Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T23:15:54.202Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattice Coverings of n-Space By Spheres

Published online by Cambridge University Press:  20 November 2018

M. N. Bleicher
M. N. Bleicher*
Affiliation:
University of California
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A lattice ∧ in euclidean n-space, En is a group of vectors under vector addition generated by n independent vectors, X1, X2 … , Xn, called a basis for the lattice. The absolute value of the n × n determinant the rows of which are the co-ordinates of a basis is called the determinant of the lattice and is denoted by d(∧). For any lattice ∧ there is a unique minimal positive number r such that, if spheres of radius r are placed with centres at all points of ∧, the entire space is covered. The density of this covering may be defined as (Jnrn)/(d(∧)) where Jn is the volume of the unit sphere in n-dimensional euclidean space. This density will be denoted by θn(∧). The density of the most efficient lattice covering of n-space by spheres, θn is the absolute minimum of θn(∧) considered as a function from the space of all lattices to the real numbers.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 632 - 650
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Bambah, R. P., Lattice coverings by spheres, Proc. Nat. Inst. Sci. India, 20 (1954).Google Scholar
2. Bambah, R. P., Lattice coverings with fcur-dimensicnal spheres, Camb. Fhil. Soc Proc, 50 (1954).Google Scholar
3. Bambah, R. P. and Davenport, H., The covering of n-dimensional space by spheres, J. Lond. Math. Soc, 27 (1952).Google Scholar
4. Barnes, E. S., The covering of space by spheres, Can. J. Math., 8 (1956).Google Scholar
5. Cassels, J. W. S., An introduction to the geometry of numbers, Berlin-Gottingen-Heidelberg (1959).Google Scholar
6. Coxeter, H. S. M., Extreme forms, Can. J. Math., 4 (1952).Google Scholar
7. Coxeter, H. S. M., L. Few, and Rogers, C. A., Covering space with equal spheres, Mathematika, 6 (1959).Google Scholar
8. Davenport, H., The covering of space by spheres, Rend. Circ Mat. Palermo, Series II, 1 (1952).Google Scholar
9. Few, L., Covering space by spheres, Mathematika, 8 (1956).Google Scholar
10. Hlawka, E., Ausfullung und Uberdeckung konvexer Korper durch konvexe Korper, Mh. Math. Phys., 53 (1956).Google Scholar
11. Kershner, R., The number of circles in a covering set, Amer. J. Math., 61 (1939).Google Scholar
12. Korkine, A. and Zolotareff, G., Sur les formes quadratiques, Math. Ann., 6 (1873).Google Scholar
13. Rogers, C. A., Lattice coverings of space, Mathematika, 6 (1959).Google Scholar
14. Voronoi, G., Recherches sur les paralléloèdres primitifs, J. Reine Angew. Math., 134 and 136 (1907 and 1909).Google Scholar