TITLE: What are polyhedrons?
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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.