. Morgan, F. (2005). Kepler’sconjecture and Hales proof – a book review, Notices of theAmerican Mathematical Society, 52(1), 44–47Google Scholar
  35. Mumford, D. (2000). The dawning age of stochasticity. In Arnold, Atiyah, Lax, Mazur (Eds.), Mathematics: frontiers and perspectives (pp. 199–218). Providence: American Mathematical societyGoogle Scholar
  36. Oesterlé, J.(1990). Empilements de sphères, Séminaire Bourbaki1989–1990. In Astérisque 189–190, 375–398Google Scholar
  37. Oesterlé, J.(1999). Densité maximale des empilements de sphères endimension 3 (d’après Thomas C. Hales et Samuel P. Ferguson),Séminaire Bourbaki 1989–1990. InAstérisque 266, 405–413Google Scholar
  38. Pöppe, C. (1999). Laconjecture de Kepler démontrée. Pour la Science,259, mai 1999, 100–104Google Scholar
  39. Reid, C. (1970). Hilbert. Beriln/Heidelberg/New York: SpringerGoogle Scholar
  40. Rigby, J. (1998). Precise colourings of regular triangular tilings. The Mathematical Intelligencer, 20, 4–11zbMATHCrossRefMathSciNetGoogle Scholar
  41. Rogers, C. (1964). Packings and coverings. Cambridge: Cambridge UniversityPressGoogle Scholar
  42. Rosenbloom, M., & Tsafsman, M. (1990). Multiplicative lattices in global fields. Inventiones Mathematicae, 101, 687–696zbMATHCrossRefMathSciNetGoogle Scholar
  43. Schiemann, A. (1997). Ternary positive definite quadratic forms are determined by their theta series, Mathematische Annalen, 308, 507–517zbMATHCrossRefMathSciNetGoogle Scholar
  44. Serre, J.-P. (1970). Cours d’arithmétique. Paris: Presses Universitaires de FranceGoogle Scholar
  45. Sullivan, J. (1994). Sphere packings give an explicit bound for the Besikovitch coveringtheorem. Journal for Geometric Analysis, 4, 219–231zbMATHGoogle Scholar
  46. Thomas, R. (1998). An updateon the four-color theorem. Notices of the American Mathematical Society, 45(7), 848–859zbMATHMathSciNetGoogle Scholar
  47. Thompson, T. (1983).From error-correcting codes through sphere packings tosimple groups. Washington: Mathematical Association of AmericaGoogle Scholar
  48. Torquato, S. & Stillinger, F., (2006). New conjectural bounds on the optimaldensity of sphere packings. Experimental Mathematics, 15(3), 307–332zbMATHMathSciNetGoogle Scholar
  49. Wills, J. (1991). An ellipsoidpacking in E3 of unexpected high density. Mathematika, 38, 318–320zbMATHCrossRefMathSciNetGoogle Scholar
  50. Wills, J. (1998). Spheres and sausages, crystals and catastrophes – and a joint packing theory, The Mathematical Intelligencer, 20(1), 16–21zbMATHCrossRefMathSciNetGoogle Scholar
  51. Zong, C. (1996). Strange phenomena in convex and discrete geometry. Berlin/Heidelberg/New York: SpringerGoogle Scholar
  52. Zong, C. (1999). Sphere packings. Berlin/Heidelberg/New York: SpringerGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Institut des Hautes Études ScientifiquesIHÉS, Bures-sur-YvetteBures-sur-YvetteFrance

Personalised recommendations