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Pólya's orchard problem is as follows:

"How thick must the trunks of the trees in a regularly spaced circular orchard grow if they are to block completely the view from the center?"

See, e.g.,

Allen, Thomas Tracy. "Polya's orchard problem." American Mathematical Monthly (1986): 98-104. JSTOR link

or this earlier MO question. This problem has been well-studied. One crude way to phrase what is known is that, if the trees/disks centered at each lattice point have radius $r$, the furthest distance one can see from the origin is $R \approx 1/r$.

Here is my question. Suppose instead of disks centered on each lattice point, we have a randomly oriented segment (a $1$-dimensional disk) centered on each lattice point, of length $2r$, i.e., of radius $r$.

Q. Is it still the case that the furthest one can see from the origin is expected to be $R=c/r$ for some constant $c$?

Here is an example, with $r=3/8$. The "Polya radius" is about $2.47$, but in this one random instance, visibility extends about $3$ times further, $R \approx 8$:

  ![Polya10][1]

In $\mathbb{R}^3$, the same question can be asked with now $2$-dimensional randomly oriented disks centered on each lattice point. And of course the question generalizes to $\mathbb{R}^d$.

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  • $\begingroup$ One might even guess that the expected size is all that matters, which would give $c=\pi/2$. $\endgroup$
    – Will Sawin
    Commented Aug 31, 2014 at 2:15
  • $\begingroup$ @WillSawin: Could you expand on your remark? I don't follow how you arrive at $\pi/2$... $\endgroup$
    – Joseph O'Rourke
    Commented Aug 31, 2014 at 2:23
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    $\begingroup$ I'm imagining looking through a long field of objects. It seems to me that the distance I see is going to be governed by the average visible angle the objects take up. For a line of radius $r$, this is $2/\pi$ times the visible angle of a disc of radius $r$. (average value of $\sin \theta$ for $0<\theta<\pi$). $\endgroup$
    – Will Sawin
    Commented Aug 31, 2014 at 2:35

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Probably this problem was not considered so far, but there are some related articles.

  1. "The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems" by Jens Marklof, Andreas Strömbergsson. In partiqular they give the number of spheres in a random direction.

  2. "Visibility and directions in quasicrystals" by Jens Marklof, Andreas Strömbergsson. (In each direction points which are closed to a straight line form a "quasicrystal".)

3."Perfect Retroreflectors and Billiard Dynamics" by Pavel Bachurin, Konstantin Khanin, Jens Marklof, Alexander Plakhov

See also Marklof's talk at ICM-2014 (lecture I09_04) for an introduction.

I think that this authors can solve your problem in arbitrary dimension.

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