For each of the 5 regular polyhedra, enumerate the number of vertices (v), edges (e), and faces (f), and then evaluate the quantity v - e + f. (One of the most interesting theorems relating to any convex polyhedron is that v - e + f = 2. -> This was given as part of the problem. I am not sure if it is of any value when solving this problem or not.)

Please use pictures to illustrate the concept above along with a detailed explanation of your response. Thank you.

Solution Preview

See attachment please!

A polyhedron is a 3-dimensional closed figure made from flat sides, or 'faces'. There are an infinite number of them. A barn, for example, has many flat surfaces or faces; it's a polyhedron. So it a textbook. A prism. The diamond in a ring. All these figures are made from flat 'faces'.

But on each one of the objects mentioned above , some of the faces are different shapes. Is there a polyhedron with all the faces the same?

Yes there is. In fact, there are just five of them! You are familiar with at least one of them ...

Solution Summary

This shows how to find the number of vertices (v), edges (e), and faces (f) of polyhedra.

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See attached file.
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Big picture: The purpose of this experimen