THAT'S MATHS:The London Underground map is a paragon of design excellence. If you know where you are and where you wish to go to, it shows you how to get there. Yet, as a map of London it is inaccurate in almost all respects. The beauty of Harry Beck's design, published in 1933, is that the key information is kept while everything else is stripped away.
The Tube map is what mathematicians call a graph. The stations are the vertices and the train lines joining them are the edges. Interchanges are shown where different lines connect. Distances and directions are distorted in the interests of clarity and simplicity. One of the earliest such graphs was drawn by the renowned Swiss mathematician Leonard Euler. Euler solved a puzzle called “the seven bridges of Königsberg” by drastically simplifying a map of that city. This made it clear that it is impossible to find a route crossing all seven bridges without re-crossing any of them.
Graph theory is a branch of topology, the area of mathematics dealing with continuity and connectivity. Topology is concerned with properties that remain unchanged under continuous deformations, such as stretching or bending but not cutting or gluing.
Topology is often called rubber-sheet geometry. If a figure such as a triangle is drawn on a sheet of rubber, certain things change but others remain unaltered as the sheet is stretched. For example, the lengths of the sides are changed but points inside the figure remain inside and points outside remain outside.
In three dimensions, a cube made of Plasticine may be distorted continuously into a ball without tearing it, so a cube and a ball are topologically equivalent. In contrast, to make a bagel, or a doughnut with a hole, a ball of Plasticene must be torn at some point. So a ball and a bagel are not equivalent.
The formal way of showing that two sets are topologically equivalent is to establish a correspondence or mapping between the two sets such that nearby points in one are mapped to nearby points in the other. If such a correspondence – called a homeomorphism – exists, the two sets are topologically equivalent.
In the familiar school geometry of Euclid we have straight lines, fixed distances between points and rigid shapes such as triangles. Since topological deformations sacrifice all of these, is there anything useful left? Yes: while the Tube map distorts distances, it preserves the order of stations and the connections between lines, so the traveller knows where to get on and off and where to change trains. It is this topological information that is critical; precise distances are of secondary importance.
The Tube map might be “corrected” by drawing it on a sheet of rubber and delicately stretching it in places, gradually but continuously, until the stations were all in the correct positions. Or it might be further distorted until the Circle line became a true circle. But the remarkable success and longevity of the map proves Beck got it just right all those years ago.
Peter Lynch is professor of meteorology at University College Dublin. He blogs at thatsmaths.com
