Friday, 31 January 2014

3D Complex Numbers - Explaining why Not

One of the things we do in part 3 of the book is to see if we can extend the 2-dimensional Mandelbrot set into a 3-dimensional object.

We don't want to simply lift it up by using the colour values as a height field. We want to see if the same iterative function z2 + c can work on some kind of 3-dimensional extension of 2-dimensional complex numbers.

Apparently the great mathematician Hilbert tried to this for years and failed. He had to settle for 4-dimensional quaternions as the next step up from complex numbers.

My challenge is to try to explain why 3-d complex numbers aren't possible in a way that makes sense readers who are not specialist professional mathematicians.

Any ideas?

If we remember that the complex i can be thought of as a geometrical transformation, rotating the unit (1+0i) counterclockwise by 90 degrees up to the (0+1i). Do it again and you get to (-1+0i) or simply -1. Why not just have a new dimension and refer to it as the j-axis, and have a similar rotation which takes the unit (1+0i+0j) to (0+0i+1j)?

This seems attractive. We've created a new kind of number with three dimensions, real, i and j. We've also succeeded in having an analogous behaviour for j just as we did for i. Importantly, we haven't created an inconsistency because the i and j dimensions are independent.

The rules for addition and subtraction are easy because the real, i and j parts are separate and don't combine. Multiplication will throw up i2, j2, ij and ji terms. The i2 and j2 can be replaced by -1 because we defined the rotational effect of these operators that way.

We don't assume that ij and ji are the same. They might be but we'll let the working out show it one way or the other. Let's see the effect of ij on the real unit 1. The i rotates 1 to (0+1i) then the j rotation doesn't have any further effect - because turning a vector pointing to (0+1i) in the same way that j turns 1 to (0+1j) simply turns it on its own axis. So ij=i, and similarly ji=j.

So what's wrong with all this? Does it matter that there is not way to rotate a point around the i-j plane? Does it matter that we have a consistent but incomplete extension of complex arithmetic from 2 to 3 dimensions?