Geodesic
A Variant of the Problem of the Thirteen Spheres
Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 1092-1100

Voir la notice de l'article provenant de la source Cambridge University Press

We use the term balls for congruent, closed spheres no two of which have interior points in common. In Euclidean n-space let Nn be the maximal number of balls which can touch a ball. Obviously, N2 = 6. R. Hoppe (see (1)) proved that N3 = 12, settling thereby a famous point of controversy between Newton and David Gregory, known as the problem of the thirteen spheres (see (3)). Simpler proofs were given by Günter (6), Schütte and van der Waerden (10), and Leech (7). 
Tóth, L. Fejes; Heppes, A. A Variant of the Problem of the Thirteen Spheres. Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 1092-1100. doi: 10.4153/CJM-1967-100-x
@article{10_4153_CJM_1967_100_x,
     author = {T\'oth, L. Fejes and Heppes, A.},
     title = {A {Variant} of the {Problem} of the {Thirteen} {Spheres}},
     journal = {Canadian journal of mathematics},
     pages = {1092--1100},
     year = {1967},
     volume = {19},
     number = {1},
     doi = {10.4153/CJM-1967-100-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1967-100-x/}
}
TY  - JOUR
AU  - Tóth, L. Fejes
AU  - Heppes, A.
TI  - A Variant of the Problem of the Thirteen Spheres
JO  - Canadian journal of mathematics
PY  - 1967
SP  - 1092
EP  - 1100
VL  - 19
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1967-100-x/
DO  - 10.4153/CJM-1967-100-x
ID  - 10_4153_CJM_1967_100_x
ER  - 
%0 Journal Article
%A Tóth, L. Fejes
%A Heppes, A.
%T A Variant of the Problem of the Thirteen Spheres
%J Canadian journal of mathematics
%D 1967
%P 1092-1100
%V 19
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1967-100-x/
%R 10.4153/CJM-1967-100-x
%F 10_4153_CJM_1967_100_x

[1] 1. Bender, C., Bestimmung der grössten Anzahl gleich grosser Kugeln, welche sick auf eine Kugel von demselben Radius, wie die übrigen, auflegen lassen, Arch. Math. Phys., 56 (1874), 302–312. Google Scholar

[2] 2. Böröczky, K. and Florian, A., Über die dichteste Kugelpackung im hyperbolischen Raum, Acta Math. Acad. Sci. Hungar., 15 (1964), 237–245. Google Scholar

[3] 3. Coxeter, H. S. M., An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size, Proceedings of Symposia in Pure Mathematics, Amer. Math. Soc. VII, Convexity (1963), 53–71. Google Scholar

[4] 4. Fejes Tóth, L., Über die Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems, Jber. Deutsch. Math. Verein. 53 (1943), 66–68. Google Scholar

[5] 5. Fejes Tóth, L., On the total area of the faces of a four-dimensional polytope, Can. J. Math., 17 (1965), 93–99. Google Scholar

[6] 6. Gunter, S., Ein stereometrisches Problem, Arch. Math. Phys., 57 (1875), 209–215. Google Scholar

[7] 7. Leech, J., The problem of the thirteen spheres, Math. Gaz., 40 (1956), 22–23. Google Scholar

[8] 8. Minkowski, H., Diskontinuitätsbereich für arithmetische Äquivalenz, J. Reine Angew. Math. 129 (1905), 220–274. Google Scholar

[9] 9. Robinson, R. M., Arrangement of 24 points on a sphere, Math. Ann., 144 (1961), 17–48. Google Scholar

[10] 10. Schütte, K. and van der Waerden, B. L., Das Problem der dreizehn Kugeln, Math. Ann. 125 (1953), 325–334. Google Scholar

Cité par Sources :