Candidate solutions to partition problems with least perimeter

This PDF document gives (mostly conjectured) minimizers for several classes of isoperimetric/packing problems in two and three dimensions:

• Minimal perimeter enclosing N cells of equal area (N ≤ 42).
• Minimal perimeter partition of the disk, the equilateral triangle, the square, the regular pentagon and the regular hexagon into N cells of equal area (N ≤ 42).
• Minimal perimeter partition of the surface of the sphere (N ≤ 32).

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.

Bidisperse free clusters

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.

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