Slicing $N$-dimensional cubes along the common direction of pairs of directions

Question

I want to understand the shape and geometry that remains after cutting an $N$-dimensional cube with all the hyperplanes from the equations $x_i = x_j$, with all possible $i, j$ with $1 \leq i \neq j \leq N$. There are $N(N-1)/2$ such hyperplanes.

For the case of $N=3$, all planes intersect along the direction $\bf{e_1} + \bf{e_2} + \bf{e_3}$ and around this axis, the cube is split in 6 slices (2 for each direction).

Is hard to visualize what happens at higher $N$, but it is clear that the vector along $\sum_i \bf{e_i}$ is always an intersection common to all the hyperplanes

Answer 1

The hypercube is cut into $N!$ simplexes, each corresponding to one of the orderings of the coordinates, since continuously moving from one ordering to another crosses the cutting hyperplane associated with the two coordinates swapping places. The simplexes are all congruent with the ”standard“ one defined by $x_1\le x_2\le\cdots\le x_N$.

Answer 2

I guess it would be divided into $n!$ pieces, the first is $$x_1\lt x_2\lt x_3\cdots\lt x_n$$ and the last one is $$x_n\lt x_{n-1} \lt x_{n-2} \cdots \lt x_1$$