For a number of larger values of N, particularly powers of 10 and hexagonal numbers, this PDF document gives conjectured minimizers for the minimal perimeter enclosing N cells of equal area.
Please tell me if you find something better, or if you think I have inadequately or incorrectly attributed results.
See the 4th edition of Frank Morgan's book, Geometric measure theory: a beginner's guide (Academic Press, 2009), to find out why these problems are interesting and the mathematics behind them.
The minimal perimeter enclosing N/2 cells of area A=2 and N/2 cells of area 1, for N=4,6,8 and 10. Conjectured least perimeters are, respectively, 13.512852, 19.050002, 24.400859 and 29.693421. See Vaz, M.F., Cox, S.J. and Alonso, M.D. (2004) Minimum energy configurations of small bidisperse bubble clusters. J. Phys: Condensed Matter. 16:4165-4175.
Conjectures for the minimal perimeter enclosing N cells with areas 1 to N are given in this PDF document for N up to 50. All solutions were calculated with Brakke's Surface Evolver. Again, please tell me if you find something better, or if you know of exact results.
Back to main page and contact details