I thought it was the case that quarternions were out of favour

That is true - each time you generalise numbers beyond imaginary numbers, you lose some useful mathematical properties.

There are certain familiar mathematical properties we had drilled into us at primary school like:

- a+0=a

- a*1=a

- a+b=b+a

- a*b=b*a (the "law" of commutivity)

- c*(a+b)=c*a+c*b
- They apply for real (1 dimensional) & imaginary (2-dimensional) numbers

- ...but they progressively lose validity as you move into quaternions (4-dimensional) and octonions (8 dimensional)

- There are many number systems used by mathematicians, but the thing I find freaky is that these are the
*only *4 mathematical spaces which behave in this way (you can't do similar mathematics in 3 dimensions, 5 dimensions or 15 dimensions!)

For example, multiplying by two complex numbers can be seen as doing a rotation in a 2-dimensional space (the complex plane). It doesn't particularly matter whether you rotate by θ

_{1} or θ

_{2} first.

However, multiplying by two quaternions can be seen as a rotation in 4-dimensional space. Here it matters greatly whether you rotate first by θ around the i axis or by φ around the j axis! So in this case, a*b ≠ b*a (ie commutivity fails in a 4-dimensional space).

Sometimes problems in the real world really

*do* need only 3, 5, 15 or 10000 dimensions, and these 4 spaces with their special mathematical properties don't apply. So mathematicians and physicists use a more general technique of

matrices. The properties of matrices are even less mathematically "nice" than quaternions. For example, in matrices, a*b may not even be defined, and even if it is, usually a*b ≠ b*a. But despite their shortcomings, matrices come in all shapes and sizes, and computers can do operations on them quite efficiently to produce computer games, weather simulations, etc.

But these higher dimensional numbers like quaternions are not

*totally *dead; matrices don't

*have *to contain real numbers - for example, matrices of imaginary numbers can be used for circuit simulation. I have even seen octonions used in theories about the fundamental nuclear particles. It has been suggested that some of the characteristics of 4 dimensional space or string theory are based on the characteristics of these higher-dimensional numbers.[/list]