We can divide $S^2$ in 2n triangles by considering the triangles that come from dividing the sphere with n meridians and the equator. I am trying to visualize a similar thing with $S^3$ and tetrahedra. I think that, as the triangles of $S^2$ share an edge, the tetrahedra of $S^3$ should share a face.
My aim is to visualize the 1-skeleton of such an object, but it seems I am 4-D blind...
Is there a way to visualize the projection of such an object in 3-D?
More generally: are there techniques to visualize 4-D object and project them in 3-D?