# Fermat point in three-space

Anglister, Shlomo shlomo.anglister at intel.com
Sun Jul 14 09:34:23 PDT 2002

```Hi,

What you refer to as a Fermat point in space, can be called a "Steiner point
in space".
There is an abundance of data in this area.
Soap bubbles is an old one, Bell used it many years ago to connect US major
cities.

Shlomo

-----Original Message-----
From: Robert H. Lewis [mailto:lewis at bway.net]
Sent: Thursday, July 11, 2002 9:13 PM
To: compgeom-discuss at research.bell-labs.com
Subject: Fermat point in three-space

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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