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SUMMARY:Spherical circle coverings and bubbles in foam - Tarnai\, T (Budap
 est University of Technology and Economics)
DTSTART:20140227T114500Z
DTEND:20140227T120500Z
UID:TALK51132@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:One of the classical problems of discrete geometry is the foll
 owing. How must a sphere be covered by n equal circles (spherical caps) so
  that the angular radius of the circles will be as small as possible? In t
 he 1980s when we started to work on this problem\, proven solutions were k
 nown only for n = 2\, 3\, 4\, 5\, 6\, 7\, 10\, 12\, 14\, and conjectured s
 olutions for n = 8\, 9\, 32. The first gaps appeared at n = 11 and 13\, fo
 r which even suggestions did not exist. We thought that the shapes of bubb
 les in foam might help. We considered the results of Matzkes experimental 
 observations\, and found that the edge graph of the only bubble with 11 fa
 ces and one of the 4 bubbles with 13 faces lead to the best coverings wher
 e the Dirichlet cells of the circle system provided the same edge graphs a
 s those of the respective bubbles. \n\nAdditionally\, we could show that f
 or 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 forme
 d 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 the range of n = 14 to 
 20\, we considered the possible shapes of coated vesicles. These are certa
 in kinds of bubbles where a part of the cellular membrane is surrounded by
  a clathrin basket  a polyhedron. With their help\, for these values of n\
 , except n = 19\, we could construct the best so far circle coverings of a
  sphere. \n\nIn the lecture\, we want to survey the results for n = 2 to 2
 0\, making comparison with soap-film-like cones\, bubbles in foam\, coated
  vesicles\, and to compare the best circle coverings with the numerical so
 lutions to the isoperimetric problem for polyhedra with n faces. This rese
 arch was supported by OTKA grant no. K801146. \n
LOCATION:Seminar Room 1\, Newton Institute
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