Vertices of a square?!
 

If the vertices of a square represent four townships and are all connected by a system of roads.

To keep costs to a minimum, what is the ideal arrangement of roads?

What insights are gained from the above to find similar cost effective systems of roadways for 5 and 6 towns i.e. those represented by the vertices of a regular pentagon and hexagon respectively.

Please help...

This is the Euclidean Steiner tree problem and there is a large research literature on it: see, for example, the Wikipedia article.

In general the shortest network is created by adding new vertices (Steiner points) and at these new points the network will have three edges meeting at 120 degree angles.

For a square it looks like

rgep you are a real star! Thanks :-)

Nat