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On the Number of Spheres Which can Hide a Given Sphere

Published online by Cambridge University Press:  20 November 2018

Aladár Heppes
Aladár Heppes*
Affiliation:
Ohio Slate University
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Several years ago H. Hornich suggested the following problem: find the minimal number of unit spheres which can hide a unit sphere in the sense that each ray emanating from the centre of that sphere meets at least one of the hiding spheres, with no two of the spheres overlapping. We shall call any set of spheres which hide a given unit sphere a cloud.

The first result concerning this and related questions can be found in a paper of Fejes Tóth (4 ; see also 5, 7, 8, 6, and 1 ). With respect to the original problem, Fejes Tóth has given a lower estimate for the minimal number N of the spheres of a cloud.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 413 - 418
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Böröczky, K., Über Dunkelwolken (lecture), Colloquium on Convexity (Copenhagen, 1965).Google Scholar
2. Danzer, L., Drei Beispiele zu Lagerungsprobleme, Archiv d. Math., 11 (1960), 159–165.Google Scholar
3. Fejes Tóth, L., Lagerungen in der Ebene aufder Kugel und im Raum (Berlin, 1953).Google Scholar
4. Fejes Tóth, L., Verdeckung einer Kugel durch Kugeln, Publ. Math. (Debrecen), 6 (1959), 234–240.Google Scholar
5. Grünbaum, B., On a problem of Fejes Tóth, Amer. Math. Monthly, 67 (1960), 882–884.Google Scholar
6. Hajós, G., Über Kreiswolken, Ann. Univ. Sci. Budapest., 7 (1965), 55–57.Google Scholar
7. Heppes, A., Kin Satz über gitterförmige Kugelpackungen, Ann. Univ. Sci. Budapest., 3-4 (1960-61), 89–90.Google Scholar
8. Heppes, A., Über Kreis- und Kugelwolken, Acta Math. Acad. Sci. Hungar., 12 (1961), 209–214.Google Scholar