Reconstructing Polytopes via Cauchy Rigidity: I

Biliana Kaneva

It is an unsolved problem to reconstruct a convex polyhedron (polytope) from a flat piece of paper and folding instructions. We approached this using Cauchy’s rigidity theorem, which says that the lengths of the edges of a triangulated polytope uniquely determine its 3D structure. I wrote C and Java programs to implement a numerical relaxation procedure (based on energy minimization) that starts with the triangulated graph structure and edge lengths, and ends with the 3D coordinates of the unique, determined polytope.

An example is shown in Figure 1, a snapshot from my Java applet. On the left is the input graph, which a group of us derived from the unfolding of a tetrahedron with one vertex clipped off. The right side of Figure 1 shows the 3D polytope as a wireframe structure produced by the relaxation algorithm. Figure 2 shows the same polytope displayed as a solid; showing that it is a five-vertex polyhedron as theory predicts. Professor O’Rourke believes this is the first example of a computed polytope folding.

 

Figure 1. Applet Input (left) and Output (right). Figure 2. Solid Polyhedron.

 

Supported by the Schultz Foundation

Advisor: Joseph O’Rourke