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Covering a sphere by equal circles, and the rigidity of its graph

Published online by Cambridge University Press:  24 October 2008

T. Tarnai  and
Zs. Gáspár
T. Tarnai
Affiliation:
University of Cambridge, Engineering Department, Trumpington Street, Cambridge CB2 1PZ
Zs. Gáspár
Affiliation:
Research Group for Applied Mechanics, Hungarian Academy of Sciences, Budapest, Műegyetem rkp. 3, H-1521, Hungary
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Abstract

How must a sphere be covered by n equal circles so that the angular radius of the circles will be as small as possible? In this paper, conjectured solutions of this problem for n = 15 to 20 are given and some sporadic results for n > 20 (n = 22, 26, 38, 42, 50) are presented. The local optima are obtained by using a ‘cooling technique’ based on the theory of bar-and-joint structures. Thus the graph of the coverings by circles is considered as a spherical cable net in which the edge lengths are uniformly decreased, e.g. due to a uniform decrease in the temperature, until the graph becomes rigid and tensile stresses appear in the cables.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 71 - 89
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Almgren, F. J. and Taylor, J. E.. The geometry of soap films and soap bubbles. Scientific American, 07 1976, 82–93.CrossRefGoogle Scholar
[2]Asimow, L. and Roth, B.. The rigidity of graphs. Trans. Amer. Math. Soc. 245 (1978), 279–289.CrossRefGoogle Scholar
[3]Brown, L. D. and Lipscomb, W. N.. Closo boron hydrides with 13 to 24 boron atoms. Inorganic Chemistry 16 (1977), 2989–2996.CrossRefGoogle Scholar
[4]Calladine, C. R.. Modal stiffness of a pretensioned cable net. Int. J. Solids Structures 18 (1982), 829–846.CrossRefGoogle Scholar
[5]Crowther, R. A., Finch, J. T. and Pearse, B. M. F.. On the structure of coated vesicles. J. Mol. Biol. 103 (1976), 785–798.CrossRefGoogle ScholarPubMed
[6]Tóth, G. Fejes. Kreisüberdeckungen der Sphäre. Studia Sci. Math. Hungar. 4 (1969), 225–247.Google Scholar
[7]Tóth, L. Fejes. On covering a spherical surface with equal spherical caps (in Hungarian). Matematikai és Fizikai Lapok 50 (1943), 40–46.Google Scholar
[8]Tóth, L. Fejes. Lagerungen in der Ebene auf der Kugel und im Raum (Springer-Verlag, 1953), p. 170.CrossRefGoogle Scholar
[9]Tóth, L. Fejes. Lagerungen in der Ebene auf der Kugel und im Raum, 2nd edition (Springer-Verlag, 1972).CrossRefGoogle Scholar
[10]Tóth, L. Fejes. Private communication (1985).Google Scholar
[11]Frank, F. C. and Kasper, J. S.. Complex alloy structures regarded as sphere packings. I. Definitions and basic principles. Acta Cryst. 11 (1958), 184–190.CrossRefGoogle Scholar
[12]Gáspár, Zs.. Some new multi-symmetric packings of equal circles on a 2-sphere. Acta Cryst. B45 (1989), 452–453.CrossRefGoogle Scholar
[13]Gáspár, Zs. and Tarnai, T.. Cable nets and circle-coverings on a sphere. Z. Angew. Math. Mech. 70 (1990), T741–T742.Google Scholar
[14]Goldberg, M.. Axially symmetric packing of equal circles on a sphere. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 10 (1967), 37–48.Google Scholar
[15]Housecroft, C. E.. Boranes and Metalloboranes (Ellis Horwood and J. Wiley, 1990).Google Scholar
[16]Jucovič, E.. Some coverings of a spherical surface with equal circles (in Slovakian). Mat.-Fyz. Časopis. Slovensk. Akad. Vied. 10 (1960), 99–104.Google Scholar
[17]Katsura, I.. Theory on the structure and stability of coated vesicles. J. Theor. Biol. 103 (1983), 63–75.CrossRefGoogle ScholarPubMed
[18]Kroto, H.. Space, stars, C60 and soot. Science 242 (1988), 1139–1145.CrossRefGoogle ScholarPubMed
[19]Matzke, E. B.. The three-dimensional shape of bubbles in foam. Amer. J. Botany 33 (1946), 58–80.CrossRefGoogle ScholarPubMed
[20]Melnyk, T. W., Knop, O. and Smith, W. R.. Extremal arrangements of points and unit charges on a sphere: equilibrium configurations revisited. Canadian J. Chemistry 55 (1977), 1745–1761.CrossRefGoogle Scholar
[21]Meschkowski, H.. Unsolved and Unsolvable Problems in Geometry (Oliver and Boyd, 1966).Google Scholar
[22]Schütte, K.. Überdeckung der Kugel mit höchstens acht Kreisen. Math. Ann. 129 (1955), 181–186.CrossRefGoogle Scholar
[23]Szabó, J.. The equation of state-change of structures. Periodica Polytechnica, Mech. Engng 17 (1973), 55–71.Google Scholar
[24]Szabó, J. and Kollár, L.. Structural Design of Cable-Suspended Roofs (Ellis Horwood, 1984).Google Scholar
[25]Tarnai, T.. Engineering methods in spherical circle-packings and circle-coverings. In 3. Kolloquium über Diskrete Geometrie (Institut für Mathematik der Universität Salzburg, 1985), pp. 253–262.Google Scholar
[26]Tarnai, T.. Geodesic domes and spherical circle-coverings. In Proceedings of the IASS International Congress, vol. 4 (Gosstroy, 1985), pp. 99–113.Google Scholar
[27]Tarnai, T. and Gáspár, Zs.. Improved packing of equal circles on a sphere and rigidity of its graph. Math. Proc. Cambridge Philos. Soc. 93 (1983), 191–218.CrossRefGoogle Scholar
[28]Tarnai, T. and Gáspár, Zs.. Covering the sphere with equal circles. In Intuitive Geometry, Colloquia Mathematica Societatis János Bolyai, vol. 48 (North-Holland, 1985), pp. 545–550.Google Scholar
[29]Tarnai, T. and Gáspár, Zs.. Covering the sphere with 11 equal circles. Elem. Math. 41 (1986), 35–38.Google Scholar
[30]Tarnai, T. and Gáspár, Zs.. Multi-symmetric close packings of equal spheres on the spherical surface. Acta Cryst. A43 (1987), 612–616.CrossRefGoogle Scholar
[31]Tarnai, T. and Gáspár, Zs.. Arrangement of 23 points on a sphere. (On a conjecture of R. M. Robinson.) Proc. Roy. Soc. Lond. Ser. A (to appear).Google Scholar