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The expected distance $d$ of randomly selected points within a unit square to the square's center is

$d = \frac{1}{6} P$

where P is the universal parabolic constant

$P = \sqrt{2} + \ln{\left(1+\sqrt{2}\right)} = 2.2955871 \dots $

see

Is this a mere coincidence or is there an (intuitive) reason why this constant shows up in the solution to this problem?

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The "reason" that the two given numbers are equal is "write up the integrals, they turn out to be the same integral".

An answer that might satisfy the "intuitive reason" criterion is that the sides of the unit square are given by axis-aligned straight lines; in polar coordinates $r=\frac{1}{cos(t)}$ or $r=\frac{1}{sin(t)}$; and $\frac{1}{cos(t)^2}=\frac{r^2}{r^2cos(t)^2}=\frac{\sqrt{x^2+y^2}}{x^2}$ is the integrand in the integral defining the arc length of the parabolic segment. Truth be told though, this isn't really much more than saying "the integrals turn out to be the same integral", so I'm not sure how much of an "intuitive" explanation this is.

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I don't see any integral in the definition of the parabolic constant. Which integrals are you talking about? – user4503 Apr 15 '15 at 19:59
    
The parabolic constant is defined (basically) as the arc length of a certain curve segment, which you find with an integral of the form $\int_a^b\sqrt{1+f'(t)^2}dt$. – Ketil Tveiten Apr 16 '15 at 8:26

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