n=4 Tetrahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.
The applet illustrates the case where 4 vertices of a regular tetrahedron "induce" the vertices of two other polyhedra:
4 ●Regular Tetrahedron← 4 ●Regular Tetrahedron →6 ●Regular Octahedron.
Description are in https://www.geogebra.org/m/y8dnkeuu and https://www.geogebra.org/m/rkpxwceh.