Consider placing $n$ lines in $d$ dimensional space in a way that the angles between any two pairs of lines is always the same (and they all pass through the origin). When $d=3$, we get configurations of lines with this property if we take any Platonic solid with triangular faces, mark the center of mass as the origin and draw lines from there to each of the vertices. Doing this with an Octahedron produces the coordinate system. Doing it with a Tetrahedron produces four lines where any pair is at $60^{\circ}$. Doing it with an Icosahedron produces $6$ lines where any pair is at about $97^{\circ}$.
Doing this with Platonic solids like the cube where the faces aren't triangles doesn't produce the same effect. This can't be a coincidence. Can we prove this will always be the case? And can the observation we made in $3$ dimensions be extended to $d$ dimensions?
This does not extend to $d$ dimensions; the $600$-cell has triangular faces and tetrahedral cells, but its $60$ axes are not all at equal angles to each other. You can see this by fixing one axis and considering the possible "altitudes" of vertices relative to the "north pole" at the top of this axis - if all lines were at the same angles to one another, there could only be two such altitudes besides the north and south poles, but in fact there are $7$ (in groups of size $12,20,12,30,12,20,12$ from north to south).