TITLE: The sparsest planar net that captures every unit segment
QUESTION [17 upvotes]: Let $\cal C = \lbrace C_i \rbrace$ be a collection
of rectifiable curves in the plane with the property that
every unit-length segment meets at least one curve
in at least one point.
Call such a collection $\cal C$
a needle net: any unit-length "needle" is captured by the net.
I would like to find the sparsest needle net, sparse in the
sense that the curves have minimum length per unit area.
That is, the limit of $L/A$ of the ratio of the length $L$ of the curves within a region
to that region's area $A$, as the region grows large, is as small as
possible.
For example, a regular grid of orthogonal parallel lines
separated by $\sqrt{2}/2$ is a needle net: the diagonal of
each square cell of the grid has length $1$.
If I've calculated correctly, the length of its
curves (lines) within each unit area region $L/A$ is $2 \sqrt{2}$.
See left below, where a unit-length
diagonal is highlighted in red, and the region of the plane
I used to compute $L/A$ is marked.

   

Again if I've calculated correctly, the equilateral-triangle tiling of
the plane obtained from three sets of parallel lines is less efficient,
and the packing arrangement of unit-diameter circles shown right above
is less efficient still.
Is the square-grid the sparsest needle net?
This feels like a question that has been addressed before, perhaps
in another guise.  If so, a pointer would be welcomed.
Thanks!
Update. Roland Bacher's more efficient needle net:

            

Is this the optimal net?

REPLY [11 votes]: Without error of my part, a paving with regular hexagons with sides of length $1/2$ gives 
$L/A=4\sqrt{3}/3\sim 2.3094$. This could very well be the optimal candidate.