There is a classic (and open) problem asking whether every polyhedron can be unfolded to give a non-overlapping net. The converse problem has been studied asking which polygons can be folded in some manner to give a polyhedron. For an overview of both problems and related discussion see:

Erik D. Demaine and Joseph O'Rourke Geometric Folding Algorithms: Linkages, Origami, Polyhedra Cambridge University Press, July 2007. ISBN 978-0-521-85757-4

##Question##

I want to ask about a more direct converse. Given a net of polygons connected at their edges when can they fold to form a polyhedron?

###Example###

As an example take the two nets shown below. By identifying the edges as shown by colour both satisfy the topological constraints to be sphere. Two nets http://www.mathematicians.org.uk/eoh/gallery/Random_Polyhedra.png

Only one, however, will fold to give a polyhedron:

Four views of a polyhedron http://www.mathematicians.org.uk/eoh/gallery/4_views_500.png

##Personal Motivation##

My motivation is to find visually appealing, simple, but non-symmetric, polyhedra. I have used equilateral triangles but would like to play with other shapes. General sufficient conditions would, therefore, be very interesting to me.