Subway networks typically serve a single city, mainly to ferry commuters in and out of the city centre from the suburbs.

For example, the London Underground consists of several lines travelling straight across the city: a train will travel from a suburb on one side, through the city centre, and then to another suburb on the opposite side, before heading back the way it came. There's also a circle line connecting up stations inside the city.

The Glasgow subway consists of a single circular route, with one line heading clockwise and another going anti-clockwise.

When updating the London Undergound map in 1931, Harry Beck realised that the exact geographical position of stations wasn't particularly important -- the most important information was which stations were connected to each other. By moving the stations on the map and straightening out the railway lines, it became much easier to see how to get from one place to another.

Thinking about networks in this way is what topology is all about: if you don't mind moving things about and stretching or bending lines, what properties about the network remain constant?

It turns out that in every subway system in the world except one, the routes passengers travel on are all either straight lines or circles.

A straight line route consists of two terminus stations between which a train shuttles back and forth. On a single round trip, a train will pass through each station twice: once heading outwards and then again when heading back to the start.

On a circle line, the train travels continuously in the same direction, passing through each station once on each round trip.

The Tyne and Wear Metro is unique, however. If you get on a train at St James station, here's what happens:

1. You pass through Monument station heading East towards the coast.
2. After going up the coast, the line comes back down through Gosforth and Jesmond, passing through Monument again heading South towards South Shields
3. Once it gets to South Shields, the train heads back the way it came. You next pass through Monument heading North to go back around the coast route.
4. Finally, after travelling down the coast, the train heads through Monument for a fourth and final time heading West towards St James.

So on a single trip and without changing trains, you can travel through the same station four times heading in four different directions before you get back to where you started!