# Networks of Minimal Length

#### MINIMUM NETWORKS What is the minimum length of a network spanning four coplanar points located at the vertices of a square? A network can be made shorter by adding other points to connect line segments. The examples at the right are networks for four coplanar points at the vertices of a square.  None of these patterns would produce a minimal network.

What would be a minimal network for four coplanar points located at the vertices of a square?

A Simpler Problem -- a network for three points

#### Extensions:

1. Four points at the vertices of a quadrilateral other than a square.

2. Four random points.

3. Five points on a convex pentagon.

4. Six points on a regular hexagon.

5. Six points in a 2 by 3 rectangular grid.

6. Nine points in a 3 by 3 rectangular grid.

7. Twelve points in a 3 by 4 rectangular grid.

8. Sixteen points in a 4 by 4 rectangular grid.

9. Twenty-five points in a 5 by 5 rectangular grid.

Conjectures - - you construct. Keep a list.

Soap Bubbles - - Try to figure . . .