The question is:
- Find functions $f(n,l)$ and $g(n,l)$ that gives the surface $(n-1)$-hypervolume and the $n$-hypervolume for any given regular geometric shape that is the n-dimensional analogue of a square and a sphere. Knowing that:
- The surface hypervolume of the first dimension n-sphere is $2$,of the second is $2\pi r$, of the third is $4\pi r^2$ and the fourth is $2\pi ^2r^3$
- The internal hypervolume of the first dimension n-sphere is $2r$,of the second is $\pi r^2$,of the third is $\frac43 \pi r^3$ and the fourth is $\frac {\pi^2}{2}r^4$
- The surface hypervolume of the first dimension n-cube is $2$, of the second $4l$, the third $6l^2$ and the fourth $8l^3$
- The internal hypervolume of the first dimension n-cube is $l$, of the second $l^2$, of the third $l^3$ and the fourth $l^4$
So, saying the spheres have functions $f(n,r)$ and $g(n,r)$ and the cube $h(n,l)$ and $i(n,l)$ we can say that $g(n,l)$ is (at least) the $r$ integral of $f(n,l)$.
Whereas $i(n,l)$ is just $l^n$ and $h(n,l)$ is $(2n)l^{n-1}$
So, can anybody help me find either $f(n,l)$ or $g(n,l)$²
