A Potential Error
It is tempting to think that every non-convex polyhedron has some vertices whose defect is negative. Here is a counterexample. Consider 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.
Negative defect indicates that the vertex resembles a saddle point, whereas positive defect indicates that the vertex resembles a local maximum or minimum.
Read more about this topic: Defect (geometry)
Famous quotes containing the words potential and/or error:
“Humanity has passed through a long history of one-sidedness and of a social condition that has always contained the potential of destruction, despite its creative achievements in technology. The great project of our time must be to open the other eye: to see all-sidedly and wholly, to heal and transcend the cleavage between humanity and nature that came with early wisdom.”
—Murray Bookchin (b. 1941)
“An error the breadth of a single hair can lead one a thousand miles astray.”
—Chinese proverb.