A question left hanging Why not a three-dimensional algebra?

One question has been left hanging: we live in a three-dimensional space, so why is 3 absent from the list of the four algebras? It is not idle curiosity. Hamilton spent years looking for numbers made of three components, and gave up only when he tried with four.

The answer comes from a theorem with a picturesque name, the hairy ball theorem (L. E. J. Brouwer, 1912), which does not speak of numbers at all: it speaks of hairs on a sphere. Let us first see what it says, then what it has to do with us.

We plant a hair at every point of the surface of a sphere and ask three things: that every hair lie flat, that is, point in a direction tangent to the surface; that in passing from a point to its neighbours the direction change continuously, without jumps; and that at no point the hair be missing. What we are describing is what mathematicians call a tangent vector field, continuous and nowhere zero; we shall say that the sphere is combed. The theorem states that on the sphere no such combing exists: however the hairs are laid out, at least one bald point remains, a tonsure.

We do not give the proof, but the result can be sensed. Imagine combing the sphere along the meridians, all the hairs from the north pole towards the south pole: the combing succeeds everywhere except at the two poles, where the meridians converge and there is no direction to choose. Two tonsures remain there. Move the hairs as we like and the tonsures move with them, but they do not disappear: they can be reduced to one, never to none.

The link with our question runs through the numbers of modulus 1. A number of an algebra of dimension 3 is made of three real numbers, so we can see it as a point of space, or as the vector running from the origin to that point: from here on we shall use the two words for the same thing, and the multiplication of the algebra applies to these vectors as to any other number. The numbers of modulus 1 are then the points at distance 1 from the origin, that is, the surface of an ordinary sphere, and they are the ones by which multiplying does not change sizes. Let us note finally that the sphere is centred at the origin, and that on such a sphere being tangent at a point means being perpendicular to that point.

The argument has this shape. Suppose the three-dimensional algebra exists. We shall show that its sphere can then be combed; but the theorem says that it cannot. The hypothesis leads to a false conclusion, so it is the hypothesis that is false: that algebra does not exist. What remains to be proved is the first step, and that is what we do now.

The point $1$ lies on the sphere. There we plant the first hair: we choose any tangent direction, that is, a non-zero vector $v$ perpendicular to $1$. It is the only choice we make by hand; all the other hairs will come from this one.

Now take any point of the sphere, that is, a number $u$ of modulus 1, and consider the operation that to every $x$ assigns $xu$. It is linear, because multiplication distributes over addition and real factors come out. And it does not change the moduli, because $|xu|=|x|\,|u|=|x|$: having taken the modulus equal to the distance from the origin, it preserves all distances. It is therefore a rigid transformation of space, and as such it preserves angles too. It carries the sphere into itself, and carries the point $1$ to the point $u$.

At the point $u$ let us then plant the hair $vu$, the product of $v$ by $u$, which is the image of $v$ under that transformation. It lies flat: $v$ is perpendicular to $1$ and angles are preserved, hence $vu$ is perpendicular to $u$. It is never missing: if $vu$ were zero it would have modulus zero, but $|vu|=|v|\neq 0$. And it changes continuously, because $vu$ depends on $u$ linearly. As $u$ runs over the whole sphere the hairs are all there, lying flat and continuous: the sphere is combed, with no tonsure.

But the sphere cannot be combed, and the hypothesis falls with it: the three-dimensional algebra does not exist. Hamilton was looking for something that was not there.

The argument runs one way only. On the circle the hairs are laid out without difficulty, all of them the same way round, but this does not prove that the algebra of dimension 2 exists: it says merely that in dimension 2 the obstacle is absent. That the complex numbers exist we know because we have them before us.

I must say, though, how far I myself get. Of this note we have proved one thing only: that if the algebra exists, its sphere can be combed. The hairy ball theorem we have taken as given, and the same holds for Hurwitz’s theorem, from which the table of the four algebras in the chapter comes. Behind both lies a vast body of mathematics: algebraic topology, the theory of fibre bundles, things one studies for years and that I do not know. I merely report the conclusions of those who built them: the practicable roads are four, and for the choice we have to make this is enough. I hope to be able to study them one day.