Topology on the Tube, London
Beck realised that in order to navigate the Tube it was most important for travellers to know what lines stations were on and how these lines were connected. Therefore he overruled the geography and instead placed stations roughly equally spaced on the lines, which he ran either horizontally, vertically or at 45 degrees. The result was that he shrank the map down to fit it into a much smaller space with a design that was much easier to read. By squeezing and stretching the tube lines Beck fitted this sprawling, twisting map into something you can fit in your pocket!
Just 500 copies of his map were trialled initially but it was so popular it became the standard way to depict the London Underground. It is now recognised as a design classic and the same principles are used for many other transport systems around the world.
Beck’s map is a great example of the importance of an area of maths called topology. Distances disappear in topology so that the size or shape of an object no longer matters – you can stretch or squeeze something and in topology’s eyes it remains unchanged. What is important is how things are connected; no cutting or tearing is allowed. A famous joke is that a coffee cup is topologically the same as a doughnut as you can smoothly deform one into the other!
A doughnut and a coffee cup might be topologically the same but you will never be able to turn either of these into a sphere. This is because a doughnut (know mathematically as a torus) has a hole, where as a sphere doesn't. Understanding which objects have holes is vital in topology as this affects how things are connected. In 1904 the French mathematician Henry Poincaré posed one of the most famous conjectures in mathematics – Poincaré stated that, topologically, the only shapes that have no holes are spheres.
This was might be obvious for objects in three dimensions but the question of whether Poincaré's conjecture was true for higher dimensions remained unanswered for nearly a century. Finally, in 2002, the Russian mathematician Grigori Perelman hit the headlines because not only did he prove that this characterisation was correct, he also refused all accolades for his incredible mathematical achievement.
From DNA to the Universe
Topology is vital in many areas. It has made huge contributions to biology where it helps to describe and understand how proteins, DNA and other molecules fold and twist. Cosmologists need topology to determine the shape of the Universe. And topology is vital in understanding the structure of graphs in network science, as we saw in our first stop on this tour.
But of course, sometimes on the ground, distances do matter. If we want to get from here, St Paul’s, to the Barbican, it's easy to see from the map of the London Underground that the shortest route is to jump on the red Central line to Bank, change to the black Northern line to Moorgate and change to the yellow Circle line to the Barbican. What isn't obvious, however, is that it's actually quicker to walk there and should only take you about 8 minutes! Topology is vital in understanding the overall structure of the Tube network, but sometimes a little local knowledge goes a long way!
Props: laminated pocket copy of tube map, laminated copy of geographically correct map, four sets of rope handcuffs.
This demonstration is a great way to introduce the concept of topology. Choose two pairs of people. For each pair (A and B), put person A’s hands in one set of cuffs and then person B’s hands in another set of cuffs so that their arms are interlinked. The two pairs of people are linked together in a similar (but importantly not the same!) way to a pair of linked rings. The aim is to see which pair can be first to separate themselves, without taking their hands out of their own cuffs or breaking the rope – no cutting or tearing is allowed in topology!
Let the two pairs try to extricate themselves, perhaps giving some clues. If no-one succeeds, draw the group together for the explanation.
The secret is to recognise that each person has a small gap between their wrists and the cuffs. If you imagine that person B is shrunk down to the size of a bangle encircling person A’s wrist (remember shrinking is allowed in topology) then that bangle can just be slipped off through the gap in the cuff. (You can see an excellent explanation at http://mathsbusking.com/shows/zeemans_ropes/.)
Topology is the secret to solving this problem, just as it was for Beck’s redesign of the Tube map – it doesn’t matter what size things are, what is important is how they are connected.
See this Site in a Tour
Geographically correct map of Zone 1 of the London Underground found at http://commons.wikimedia.org/wiki/File:London_Underground_Zone_1_with_street_map.jpg / CC BY-SA 3.0
Geographically correct map of the whole London Underground found at http://commons.wikimedia.org/wiki/File:London_Underground_Zone_1_with_street_map.jpg / CC BY-SA 3.0
Animated GIF of coffee cup and doughnut by Mike Pearson from Plus magazine, used with permission
Photograph of topology demonstration by Rob Judges / CC BY-SA 3.0