What is the 'optimal' equal-area partition of a circle?

Question

What is the (an?) n-partition of a circle that meets the following criteria:

1. The boundaries of each partition can be represented as a union of finitely many finite-piecewise-smooth simple closed contours. No weird unrealizable infinite cantor nonsense.
2. The integral over each partition is equal.
3. The sum of arclengths of each smooth contour is minimized

Answer 1

For small numbers of cells $3,4,5,..$ there may be a possibility of finding explicitly the optimal partition (the Y 120 degrees partition in the case $n=3$, etc). For large numbers of cells in the partitions, analytical results are not known. In order to find candidates for the optimal partitions, some numerical studies were performed:

• Cox and Fikkema: article They studied various polygons and the circle, for up to $42$ cells. The software Evolver was used (specifically designed for the study of bubble configurations). Recently studies were made for bubble configurations up to $200000$ cells (link). For large numbers of cells, hexagonal configurations are observed. In fact, for $n$ large the configuration converges to the honeycomb, due to the theorem of T. Hales.

• E. Oudet: link a method based on a relaxation argument. The results are not optimal for large number of phases.

• B. Bogosel: link a modification of the above method which avoids better the local minima. In particular it obtains similar results to the ones obtained by Cox and Fikkema.