Consider the sphere $S^{n-1} = \{ (x_1,\ldots, x_n)\in\mathbb{R}^n:\ x_1^2+\ldots +x_n^2=1 \}$ and denote by $S_i$ (for $i=1\ldots n$) the $n-2$ dimensional subsphere of $S^{n-1}$ orthogonal to $e_i$. More precisely, $S_i = \{ (x_1,\ldots, x_{i-1},0,x_{i+1},\ldots, x_n):\ x_1^2+\ldots+x_{i-1}^2+x_{i+1}^2+\ldots+x_n^2=1 \}$.
The image below gives an illustration in the 3 dimensional case.
In dimension 3, it's easy to see the number of intersection between the $S_i$'s. We have that
$\sharp (S_1\cap S_2) = \sharp(S_1\cap S_3) = \sharp(S_2\cap S_3) = 2,$
$\sharp(S_1\cap S_2\cap S_3) = 0$.
I'm interested in the number of intersections in the general case. My guess is the number of intersections will be a power of 2, depending on how many $S_i$'s I am intersecting. I don't know how to make this calculations and couldn't find any article or textbook about this. My last hope is to share my problem here. Thank you!
$\newcommand{\Reals}{\mathbf{R}}$Generally, $0 \leq k \leq n$ distinct coordinate hyperplanes in $\Reals^{n}$ mutually intersect in a subspace of dimension $n - k$, whose intersection with the unit sphere $S^{n-1}$ is a great sphere $S^{n-k-1}$. The number of points in this set is equal to:
Infinity if $k < n - 1$;
Two if $k = n - 1$;
Zero if $k = n$.
For example, the intersection of the unit $3$-sphere $S^{3} \subset \Reals^{4}$ with the hyperplanes $\{x_{1} = 0\}$ and $\{x_{2} = 0\}$ is the unit circle lying in the $(x_{3}, x_{4})$-plane, which contains infinitely many points.