Holes in the F1 regulations

Once again, Formula One has tied itself into something of a knot over the question of holes. As with the double-diffuser controversy of 2009, the conflict concerns the very definition of a hole. This time, the argument pertains to the slot in the step-plane which Red Bull have incorporated in front of the rear wheel. As Craig Scarborough explains, this arguably contravenes the following regulation:

"All parts lying on the reference and step planes, in addition to the transition between the two planes, must produce uniform, solid, hard, continuous, rigid (no degree of freedom in relation to the body/chassis unit), impervious surfaces under all circumstances."

Teams such as Sauber have achieved much the same effect as Red Bull's slot, but do so by deforming the boundary of the step plane, not by creating a hole. In this respect, there is a very clear topological definition of a hole, which goes as follows:

A surface possesses a hole if there is loop through any point on the surface which cannot be continuously shrunk to a point.

If you draw a loop around a hole, then the hole prevents the loop from being shrunk to a point. This is a concept from algebraic topology, and in technical terms one can say that a surface has a hole (or holes) if the zeroth homotopy group (the 'fundamental group') is non-trivial, i.e., contains more than the identity element. A surface with a trivial fundamental group is said to be simply connected.

The slot on the Red Bull clearly prevents a loop around it from being shrunk to a point, while any loop drawn on the step-plane of the Sauber can still be shrunk to a point.

However, the actual regulation fails to provide such a clear, topological definition of a hole. In fact, it draws upon a variety of quite different concepts. The step-plane must be "uniform" (a statistical concept); "solid" (a tricky concept in physics, sometimes distinguished from a fluid by the symmetry group of the medium, or by the constitutive relationship of the medium, i.e., the expression relating stress to strain); "hard" (a concept usually defined in terms of the local elastic modulus of the material, or its resistance to indentation tests); "continuous" (a topological concept, but one which is ultimately an idealisation, given that all materials are composed of discrete particles); "rigid" (usually meaning a high elastic modulus, but here qualified by "no degree of freedom in relation to the body/chassis", so actually meaning "rigidly attached"; and "impervious", a concept pertaining to the transmissibility of other media through a material.

None of these concepts actually capture the intended requirement for the absence of holes. In particular, in topological terms, a surface with holes can be perfectly continuous. Continuous surfaces can be simply connected (without holes) or multiply connected (with holes).

As Craig Scarborough also points out, there is another regulation intended to prohibit discontinuities of all types, in all but the outer 50mm of the floor:

“In an area lying 650mm or less from the car centre line, and from 450mm forward of the rear face of the cockpit entry template to 350mm forward of the rear wheel centre line, any intersection of any bodywork visible from beneath the car with a lateral or longitudinal vertical plane should form one continuous line which is visible from beneath the car.”

Whilst all the teams, Red Bull included, seem to accept that this regulation prohibits the existence of discontinuities in all but the outer 50mm of the floor, one could argue that this is still open to interpretation. If an aperture is placed on a lateral vertical plane, created by a raised lip in the step-plane, so that the hole is not visible from below, then the intersection of a longitudinal vertical plane will still trace out a continuous line when viewed from beneath the car (i.e., when projected down onto a horizontal plane).

In conclusion, rather than adopting such a scatter-gun approach to the regulations, it might be better if the regulations defined a hole in topological terms, and then derived the empirically applicable criteria from the general definition.