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A recent question on puzzling relied on hyper-sphere having maximal area at dimension 7.

Which is shown here.

What I can't shake off is the feeling that one can't compare the surface of unit spheres of different dimensions. It seems the units would be totally unrelated. A sphere has a surface in squared units, a circle has a length in units and a 4-d hyper-sphere has an area in cubic units (typically a volume, for us in 3d).

But the fact that someone went to the effort of making this wikipedia graph implies there's some usefulness or value there. Or even just that the values are comparable in some way.

What am I missing?

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  • $\begingroup$ Yea, you can't compare their measures (the proper terminology) like that, as the measures are defined on different spaces. For example, the measure of the unit sphere in $\mathbb R^3$ would be some non-zero value, while it is zero in all $\mathbb R^{4+}$ spaces. $\endgroup$ Oct 3, 2019 at 14:28
  • $\begingroup$ Not an answer, but related and interesting: math.stackexchange.com/questions/2644700/… $\endgroup$ Oct 3, 2019 at 16:31
  • $\begingroup$ Like Vasily points on in his useful answer, one can still ask about the ratio of the $n$-sphere to the $n$-cube. Alternatively, the volume of an $n$-dimensional hypersphere is $$V_r(n) = \frac{\pi^{n / 2}}{\Gamma\left(\frac{n}{2} + 1\right)} r^n, $$ and all of the quantities $$V_r(n)^{1 / n} = \frac{\sqrt{\pi}}{\Gamma\left(\frac{n}{2} + 1\right)^{1 / n}} r \sim \sqrt{2 \pi e} \,r \cdot \frac{1}{\sqrt n} + O\left(\frac{\log n}{n^{3 / 2}}\right)$$ have the same units, though I don't claim that this has any particularly helpful interpretation either. $\endgroup$ Oct 3, 2019 at 17:26
  • $\begingroup$ This is less interesting than just considering the coefficients $V_1(n)$ in that it is a strictly decreasing function of n. Amusingly, if we let $n$ take on real values, we have $$\lim_{n \searrow 0} V_1(n)^{1 / n} = \sqrt{\pi e^\gamma} ,$$ where $\gamma$ is the Euler-Mascheroni constant. $\endgroup$ Oct 3, 2019 at 17:29

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You cannot compare measures itself, but you can compare coefficients before $r^n$ in a corresponding formula.

In other words you can compare how large the volume of unit n-sphere is in comparison to the volume of unit cube. I will leave the usefulness of this aside.

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I don't think the purpose was to make a meaningful comparison between the volumes. The statement is simply that if you set the radius of the sphere to be $1$ then the quantity representing the volume of the boundary is largest in $7$ dimensions. This is an interesting statement even if the comparison between the quantities is not meaningful, because they are at least analogous in being the volume of the unit sphere.

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