Louvres, holes and algebraic topology

Autosport's technical triumvirate of Mark Hughes, Gary Anderson and Giorgio Piola, have spotted a beautiful touch on the engine cover McLaren were intending to introduce with their new exhaust-blown diffuser at last week's British Grand Prix. Whilst the teams have in recent years been forced, by regulation, to substitute single exit orifices in place of radiator exit louvres, McLaren have cleverly realised that if they cut the bodywork between each louvre in half, then the result is topologically identical to a single hole.

To recall, topology is the mathematical study of the connectedness and continuity of shapes and surfaces, irrespective of their geometry. Thus, the surface of a tea-cup is often said to be topologically identical to the surface of a doughnut, and the London Underground Map is said to preserve the topology of the capital city's subterranean transportation network, if not the actual length and shape of the tracks.

Now, the number of holes in a shape or surface M is typically characterised by an object from algebraic topology called the fundamental group π₁(M). Algebraic topology is essentially the use of groups to characterise the topological characteristics of shapes and surfaces. Recall that a group is a set of elements which is equipped with a binary product operation, a unary operation called the inverse, and a special element called the identity element.

To understand what the fundamental group is, we need another concept, called homotopy equivalence. Basically, two curves or loops are said to be homotopically equivalent if one can be continuously deformed into another. If a pair of curves cannot be deformed into each other, then they belong to different homotopy equivalence classes.

Now, if we consider the set of all loops beginning and ending at the same point p in a shape or surface, then we can tag one loop onto the end of another to form a new loop. This concatenation operation gives us a product operation between different homotopy equivalence classes of loops at a point. Furthermore, by simply running around a curve in the opposite direction, we have an inverse operation, and the degenerate loop consisting of the point p, serves as the identity element e of a group. The homotopy equivalence classes of loops at a point can thus be treated as a group, and this group is called the fundamental group π₁(M). (If the shape or surface is connected, it can be shown that the fundamental group at each point is isomorphic, hence the point chosen is arbitrary).

If a shape or surface has no holes in it, then all loops through an arbitrary point p can be continuously shrunk down to the point itself, hence the fundamental group consists of a single element π₁(M) = {e}. However, if the shape or surface has a single hole, then there will be at least two homotopy classes of loops through a point: those which can be deformed down to the point, and those which cannot, because they circle the hole, and cannot be shrunk any smaller than the hole. In this case, the fundamental group consists of at least two elements. If there are two holes, then the loops around both holes cannot be shrunk to a loop around one hole, and the loops around one hole cannot be shrunk to a point, hence the fundamental group will contain at least three elements.

In the case of a radiator exit with n louvres, the fundamental group of the surface will contain at least n+1 elements. By cutting through the bodywork between the louvres, however, it becomes impossible to form a loop around anything but the entire louvre collection. Thus, topologically speaking, there is only a single exit orifice. Ingenious.