Find all Equiangular Platonic triangles

Question

A spherical triangle A is called equiangular if its 3 angles are equal. A is called Platonic if copies of A tile the unit sphere. I need to find all such triangles. Don't we have an infinite amount of them? Any input would be appreciated.

Answer 1

HINT

There are three Platonic solids that have equilateral triangles as faces: the tetrahedron, the octahedron, and the icosahedron. All these bodies can be circumscribed by spheres. On the circumscribing spheres we can naturally define the spherical triangles corresponding to the faces of the cirumscribed solids. There cannot be other regular coverings of the sphere because then there would be other Platonic bodies.

The task is to calculate the spherical angles of the generated spherical triangles.

Answer 2

As a warning to other potential posters this is a current homework problem:

http://homepages.warwick.ac.uk/~masgar/Teach/2015_MA243/examples6.pdf

Which coincides with the timing of your post. So any posters should refrain from posting full solutions as of yet.

HINT

To aid intuition, first consider the tessellation of the plane by some p-gons and try to derive an equation on the total angle sum about any point where the vertices of these p-gons meet. There are similarities between this and the tiling of the sphere.