Minimal Networks - Steiner's Problem Resize


Steiner's Problem

This problem was named the Steiner problem for the first time in 1941, in the popular book What is mathematics?, by R. Courant and H. Robbins.

However, long before Steiner, other mathematicians like Fermat (1601 - 1665), Torricelli, Cavalieri and Simpson had approached the first variants of this problem.

It all started with a simple problem proposed by Fermat:

"To find a point in the plane such that the sum of the distances to three given points \(A\), \(B\) and \(C\) is minimum."

Torricelli found a solution to this problem in the case in which the internal angles of the triangle formed by the three given points \(A\), \(B\) and \(C\) are all smaller than or equal to 120º.

TORRICELLI's METHOD:

On the three sides of the triangle \([ABC]\), construct three equilateral triangles outside of \([ABC]\).
Draw the circles that circumscribe each of these three triangles.
The point where these three circles intersect is called Torricelli point and it solves Fermat's problem.

When one of the inner angles of the triangle \([ABC]\), for instance the angle corresponding to vertex \(A\), is greater than or equal to 120º, the solution to Fermat's problem is unique and coincides with point \(A\). (Note that in such cases the Torricelli point is outside the triangle \([ABC]\) and thus it cannot be a solution to Fermat's problem.)

Cavalieri found an important property of the Torricelli point, which was published in his book Exercitationes Geometricae in 1647:

When the Torricelli point is inside the triangle, the angles between the segments that link the Torricelli point and the vertices \(A\), \(B\) and \(C\) are all equal to 120º.

Later, Simpson found another way of constructing the Torricelli point and published it in his book Doctrine and Application of Fluctions in 1750.

SIMPSON's METHOD:

On the three sides of the triangle \([ABC]\) construct three equilateral triangles outside of \([ABC]\).
Draw the three line segments that link each vertex of the triangle \([ABC]\) and the opposite vertex of the equilateral triangle constructed in the opposite side. These line segments are called Simpson segments.
The three Simpson segments intersect in a point that coincides with the Torricelli point and which solves Fermat's problem.

Also this method only solves Fermat's problem when the three internal angles of the triangle \([ABC]\) are smaller than or equal to 120º.

Another important property, mentioned by Heinen in 1834, is the following:

The lengths of the Simpson segments are all equal to the sum of the distances between the Torricelli point and the vertices \(A\), \(B\) and \(C\).

Steiner was attracted to this problem while trying to solve a Fermat's problem generalization exercise, proposed by Simpson in his book Fluxions:

"To find a point in the plane (or in the d-dimensional Euclidian space) such that the sum of the distances to \(n\) given points \(A1, ... , An\) is minimum."

Another variant of the problem is the proposal by Jarnik and Kössler in 1934:

"Finding the shortest network that links \(n\) points in the plane."

In this section you can find pages with sketches about this problem.