Deformations of Polyhedra and Polygons by the Unitary Group

Understand

We introduce the set of framed convex polyhedra with N faces as the symplectic quotient C^2N//SU(2).

• A framed polyhedron is then parametrized by N spinors living in C^2 satisfying suitable closure constraints and defines a usual convex polyhedron plus a phase for each face.
• We show that there is an action of the unitary group U(N) on this phase space, which changes the shape of faces and allows to map any polyhedron onto any other with the same total area.
• This realizes the isomorphism of the space of framed polyhedra with the Grassmannian space U(N)/SU(2)*U(N-2).