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

**z**

^{2}+

**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

**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**

*i***-axis, and have a similar rotation which takes the unit (1+0i+0j) to (0+0i+1j)?**

*j*This seems attractive. We've created a new kind of number with three dimensions,

**,**

*real***and**

*i***. We've also succeeded in having an analogous behaviour for**

*j***just as we did for**

*j***. Importantly, we haven't created an inconsistency because the**

*i***and**

*i***dimensions are independent.**

*j*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

*i*^{2},

*j*^{2},

**and**

*ij***terms. The**

*ji*

*i*^{2}and

*j*^{2}can be replaced by -1 because we defined the rotational effect of these operators that way.

We don't assume that

**and**

*ij***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**

*ji**on the real unit 1. The*

**ij****rotates 1 to (0+1i) then the**

*i***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**

*j***=**

*ij***, and similarly**

*i***=**

*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?