Candidate solutions to partition problems with least perimeter
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
- 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.
All solutions were calculated with Brakke's Surface Evolver.
Further details about the calculations can be found in
Here are conjectured minimizers for patterns in which bubbles take one of two possible areas:
- Free clusters:
The minimal perimeter enclosing N/2 cells of area A=2 and N/2 cells of area A=1, for N=4,6,8 and 10.
Conjectured least perimeters are, respectively, 13.512852, 19.050002, 24.400859 and 29.693421.
For details see Vaz, M.F., Cox, S.J. and Alonso, M.D. (2004)
configurations of small bidisperse bubble clusters. J. Phys: Condensed
- Confined clusters: this article (F.J. Headley and S.J. Cox (2019) Euro. Phys. J. E. 42: 92.) describes an exhaustive
method to generate conjectured minimizers for the least-perimeter partition of the disc into N ≤ 10 regions of two different areas,
for all area ratios. The conjectured minimizers are illustrated here
and an example of a Surface Evolver file is available here.
- Space-filling 2D foams: bidisperse tilings of the plane, with equal numbers of bubbles of each
of two areas and small unit cells, have been catalogued by M.A. Fortes and P.I.C. Teixeira: Minimum
perimeter partitions of the plane into equal numbers of regions of two different areas, Eur. Phys. J. E,
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.
tell me if you find something better, or if you know of exact results.
Copyright Simon Cox.
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