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?

Complex Numbers for the Wary

Complex numbers have a terrible name. They're not complicated but the name puts people off.

They're actually simple, the name is an accident of history and if anything refers to the fact that they are made of up 2 independent parts. They would have been better called composite numbers but that name is taken!

This blog entry is a gentle and interesting introduction to complex numbers

I like it because it introduces ideas by relating them to things we are all familiar and comfortable with. And like the best tutorials it makes excellent use of visual explanations.

In particular I like his gentle introduction to arithmetic as geometric transformations. Multiply by 3 means magnify by 3. Multiply by -1 means flip across the origin point on the number line. Multiply by -3 means do both scaling and flipping.

It's natural then to ask which transformation turns 1 into -1 when applied twice. A 90 degree rotation counterclockwise in a two dimensional plane works, and is in fact a great way to think of the complex i operator. Suddenly i2 = -1 makes so much more sense!

What a great introduction to the otherwise strange i, and a great encouragement to think about apparently innocent looking mathematics as geometrical transformations.

Sunday, 5 January 2014

Make Your Own Mandelbrot

This blog accompanies the Make Your Own Mandelbrot guide.

You can suggest new ideas, corrections and requests to explain some of the mathematical concepts in more detail or in different ways.