Candidate solutions to partition problems with least perimeter

candidates for N=32

Monodisperse case

This PDF document gives (mostly conjectured) minimizers for several classes of equal area isoperimetric/packing problems in two and three dimensions, in some sense generalizing the celebrated Honeycomb Conjecture for the tiling of the plane proved by Hales. These cover:

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.

All solutions were calculated with Brakke's Surface Evolver. Further details about the calculations can be found in

Bidisperse case

Here are conjectured minimizers for patterns in which bubbles take one of two possible areas:

Polydisperse free clusters

Conjectures for the minimal perimeter enclosing N cells with areas 1 to N are given in this PDF document for N up to 50. Again, please tell me if you find something better, or if you know of exact results.

Copyright Simon Cox.

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