Why can't you form a sixth Platonic solid by identifying faces of tetrahedrons?

[ehrenfest]

[SOLVED] platonic solids

Homework Statement

Let f_1 be a face of one tetrahedron and let f_2 be a face on a second same-sized tetrahedron. Why do you not get a sixth Platonic solid when you identify f_1 and f_2, which you can do because they are the same triangle?

Homework Equations

The Attempt at a Solution

 

[Dick]

For a platonic solid, the same number of faces have to meet at each vertex.