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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Spherical circle coverings and bubbles in foam - T
arnai\, T (Budapest University of Technology and E
conomics)
DTSTART;TZID=Europe/London:20140227T114500
DTEND;TZID=Europe/London:20140227T120500
UID:TALK51132AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/51132
DESCRIPTION:One of the classical problems of discrete geometry
is the following. How must a sphere be covered by
n equal circles (spherical caps) so that the angu
lar radius of the circles will be as small as poss
ible? In the 1980s when we started to work on this
problem\, proven solutions were known only for n
= 2\, 3\, 4\, 5\, 6\, 7\, 10\, 12\, 14\, and conje
ctured solutions for n = 8\, 9\, 32. The first gap
s appeared at n = 11 and 13\, for which even sugge
stions did not exist. We thought that the shapes o
f bubbles in foam might help. We considered the re
sults of Matzkes experimental observations\, and f
ound that the edge graph of the only bubble with 1
1 faces and one of the 4 bubbles with 13 faces lea
d to the best coverings where the Dirichlet cells
of the circle system provided the same edge graphs
as those of the respective bubbles. \n\nAdditiona
lly\, we could show that for n = 2 to 12\, except
11\, the edge network of the Dirichlet cells of th
e best circle covering is topologically identical
to the minimal net formed by the intersection of n
soap-film-like cones by a sphere (determined by A
. Heppes\, F.J. Almgren and J.E. Taylor). \n\nIn t
he range of n = 14 to 20\, we considered the possi
ble shapes of coated vesicles. These are certain k
inds of bubbles where a part of the cellular membr
ane is surrounded by a clathrin basket a polyhedr
on. With their help\, for these values of n\, exce
pt n = 19\, we could construct the best so far cir
cle coverings of a sphere. \n\nIn the lecture\, we
want to survey the results for n = 2 to 20\, maki
ng comparison with soap-film-like cones\, bubbles
in foam\, coated vesicles\, and to compare the bes
t circle coverings with the numerical solutions to
the isoperimetric problem for polyhedra with n fa
ces. This research was supported by OTKA grant no.
K801146. \n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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