That’s Maths in this week’s Irish Times ( TM013 ) is about the branch of mathematics called topology, and treats the map of the London Underground network as a topological map.
Topology is the area of mathematics dealing with basic properties of space, such as continuity and connectivity. It is a powerful unifying framework for mathematics. Topology is concerned with properties that remain unchanged under continuous deformations, such as stretching or bending but not cutting or gluing.
The Tube Map
Graph theory is a branch of topology, and the London Underground 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. The Tube Map is a splendid example of excellent design. If you know where you are and where you wish to go to, it shows you how to get there, and where to change trains if necessary.
The formal way of showing that two sets are topologically equivalent is to establish a homeomorphism between them. We will define homeomorphism below but, for now, it is a correspondence between pairs of points, one in each of two sets, such that nearby points are mapped to nearby points.
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 homeomorphic. In contrast, to make a bagel, or a doughnut with a hole, a ball of Plasticine must be torn at some point. So, a ball and a bagel are not topologically equivalent. This leads to the cheesy joke: a topologist is a mathematician who doesn’t know the difference between a doughnut and a coffee cup.
In topology we treat two sets X and Y as equivalent if there is a homeomorphism between them. Technically, a homeomorphism is a bicontinuous bijection. What does this mean? A bijection is a map that is one-to one and onto. That is, every point in each of the two sets corresponds to precisely one unique point in the other set.
Bijections are invertible; we can define the reverse map from Y to X. Bicontinuous means that both the map and its inverse are continuous: nearby points in one set correspond to nearby points in the other.
Homeomorphisms allow continuous deformations, such as stretching or bending but not cutting or gluing. Topology is concerned with properties that are preserved under such continuous deformations. It has many branches.
Point-set Topology studies the inherent properties of spaces, line compactness and connectedness. Algebraic Topology uses constructs like homotopy and homotopy groups, first introduced by Poincaré. Graph Theory examines networks of connections between nodes. And Knot Theory looks at the entanglement of structures embedded in higher dimensions.
In the familiar Euclidian geometry learnt at school, we have straight lines, fixed distances between points and rigid shapes like triangles. Since the deformations of a homeomorphism sacrifice all 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. 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 that the designer of the map, Harry Beck, got it just right all those year ago.