There are six regular convex polytopes in 4D, which are analogous to
the five Platonic solids in 3D. This is the second, analogous to the octahedron, called the cross polytope.
This is close to a vertex-first projection, but rotated a little so
the central vertices don't quite overlap and you can see all 16
tetrahedral cells. The cross polytope is dual to the hypercube, so its 16 cells correspond to the 4-cube's 16 corners.
A different projection is here