Two Theorems

The Hairy Dog Theorem

There are some quite hairy theorems in mathematics, not the least of which is the hairy dog theorem. This states that if you comb the hair on a dog, there must be at least one point where the hair radiates outwards leaving the skin visible. On a real dog, of course, this point is at its nose.

As with many mathematical theorems, you might think that this one is marginally interesting, but totally useless. But you would be very wrong. It is crucial to the design of fusion power generators.

In fusion reactors, the reacting plasma has to be kept at about 100 000 000 K, about 7 tiimes the temperature at the centre of the sun. At this sort of temperature it would instantly vaporize a material container. So magnetic containment is used. The container is a 3D space which has to have the magnetic field lines on its surface everywhere parallel to the surface. Being parallel to the surface stops the plasma from escaping.

But the hairy dog theorem tells us that such a container cannot be built: there must be at least one place where the magnetic field is zero or perpendicular to the surface. The plasma would leak out at this point (squirt out in fact).

However, this assumes that the hairy dog is topologically equivalent to a sphere and in fact the hairy dog theorem applies only to such shapes.

A donut has a hole through it and so is not topologically a sphere. Consequently, the hair on a donut-shaped dog can be combed without producing a bald spot – you go up through its hole and down round its outside. In the same way a donut-shaped magnetic fusion container can be made with the magnetic field everywhere parallel to its surface. So fusion containers are always toroidal or donut-shaped.

The Ham and Cheese Sandwich Theorem

An interesting mathematical theorem is the ham and cheese sandwich theorem.

The theorem states that if you make a ham and cheese sandwich for two people, it doesn’t matter how badly you make it, you will always be able to cut it into two pieces with one planar cut such that both pieces will have the same amount of bread, the same amount of ham and the same amount of cheese.

The bread pieces can be any shape and size, the cheese can be put anywhere, for instance all on one side of the sandwich and likewise with the ham – the ham could even be on the floor nowhere near the bread or in pieces all over the wall. You will still be able to find a plane that divides it fairly.

This theorem illustrates the mathematical concept of degrees of freedom. There are 3 degrees of freedom in specifying a plane in 3 dimensional space. There are 3 constraints in having to fairly divide the bread, the ham and the cheese. So, there is one way to choose a plane that does the job.