One of the things I’ve been working on this week is : Moduli Space of Shapes of a Tetrahedron with Faces of Equal Area.
This follows on the work done in the posts:
- Numerical work with sagemath 15: Holomorphic factorization
- Holomorphic Factorization for a Quantum Tetrahedron by Freidel, Krasnov and Livine
The space of shapes of a tetrahedron with fixed face areas is naturally a symplectic manifold of real dimension two. This symplectic manifold turns out to be a Kahler manifold and can be parametrized by a single complex coordinate Z given by the cross ratio of four complex numbers obtained by stereographically projecting the unit face normals onto the complex plane.
This post shows how this works in the simplest case of a tetrahedron T whose four face areas are equal. For convenience, the cross-ratio coordinate Z is shifted and rescaled to z=(2Z-1)/Sqrt so that the regular tetrahedron corresponds to z=i, in which case the upper half-plane is mapped conformally into the unit disc w=(i-z)/(i+z).The equi-area tetrahedron T is then drawn as a function of the unit disc coordinate w.