Abstract
We address the problem of covering ℝn with congruent balls, while minimizing the number of balls that contain an average point. Considering the 1-parameter family of lattices defined by stretching or compressing the integer grid in diagonal direction, we give a closed formula for the covering density that depends on the distortion parameter. We observe that our family contains the thinnest lattice coverings in dimensions 2 to 5. We also consider the problem of packing congruent balls in ℝn, for which we give a closed formula for the packing density as well. Again we observe that our family contains optimal configurations, this time densest packings in dimensions 2 and 3.
Keywords
This research is partially supported by DARPA under grant HR0011-09-0065 and NSF under grant DBI-0820624.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bambah, R.P.: On lattice coverings by spheres. Proc. Natl. Inst. Sci. India 20, 25–52 (1954)
Blichfeldt, H.F.: The minimum values of quadratic forms in six, seven, and eight variables. Math. Zeit. 39, 1–15 (1935)
Cohn, H., Kumar, A.: The densest lattice in twenty-four dimensions. Electronic Research Announcements of the Amer. Math. Soc. 10, 58–67
Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. Springer, New York (1988)
Delone, B.N., Ryskov, S.S.: Solution of the problem of least dense lattice covering of a four-dimensional space by equal spheres. Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 4, 1333–1334 (1963)
Edelsbrunner, H.: Geometry and Topology for Mesh Generation. Cambridge Univ. Press, Cambridge (2001)
Edelsbrunner, H., Kerber, M.: Dual complexes of cubical subdivisions of ℝn. IST Austria, Klosterneuburg, Austria (2010) (manuscript)
Fejes Tóth, L.: Lagerungen in der Ebene, auf der Kugel und im Raum. Grundlehren der mathematischen Wissenschaften, vol. 65. Springer, Berlin (1953)
Freudenthal, H.: Simplizialzerlegung von beschränkter Flachheit. Ann. of Math. 43, 580–582 (1942)
Gauss, C.F.: Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber. Göttingische Gelehrte Anzeigen (1831); reprinted in Werke II, Königliche Gesellschaft der Wissenschaften, pp. 188–196 (1863)
Hales, T.: A proof of the Kepler conjecture. Ann. Math., Second Series 162, 1065–1185 (2005)
Kershner, R.: The number of circles covering a set. Amer. J. Math. 61, 665–671 (1939)
Korkine, A., Zolotareff, G.: Sur les formes quadratiques positives. Math. Ann. 11, 242–292 (1877)
Kuhn, H.W.: Some combinatorial lemmas in topology. IBM J. Res. Develop. 45, 518–524 (1960)
Leech, J.: Notes on sphere packings. Canad. J. Math. 19, 251–267 (1967)
Rogers, C.A.: Packing and Covering. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 54. Cambridge Univ. Press, Cambridge (1964)
Ryskov, S.S., Baranovskii, E.P.: Solution of the problem of least dense lattice covering of a five-dimensional space by equal spheres. Doklady Akademii Nauk SSSR 222, 39–42 (1975)
Schürmann, A., Vallentin, F.: Local covering optimality of lattices: Leech lattice versus root lattice E8. Int. Math. Res. Notices 32, 1937–1955 (2005)
Schürmann, A., Vallentin, F.: Computational approaches to lattice packing and covering problems. Discrete Comput. Geom. 35, 73–116 (2006)
Szpiro, G.G.: Kepler’s Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World. Wiley, Hoboken (2003)
Thue, A.: Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene. Norske Vid. Selsk. Skr. 1, 1–9 (1910)
Witt, E.: Collected Papers. Gesammelte Abhandlungen. Springer, Berlin (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Edelsbrunner, H., Kerber, M. (2011). Covering and Packing with Spheres by Diagonal Distortion in ℝn . In: Calude, C.S., Rozenberg, G., Salomaa, A. (eds) Rainbow of Computer Science. Lecture Notes in Computer Science, vol 6570. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19391-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-19391-0_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19390-3
Online ISBN: 978-3-642-19391-0
eBook Packages: Computer ScienceComputer Science (R0)