```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, beinglargely ignorant of all these triangle centers, which seem to form such arich subject.For a meaningful solid-angle quadrisection, all I can imagine at thispoint, is, fixing one's attention upon a single vertex of a tetrahedron,let the three faces surrounding be bisected at that vertex. Then the threelines of bisection, with the three edges of the tetrahedron, define foursolid angles meeting at the original vertex: three on "the exterior," and asingle one interior.Completing the construction, one obtains a dissection of the originaltetrahedron into 11 tetrahedra, one on each of the six original edges, fourhidden 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? Intercepteach tetrahedral region by a sphere on the common vertex, and give up on"bisecting" the angles of the original tetrahedron, allowing the divisionsto fluctuate until the volumes of the tetrahedral regions meeting at avertex are equal, or ...?--Russell TowleRussell TowleGiant Gap Press:  books on California history, digital topographic mapsP.O. Box 141Dutch Flat, California 95714------------------------------Voice:  (916) 389-2872e-mail:  rustybel@foothill.net------------------------------
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