1. Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College

2. Outline: Reconstruction of Convex Polyhedra zCauchy to Sabitov (to an Open Problem) yCauchys Rigidity Theorem yAleksandrovs Theorem ySabitovs Algorithm

3. zgraph zface angles zedge lengths zface areas zface normals zdihedral angles zinscribed/circumscribed Reconstruction of Convex Polyhedra Steinitzs Theorem Minkowskis Theorem }

5. zgraph zface angles zedge lengths zface areas zface normals zdihedral angles zinscribed/circumscribed Reconstruction of Convex Polyhedra Cauchys Theorem }

6. Cauchys Rigidity Theorem zIf two closed, convex polyhedra are combinatorially equivalent, with corresponding faces congruent, then the polyhedra are congruent; zin particular, the dihedral angles at each edge are the same. Global rigidity == unique realization

7. Same facial structure, noncongruent polyhedra

8. Spherical polygon

9. Sign Labels: {+,-,0} zCompare spherical polygons Q to Q zMark vertices according to dihedral angles: {+,-,0}. Lemma: The total number of alternations in sign around the boundary of Q is 4.

10. The spherical polygon opens. (a) Zero sign alternations; (b) Two sign alts.

11. Sign changes Euler Theorem Contradiction Lemma 4 V

12. Flexing top of regular octahedron