  We prove a novel method for the embedding of a 3-fold rotationally symmetric
sphere-type mesh onto a subset of the plane with 3-fold rotational symmetry.
The embedding is free-boundary with the only additional constraint on the image
set is that its translations tile the plane, in turn this forces the angles at
the embedding of the branch points in the construction. These parameterizations
are optimal with respect to the Dirichlet energy functional defined on
simplicial complexes. Since the parameterization is over a fixed area domain,
it is conformal (i.e. a minimizer of the LSCM energy). The embedding is done by
a novel construction of a torus from 63 copies of the original sphere. As a
foundation for this result we first prove the optimality of the embedding of
disk-type meshes onto special types of triangles in the plane, and rectangles.
The embedding of the 3-fold symmetric torus is full rank and so cannot be
reduced by simpler constructions. 3-fold symmetric surfaces appear in nature,
for example the surface of the 3-fold symmetric proteins PIEZO1 and PIEZO2
which are an important target of current studies.
