On Thu, 17 Sep 1998, Russell Towle wrote: > > John, what intrigues me about this, is, do such patterns operate in higher > spaces? Do we see such behavior in tetrahedra, for instance, with a regular > Platonic tetrahedron arising from the trisection or quadrisection of a > solid angle?
Let me rather obliquely say that I have long wondered whether the original Morley theorem is itself in some sense 3-dimensional, since its figure is topologically a Schlegel diagram of an octahedron.
But as to a version for tetrahedra, I can't think of any reasonable type of solid-angle quadrisection that could even take part in a meaningful statement, let alone a true one! [Let me say that although lots of triangle-geometry does admit extensions to tetrahedra, there's lots that doesn't even among the very simple stuff - for instance the general tetrahedron doesn't have an orthocenter.]
Your question has suddenly produced an interesting thought - maybe the 3D version involves a Schlegel diagram for the orthoplex (my preferred name for the cross-polytope) in which the vertex figures at the vertices of the outer tetrahedron are versions of the Morley figure? I don't think it can work for a Euclidean tetrahedron, because there doesn't seem to be a spherical version of Morley; but it could conceivably work for an ideal tetrahedron in hyperbolic space.