Imagine you want to make a net out of string to filter and catch objects of a certain size, minimizing the length of string employed. (This actually arises in filtering biological impurities from water.) One way to model the problem is as follows.

A *net* $N$ is a connected, infinite plane graph with no vertex of degree $1$.
The objects to be caught are all convex shapes $C$ of area $\ge 1$:

Above, the disk has radius $r=1/\sqrt{\pi} \approx 0.56$, the square is $1 \times 1$, etc.

A shape $C$ can pass through the net if there is some positioning
of $C$ such that the closed shape $C$ shares no point with $N$.
Otherwise $C$ is *caught* by the net $N$.
Let $\cal C$ be the set all convex shapes $C$ of area $\ge 1$.

A natural net is an orthogonal grid, a tiling of the plane by $1 \times 1$ squares. It seems clear this catches all shapes in $\cal C$. Its efficiency can be measured by the length of string per unit area of the plane. For this grid, this $L/A$ ratio is $2$.

But a tiling of the plane by unit-area hexagons is more efficient. The edge length of such a hexagon is $\frac{\sqrt{2}}{3^{3/4}} \approx 0.62$; its incircle radius is $\frac{1}{\sqrt{2} \sqrt[4]{3}} \approx 0.54$. Here it is shown superimposed on the unit-area disk, which it clearly catches.

This hexagonal tiling seems to catch all shapes in $\cal C$, and if I've calculated correctly, its $L/A$ ratio is $\frac{3^{3/4}}{\sqrt{2}} \approx 1.61$.

Q. Is there a more efficient net than this hexagonal tiling?

I asked an analogous question two years ago, "The sparsest planar net that captures every unit segment," whose answer might also be a hexagonal tiling. The current question is more like an isoperimetric problem, and might have been explored already in that context. If anyone knows, I'd appreciate a pointer. Thanks!

**Answered**by

*Yoav Kallus*. My problem is "equivalent to asking what is the least mean length partition of the plane into cells of area at most $1$." It is therefore "equivalent to the Honeycomb problem, which was solved by [Thomas] Hales" in 1999, just 2035 years since it was "conjectured" by Varro :-). So the hexagonal tiling described above is indeed optimal, but certainly not obviously so. The answer to

**Q**is therefore:

*No*.