
Re: A Theorem concerning the Trisectors of a Triangle
Posted:
Sep 17, 1998 12:30 PM


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 3dimensional, 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 solidangle quadrisection that could even take part in a meaningful statement, let alone a true one! [Let me say that although lots of trianglegeometry 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 crosspolytope) 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.
JHC

