Additionally, the same net may have multiple valid gluing patterns, leading to different folded polyhedra. However, the polyhedron formed in this way may have different faces than the ones specified as part of the net: some of the net polygons may have folds across them, and some of the edges between net polygons may remain unfolded. If a net is given together with a pattern for gluing its edges together, such that each vertex of the resulting shape has positive angular defect and such that the sum of these defects is exactly 4 π, then there necessarily exists exactly one polyhedron that can be folded from it this is Alexandrov's uniqueness theorem. Conversely, a given net may fold into more than one different convex polyhedron, depending on the angles at which its edges are folded and the choice of which edges to glue together. ![]() The edges that are cut from a convex polyhedron to form a net must form a spanning tree of the polyhedron, but cutting some spanning trees may cause the polyhedron to self-overlap when unfolded, rather than forming a net. ![]() Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated.
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