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The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$ and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$

both attain maximum values for finite $n$. We can see from the Wikipedia page on Unit sphere that (for integer $n$ at least), those values are 5 and 7, respectively.

Now, I know that in a strict sense, these values are not really comparable, because they represent measures in different dimensions. But I was wondering if there is nonetheless some good intuitive reason why this happens, both why they both decrease eventually, and why the values where they attain their maximums are those (apparently the actual maxima are fractional, from WolframAlpha, though it didn't give a closed form).

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    For the volume, this has been extensively discussed on MO: http://mathoverflow.net/questions/8258/whats-a-nice-argument-that-shows-the-volume-of-the-unit-n-ball-in-rn-approaches2012-11-14
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    Ah, I rather suspected there might be something there. That site tends to have more upper-level math questions than this one.2012-11-14
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    And maybe I missed it, but I didn't see any intuition on the special values of 5 and 7.2012-11-14
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    The MO explanation includes a discussion of how fast the volume decreases (i.e., very). Note that $A_n = nV_n$ and it's then easy to see why the surface area is going to $0$ as well.2012-11-14
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    The maximum of $\pi^{n/2}/\Gamma(1+n/2)$ for is at approximately $ 5.256946405$, which is where $\Psi(1+n/2) = \ln(\pi)$. I would not expect this to have a closed-form solution.2012-11-14
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    And $\Psi(n/2)=\ln(\pi)$. Does $\Psi$ have an inverse?2012-11-14
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    You should see the discussion here: http://math.stackexchange.com/questions/15656/volumes-of-n-balls-what-is-so-special-about-n-5/15798#157982013-04-22
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    [This post](https://divisbyzero.com/2010/05/09/volumes-of-n-dimensional-balls/), and the comments on it, have some good answers to this.2017-05-26

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