The lore of the game Numenera mentions "an irrational number that may be a four-dimensional equivalent of $\pi$". What could this mean? There's an RPG (Role-Playing Game) called Numenera, set on the Earth a billion years in the future, which is covered in the partially-functional and generally weird and mysterious technological ruins of a billion years of advanced civilizations. In one of its sourcebooks, it lists possible fragmentary transmissions that a player character might receive from its global "datasphere", one of which is listed as the following:

An irrational number that may be a four-dimensional equivalent of $\pi$.

When I saw this, my first thought was "I'm pretty sure that's probably just pi multiplied by a constant", followed by "If it's not, I'm sure mathematicians have already worked it out." When I did I Google search, I couldn't find anything obvious.

So, what is the equivalent to pi for a four-dimensional hypersphere?

 A: If you mean the sphere in the 4-dimensional Euclidean space, then the ratio between its surface area and its radius cubed is $2\pi^2$
A: You're close.  The "volume" of a $4$-dimensional ball is given by
$$
V = \frac{\pi^2}{2}R^4
$$
and its "surface area" is given by
$$
S = 2 \pi^2 R^3.
$$
If we take the $n$-dimensional equivalent of $\pi$ to be the ratio between the volume of the $n$-ball and $R^n$ (the volume the $n$-cube with side length $R$), then the 4-D equivalent of $\pi$ is $\frac{\pi^2}{2}$.
More generally, we would have (for positive integers $k$)
$$
\begin{align}\pi_{2k} &= \frac{\pi^k}{k!}, \\
\pi_{2k+1} &= \frac{2^{k+1}\pi^k}{(2k+1)!!} = \frac{2(k!)(4\pi)^k}{(2k+1)!}.\end{align}
$$
where $\pi_n$ is the $n$-dimensional equivalent of $\pi$.  Because $\pi$ is known to be transcendental, we can conclude that these are all irrational (and transcendental as well).
A: Another possible generalization, that extends the "circumference to diameter" concept more directly, is to consider the ratio of the surface volume to the volume of a diametric cross-section, the latter of which is a sphere. In three dimensions, the equivalent is the ratio of surface area of the sphere to the area of a planar cross-section taken through the center (which is a circle), and to see the relationship to the two-dimensional case, note that "circumference" can be thought of as the "surface length" and "diameter" as the length of a linear cross-section through the center.
That is, in effect, the "$\pi$" is the ratio of the surface volume of a 4D ball of radius $R$ to the regular volume of a 3D ball of radius $R$. Thus...
The surface volume of the 4-dimensional ball is
$$\mathrm{SV} = 2\pi^2 R^3$$
and the volume of the 3-dimensional ball is
$$V = \frac{4}{3} \pi R^3$$
so the ratio is just
$$\pi_4 = \frac{\mathrm{SV}}{V} = \frac{2\pi^2 R^3}{\frac{4}{3}\pi R^3} = \frac{3}{2} \pi$$
Hence, "four-dimensional pi" is $\frac{3}{2} \pi$. You're welcome. And yes, it is $\pi$ times a constant (rational number, as @rghome mentions in the comments).
A: Define $\pi$ as the unique real number such that
$$\int_{-\infty}^\infty e^{-\pi x^2}\,\mathrm dx=1.$$
This is the same in $1,2,3,4,\dots$ dimensions.
Indeed, $\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-\pi (x^2+y^2+z^2+w^2)}\,\mathrm dx\,\mathrm dy\,\mathrm dz\,\mathrm dw=1.$
