The Platonic Solids and one odd-ball Polyhedron

The Platonic Solids can be named by an integer pair {n,m} where "n" and "m" are the number of edges on each face and the number of faces that meet at each corner, respectively.

The polyhedra are {x,3} and {3,y} in the ascending and descending arcs of triangluar pattern, respectively, and the values x and y increase as you move from the tetrahedron, {3,3}.

Each Platonic Solid has a "dual", a reflection of itself in the set of Platonic Solids. The dual of {x,y} is {y,x}, and thus the tetrahedron is its own dual.

• Now for some thought excercises.
Imagine that each of the Platonic Solids is a real, tangible object. With a very sharp (and imaginary) knife slice one corner off each of the solids.

For each of the Platonic solids {n,m}, the newly created face has how many sides? And how many original faces were altered?

m&m.     m faces were altered and each new face has m sides.

• The denouement.
For each of the Platonic solids, slice off all of the corners, with planar cuts passing through the center of each face adjacent to each corner. For each of the solids, what do you have now?

Its dual!

Cut off all the corners of an icosahedron with planar cuts such that center third of each edge is left intact, and the rest is removed. Now what do you have?

Well, it looks like a soccerball, but its called a truncated icosahedron.