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The ring lemma in three dimensions

Jonatan Vasilis (Institutionen för matematiska vetenskaper, matematik)
Geometriae Dedicata (0046-5755). Vol. 152 (2011), 1, p. 51-62.
[Artikel, refereegranskad vetenskaplig]

Suppose that n cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc. The sharp ring lemma in two dimensions states that no disc has a radius below c (n) (R (2)) = (F (2n-3)-1)(-1)-where F (k) denotes the kth Fibonacci number-and that the lower bound is attained in essentially unique Apollonian configurations. In this article, generalizations of the ring lemma to three dimensions are discussed, a version of the ring lemma in three dimensions is proved, and a natural generalization of the extremal two-dimensional configuration-thought to be extremal in three dimensions-is given. The sharp three-dimensional ring lemma constant of order n is shown to be bounded from below by the two-dimensional constant of order n - 1.

Nyckelord: Ring lemma, Circle packing, Sphere packing, Apollonian, apollonian circle packings, geometry, theorem



Denna post skapades 2011-06-07.
CPL Pubid: 141393

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)

Ämnesområden

Matematik

Chalmers infrastruktur