The shortest road network joining four towns

Question

Four towns are located at the corners of a square of side 1 km. Roads are to be built so that it is possible to travel between any two towns along roads. Roads may meet at junctions anywhere, not just at towns.

Three sides of the square give a network of length 3. The two diagonals give a network of length 2 root 2, about 2.83.

What is the shortest possible total road length, and what does the network look like?

Accepted Answer

1 + root 3 km, about 2.732, using two junction points inside the square. The shape. Put the towns at (0, 0), (1, 0), (0, 1) and (1, 1). Place two junctions on the horizontal line through the centre of the square, at height 1/2, symmetric about the centre. Join the left junction to the two left-hand towns, the right junction to the two right-hand towns, and the two junctions to each other by a horizontal road. At each junction three roads meet at 120 degrees: the horizontal connector, and two town-roads that each make a 60-degree angle with it. The lengths. A town-road rises 1/2 (from a town a…

The shortest road network joining four towns

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