# Candidate solutions to partition problems with least perimeter

### Monodisperse case

This PDF document gives
(mostly conjectured) minimizers for several classes of equal area
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).

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.

### 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.

### 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.
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.

Copyright Simon Cox.
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