Someone told me that the formulas of the volume and surface area of an n-dimensional sphere get simplified a lot if we express them in terms of $\eta = \frac{\pi}{2}$ instead of $\pi$.
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In terms of $\pi$ we have: $$V_n = \frac{\pi^\frac{n}{2} r^n}{\Gamma\left(1+\frac{n}{2}\right)}$$ Which is not very simple since we have the $\Gamma$ function.
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In terms of $\tau = 2\pi$ we have: $$V_n = \frac{\tau^{\left\lfloor\frac{n}{2}\right\rfloor} r^n}{n!!}$$ × 1 if n is even ; × 2 if n is odd.
Which is much simpler, but it unfortunately depends on the parity of $n$...
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In terms of $\eta = \frac{\pi}{2}$ we have: $$V_n = \frac{2^n \eta^{\left\lfloor\frac{n}{2}\right\rfloor} r^n}{n!!}$$ Which is beautifully simple.
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Note that here $\lfloor x \rfloor$ denotes the floor function: $\lfloor 4.2\rfloor = \lfloor 4.9\rfloor = 4$
The formulas for the surface area are almost the same.
He showed me an article on a website which seemed to confirm what he said (though that article was primarily about $\tau = 2\pi$).
But then I made some searches by myself on Google, and it turns out that there is absolutely no other website which mentions this incredible thing. So I'm not totally sure of the credibility of all this... Surely if this were true, it would be mentioned in every article speaking of volume and surface area of n-dimensional spheres ; but it's not, so there's something fishy going on there...
I view the formulas of the volume and surface area of n-dimensional spheres as the most fundamental, the most important and the most elemental of all equations, because they are even above the dimensional continuum, they exist independently of how many dimensions we are in.
So if it is indeed true that these formulas are written in their simplest form when they are expressed in terms of $\eta = \frac{\pi}{2}$, then I would view $\eta = \frac{\pi}{2}$ as the true most fundamental circle constant.
There's a discussion of $\eta$ at http://blogs.adelaide.edu.au/maths-learning/2011/06/08/pi-tau-and-eta/ .
In the $n$-sphere formulas, I regard the use of the floor function (especially in an exponent) as somewhat ugly. By writing $\eta^\left\lfloor\frac{n}{2}\right\rfloor$ (or $\tau^\left\lfloor\frac{n}{2}\right\rfloor$) you are already causing me to think separately about the cases for $n$ odd and $n$ even:
$$\left\lfloor\frac{n}{2}\right\rfloor = \frac{n}{2} \ \mbox{for $n$ even,}$$ but $$\left\lfloor\frac{n}{2}\right\rfloor = \frac{n-1}{2} \ \mbox{for $n$ odd.}$$
This means that when we write out the explicit formulas for $n = 1, 2, \dots,$ the exponent of $\eta$ for dimension $n$ will be the same as for dimension $n-1$ if $n$ is odd, but will be one greater than for dimension $n-1$ if $n$ is even.
The double factorial, $n!!$, also implies separate even and odd cases:
$$n!! = n \cdot (n-1) \cdot (n-2) \cdot \cdots \cdot 2 \ \mbox{for $n$ even,}$$ but $$n!! = n \cdot (n-1) \cdot (n-2) \cdot \cdots \cdot 1 \ \mbox{for $n$ odd.}$$
In fact, if you write out the even and odd cases of $V_n$ separately, it is very easy to confirm that the formula given for $V_n$ in terms of $\eta$ is exactly equal to the one in terms of $\tau$, and not hard to show that it is equal to the formula given in terms of $\pi$ and the gamma function. As for why you are finding only two of these formulas on Web sites, I think it is because the standard formulas use $\pi$ and the advocates of $\tau$ have promoted their views diligently; the publications using $\eta$ are simply lagging behind those that use the other two symbols.