What are polyhedrons? Polyhedrons or three dimensional analogues of polygons were studied by Euler who observed that if one lets $f$ to be the number of faces of a polyhedron, $n$ to be the number of solid angles and $e$ to be the number of joints where two faces come together side by side $n-e+f=2$.
It was later seen that a serious defect in this definition (and in the proof supplied by Euler) is that it is not at all clear what is a polyhedron in the first place. For example if we consider a cube nested within another cube as a polyhedron then $n-e+f=4$, a counter example to Euler's result.
What will be the modern definition of that polyhedron which will comply with Euler's result?

It simplifies matters to insist that all faces ("components" below)
be convex polygons, and permitting coplanar faces (partitioning
any nonconvex face into convex pieces). Then 



I have a discussion expanding these three conditions on pp.101-104 of
Computational Geometry in C, which is available via Google Books.
