
> 
> **Question:** Can I have an arbitrarily large finite family of lines $\ell\_1,\dotsc,\ell\_n\subset\Bbb R^3$ so that the average angle between two (distinct) lines is $\ge \pi/3$?
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We can assume that all lines pass through the origin, and by angle I mean the smaller one of the two angles defined by two intersecting lines.


If my integral calculus hasn't failed me, then the expected angle between two random lines with uniformly chosen directions is exactly $1\, \mathrm{rad}$, wheras $\pi/3\approx 1.0472 >1$.
So generic arrangements won't do it. Can I be more clever? Or is there perhaps some standard measure theory argument that tells me that in the limit, uniformly chosen lines is the best I can do? If so, can we put a bound on $n$ for where this stops being possible?


I know from random experiments that $n=17$ is possible, but already for $n=18$ I have no examples so far.




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**Update**


The [comment](https://mathoverflow.net/questions/428555/are-there-arbitrarily-large-families-of-lines-in-bbb-r3-with-average-angle#comment1102298_428555) by alesia answers the question as posed (and I am looking forward to accept their [answer](https://mathoverflow.net/a/428607)): there are arbitrarily large families of lines with average angle $> \pi/3$.
Though one should note that the bound $\pi/3$ still seems kind of substantial: the more lines I have, the close the average will be to $\pi/3$.
I won't change the question (I might post a separate one), but the following seems to be the more fundamental question:



> 
> Is there a probability distribution on lines in $\Bbb R^3$ so that the expected value for the angle between two (independently sampled) lines is $>\pi/3$?
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I think *No* and I am looking forward to ideas. The value $\pi/3$ is attained by alesia's construction (uniformly
choosing directions from an orthogonal basis).



Are there arbitrarily large families of lines in $\Bbb R^3$ with average angle $\ge \pi/3$?
Start with $n$ times each coordinate axis, so $3n$ lines in total. The average angle between non necessarily distinct lines is $1/3\*0 + 2/3\*\pi/2 = \pi/3$. So the average angle between "distinct" lines (here, distinct lines may coincide geometrically due to multiplicities) is greater than $\pi/3$.


You can then perturb the lines to make them geometrically distinct while keeping an average angle larger than $\pi/3$.


Although that's not the question, I'm not sure what happens if one wants an infinite number of lines.


