TITLE: The lore of the game Numenera mentions "an irrational number that may be a four-dimensional equivalent of $\pi$". What could this mean?
QUESTION [25 upvotes]: 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?

REPLY [2 votes]: 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, $\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-\pi (x^2+y^2+z^2+w^2)}\,\mathrm dx\,\mathrm dy\,\mathrm dz\,\mathrm dw=1.$