Date: Sep 17, 1998 1:59 PM
Author: Russell Towle
Subject: Re: A Theorem concerning the Trisectors of a Triangle

John Conway wrote,

> 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.]

Hmmm. Well, I was shooting from the hip, really all I'm capable of, being
largely ignorant of all these triangle centers, which seem to form such a
rich subject.

For a meaningful solid-angle quadrisection, all I can imagine at this
point, is, fixing one's attention upon a single vertex of a tetrahedron,
let the three faces surrounding be bisected at that vertex. Then the three
lines of bisection, with the three edges of the tetrahedron, define four
solid angles meeting at the original vertex: three on "the exterior," and a
single one interior.

Completing the construction, one obtains a dissection of the original
tetrahedron into 11 tetrahedra, one on each of the six original edges, four
hidden but touching each original vertex, and one central tetrahedron,
touching the centroid of each face. Four solid angles meet at each vertex.
What might one have to do, to make the four solid angles equal? Intercept
each tetrahedral region by a sphere on the common vertex, and give up on
"bisecting" the angles of the original tetrahedron, allowing the divisions
to fluctuate until the volumes of the tetrahedral regions meeting at a
vertex are equal, or ...?

--Russell Towle

Russell Towle
Giant Gap Press: books on California history, digital topographic maps
P.O. Box 141
Dutch Flat, California 95714
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