At first, you might think this is obvious – just “clip” off each corner of the tetrahedron to create the truncated tetrahedron (by essentially creating a triangle from each of these clipped corners – see below for the associated graph). Then just map each such triangle to the corresponding vertex of the tetrahedron. No, it’s not obvious because the map just described is not a covering. This post describes one way to think about how to construct any covering.

First, color the vertices of the tetrahedron in some way.

The coloring below corresponds to a harmonic morphism :

All others are obtained from this by permuting the colors. They are all covers of – with no vertical multiplicities and all horizontal multiplicities equal to 1. These 24 harmonic morphisms of are all coverings and there are no other harmonic morphisms.