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Topic: A Theorem concerning the Trisectors of a Triangle
Replies: 24   Last Post: Nov 10, 1998 12:19 AM

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 John Conway Posts: 2,238 Registered: 12/3/04
Re: A Theorem concerning the Trisectors of a Triangle
Posted: Sep 17, 1998 12:30 PM

On Thu, 17 Sep 1998, Russell Towle wrote:
>
> 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.

JHC

Date Subject Author
9/12/98 Den Roussel
9/12/98 John Conway
9/13/98 Larry Cusick
10/27/98 John Conway
9/13/98 steve sigur
9/14/98 John Conway
9/15/98 John Conway
9/15/98 Richard Guy
9/15/98 John Conway
9/15/98 Richard Guy
9/15/98 Richard Guy
9/16/98 Floor van Lamoen
9/16/98 John Conway
9/17/98 Floor van Lamoen
9/17/98 John Conway
9/17/98 Floor van Lamoen
9/17/98 Russell Towle
9/17/98 John Conway
9/17/98 Russell Towle
9/17/98 Douglas J. Zare
9/19/98 Russell Towle
9/20/98 John Conway
9/20/98 John Conway
9/18/98 Antreas P. Hatzipolakis
11/10/98 Den Roussel