Five tetrahedra with an edge in common
A simple way to understand why there are only five regular polyhedra is to examine how regular polygons can be disposed around a vertex so that the sum of their angles is smaller than 360°. In the same way, if we want to understand how many 4-dimensional regular politopes there are, we can examine how regular polyhedra can be disposed around an edge so that the sum of their dihedral angles is smaller than 360°. Here we see 5 tetrahedra around an edge (this combination produces the 600-cell); compare this image with the photo of a model and see the other possibilities.
immagine di Francesca Lazzaroni