Editing Angular defect (section)

==Positive defects on non-convex figures==
It is tempting to think that every non-convex polyhedron must have some vertices whose defect is negative, but this need not be the case if the Euler characteristic is positive (a topological sphere).  

{| class=wikitable
|+Polyhedra with positive defects
|[[Image:Polydera with positive defects convex.svg|180px]]
|[[Image:Polydera with positive defects concave.svg|180px]]
|}

A counterexample is provided by a [[cube]] where one face is replaced by a [[square pyramid]]: this [[elongated square pyramid]] is convex and the defects at each vertex are each positive. Now consider the same cube where the square pyramid goes into the cube: this is concave, but the defects remain the same and so are all positive. 

Two counterexamples which are self-intersecting polyhedra are the [[small stellated dodecahedron]] and the [[great stellated dodecahedron]], with twelve and twenty convex points respectively, all with positive defects.