I recently visited a friend in a neighbouring city, but wasn't sure where he lived so asked him for directions. He asked where I would be entering the city. I wrote back to say that I wasn't sure, as I didn't know the city at all. His reply quite shocked me. Instead of a lot of different directions that depended on where I was coming from, he just sent me a map of the city and one set of directions. “Don't worry,” he wrote. “If you follow these instructions, wherever you are in the city you'll eventually arrive at my house.”

I couldn't quite believe that one set of instructions was going to be enough. I could be anywhere in the city. How would taking the second right followed by the first left work wherever I was? But when I opened up the package with the map and instructions, I realised that these weren't the usual directions that you hear the lady on the GPS blurting out. My friend, after all, is a mathematician.

Mathematicians like to throw away unnecessary information and get to the bare bones of a problem. The city consists of lots of roads meeting at various junctions. The distances between each of the junctions are unimportant. Once I have started driving down a road, I will end up at the next junction and need a new instruction. So the map just consisted of lots of dots, the junctions, and lines emanating from these dots, representing the roads meeting at this junction.

In fact, anyone travelling around London or any other major city with an underground system will be quite used to this sort of map. A physical map showing the geographic locations and routes on the London Underground is not a very helpful picture for negotiating your way around the city. Instead Harry Beck's iconic map of the London Underground, which he developed in 1933, isolates the way in which the network is connected, while ignoring physical dimensions. The fact that the same length of line is used to represent the connection between Covent Garden and Leicester Square as that between Finsbury Park and Seven Sisters does not mean that the actual distances between them are the same. For a commuter, knowing there is such a connection is much more important than knowing the distance between stations.

This is an example of a new way of looking at the world that was introduced in 20th-century mathematics. The word geometry comes from the Greek for measuring the Earth. But often the exact distances between objects are not important. It is how they are connected that is key to the identity of the shape. This new way of looking at the world is called topology.

Each road in the map that I had was represented by two lines with arrows on: one line represented the side of the road on which traffic was oncoming, the other line was the lane going in the opposite direction. If the road was one-way, there was only one line with an arrow indicating which direction to travel.

Then came the really clever bit. My friend had indicated each of these lines in different colours. The instructions consisted of the following directions: first take a blue road; at the next junction take a red road; at the next junction take a blue road. Remember, I can go up a road only if the direction allows me. Amazingly, wherever I start in the city, if I obey these instructions the journey will end outside my friend's house.

Whether such instructions were possible for certain types of maps was a subject of conjecture until last year, when a 63-year-old former security guard in Israel came up with an elegant proof. Avraham Trakhtman gives the lie to the belief that maths is only for the young.

Of course, not only cities contain networks of roads. The internet too can be considered a huge network of interconnected hubs and links. Just imagine an e-mail that has been lost in the system and can't find its recipient. Thanks to Trakhtman's theorem, wherever the e-mail is in the system, one set of instructions could steer it through the network so that it would arrive at its intended destination.

So you stick with your GPS or your A-Z if you want to. Me, I'm getting out my coloured pens.

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