Maps on the platonic solids have been known for a long time. They can be equal area or conformal. The gnomonic projection is particularly easy to do on the surface of these solids.
Apparently they have never been done on the surface of a pyramid. The next illustration is a special case of an equal area projection having N pointed triangular protrusions on an N sided base. For three lobes the base is a triangle. With four lobes we get the pyramid. For six lobes the base is a hexagon, etc. All can be drawn with one computer program, with N as a parameter. Conformal versions are also possible.