Fermat point in three-space
Robert H. Lewis
lewis at bway.net
Thu Jul 11 15:12:55 PDT 2002
Hello,
The Fermat point of a triangle ABC is the point, say P, that
minimizes the sum of the distances d(z,A) + d(z,B) + d(z,C) where z is
any point in the plane. There was a good article on this in a recent
(May, I think) issue of the American Mathematical Monthly. If all the
angles in ABC are less than 120 degrees, P is inside ABC. Otherwise
it's one of the vertices.
I couldn't find there any reference to the generalization to
3-space. It seems there are two generalizations: Given a tetrahedron
ABCD,
(1) find the point that minimizes the sum d(z,A) + d(z,B) + d(z,C) +
d(z,D).
(2) find the point that minimizes the sum of the areas of the
triangles formed by z and the six edges of ABCD (assume z is inside ABCD).
Searching the web, I found only a comment (no proof) that soap films
assume the minimizing form on a wire frame tetrahedron. That would be
problem (2).
Any information on this problem would be of interest.
Robert H. Lewis
Mathematics
Fordham University
Bronx NY
rlewis at fordham.edu
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