Three icosahedra can't share an edge
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 that we can't use icosahedra because if we put three of them around an edge, the sum will exceed 360°. You can compare this image with what happens with the other regular polyhedra.
© matematita
immagine di Francesca Lazzaroni
The image belongs to the sections...:
Other costructions (3D geometry)
Images of the exhibition (From the exhibitions of matematita)
Various matters (4D geometry)
