The Coffee Cup and the Doughnut

Königsberg sat on both banks of the Pregel, with two islands, connected by seven bridges. The question, apparently a genuine local pastime, was whether you could walk a route crossing each bridge exactly once.

Euler answered it in 1736, and his answer is remarkable mostly for what it throws away. The lengths of the bridges are irrelevant. So is the shape of the islands, the width of the river, the distance between anything and anything. The only thing that matters is which landmass connects to which.

Reduce the city to four dots and seven lines. Now count the lines at each dot. Every time your walk passes through a landmass it uses one bridge to arrive and one to leave, so the bridges at that landmass pair up and the count must be even. The only exceptions are where you start and where you finish.

In Königsberg all four landmasses have an odd number of bridges. You are allowed two odd ones at most. The walk is impossible, and no amount of ingenuity will produce it.

This is usually called the first paper in topology, and also the first in graph theory, and it demonstrates the move that defines the subject: discard everything metric and see what is left. What is left is connection.

Topology studies the properties that survive continuous deformation, stretching, bending, twisting, but never tearing or gluing. Lengths go. Angles go. Areas go. What remains is whether a shape is in one piece, how many holes it has, whether a loop can be shrunk to a point, whether a knot is genuinely knotted.

Hence the standard joke, which is exact rather than funny. A topologist cannot tell a coffee cup from a doughnut, because each is a solid lump with one hole through it, and one can be deformed into the other by pushing the material around without cutting.

Proving two shapes are the same is easy in principle: exhibit the deformation. Proving they differ is harder, because no failed attempt shows that all attempts must fail. What you need is an invariant (a quantity that cannot change under deformation) and then you compute it for both.

The oldest one is Euler's. For a polyhedron, count vertices, subtract edges, add faces. A cube: 8 − 12 + 6 = 2. A tetrahedron: 4 − 6 + 4 = 2. An icosahedron: 12 − 30 + 20 = 2. The answer is always 2 for anything deformable into a sphere, and 0 for anything deformable into a doughnut. So a sphere and a doughnut are not the same shape, and now you have a proof rather than an intuition.

Most of modern algebraic topology is the manufacture of better invariants for harder cases, which is why a subject founded on a walking puzzle turned into machinery.

The everyday appearance of topological thinking is the transit map. Harry Beck was an engineering draughtsman at the Underground, and in 1931 he drew the London network as a circuit diagram: straight lines, angles at 45 and 90 degrees, stations spaced evenly, the central area enlarged and the suburbs compressed. Every distance is wrong. Most directions are wrong. The connections and the order of stations are exact.

The publicity department thought it too radical and turned it down. It was printed as a trial in 1933, and passengers took it immediately, because the map answers the question they actually have (where do I change, and how many stops) and discards the one they do not, which is how far apart the platforms really are. Nearly every metro system in the world now uses the idea.

One more result, because it makes the flavour of topological conclusions clear. The hairy ball theorem says any continuous tangent field on a sphere must vanish somewhere: you cannot comb a hairy ball flat without leaving a cowlick.

Horizontal wind at the surface of the earth is a continuous tangent field on a sphere. So at every moment there is at least one point on the planet where the horizontal wind is exactly zero.

That is a fact about the weather proved without any meteorology, and it comes with the characteristic limitation of the method. Where is the calm point? The theorem does not say. How many are there? At least one. Topology is extremely strong about existence and almost entirely silent about location, which is both its power and the reason it needs the rest of mathematics.