# Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points.

Fix a point $p$ on a plane and consider circles with center at the point $p$ and radius $r$. Let $N(r)$ be the number of regions inside the circle and $L(r)$ the total length of all curves inside this circle.

What is best possible arrangement of curves if we want to minimize $$\lim_{r\rightarrow\infty}\frac{L(r)}{N(r)}.$$

I guess that the curves should be edges of hexagons, i.e. they should form the honeycomb structure. In that case $\lim_{r\rightarrow\infty}\frac{L(r)}{N(r)}=\frac{4}{\sqrt[4]{27}}$ because on average we need to draw $4$ new edges of hexagon to create every new region and the length of the edge of hexagon which has volume $1$ is $1/\sqrt[4]{27}$.

Actually, if we assume that all regions have to be of the same shape then the only possible choices are: triangular, square or hexagonal lattice. It is easy to show that hexagonal lattice is the best. But do the regions really have to have all the same shape?

This is probably a standard problem. In fact it was solved by bees long time ago, but do they find the optimal solution?

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This reminds me of the type of problems in Burago-Burago-Ivanov's "Metric Geometry"; sorry I can't think of a chapter, nor page. – user99680 Jun 26 '14 at 19:19