TITLE: Optimal stacking of split logs
QUESTION [7 upvotes]: Consider firewood logs as unit-radius cylinders of the same length.
Each log is split into $k$ pieces by equiangular sectors
meeting in the circle center:
$k=2$ leads to semicircles, $180^\circ$;
$k=3$ results in $120^\circ$ sectors, etc.

          



Q. What is the optimal packing of a halfplane 
  by equiangular circular sectors, for each $k$?

I am hoping that for some $k>1$, the optimal packing is known.

          


                   


A practical application, for small $k$ (say, at most $6$, as above [gray]), is how
to optimally stack firewood for the winter.

REPLY [3 votes]: I don't know if this is the optimal packing of 1/6-circle wedges, but I do know that the following is the optimal packing of the shape formed by gluing two of those wedges along a full edge:

The two-wedge shape is convex and 2-fold symmetric, so its optimal packing is the optimal lattice packing, which can be solved for numerically. The density of this packing is $0.978769$. The horizontal edges have a small segment of length $0.0291219$ (if the radius is 1) hanging off and not touching the neighboring wedge. If the horizontal edges coincided fully, the density would only be $0.978263$. Compare to the density of circle packing, $0.906900$.