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On the ring lemma

Jonatan Vasilis (Institutionen för matematiska vetenskaper)
Göteborg : Chalmers University of Technology, 2008. - 27 s.

The sharp ring lemma states that if n ≥ 3 cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc, then no disc has a radius below cn = (F2n-1 + F2n-2 - 1)-1 – where Fk denotes the kth Fibonacci number – and that the lower bound is attained in essentially unique Apollonian configurations.

Here we give a proof by transforming the problem to a class of strip configurations, after which we closely follow a method of proof due to Aharonov and Stephenson.

Generalizations 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

Peter Kumlin (Matematiska vetenskaper, Chalmers och Göteborgs universitet) var diskussionsinledare vid licentiatseminariet.

Denna post skapades 2008-01-15. Senast ändrad 2008-09-17.
CPL Pubid: 66601


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Institutioner (Chalmers)

Institutionen för matematiska vetenskaperInstitutionen för matematiska vetenskaper (GU)


Matematisk analys

Chalmers infrastruktur


Datum: 2008-02-06
Tid: 13:15
Lokal: Pascal, Chalmers Tvärgata 3

Ingår i serie

Preprint - Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University 2007:43