If you are not familiar with matrix notation, don't worry, I expanded it out during the thread.
Previously, I evilly hijacked a thread
with my idea of doing Matrix-based Mandelbrot sets.
In the last two days, I finally started playing around with different matrix configurations to find the ones that reproduce behaviour of the complex, the split complex and the dual numbers. - Later I looked it up at Wikipedia and found that I was correct, which is somewhat satisfying already
First, I'll show you the three kinds of matrices that do this and then I'll state the obvious and show a way, how to extend that to a generic algebra
So here comes the complex matrix which behaves just like complex numbers:
Here is the Split complex one:
And there you have the dual one:
Note, that in the upper right corner, the 2,1-position, you find the imaginary part multiplied by the square of the imaginary unit in either case.
So you could also write those as:
So the obvious way to extend this is to do
- where n² is what ever value your additional dimension's unit should square to.
Adding two matrices together is done component-wise, so that's very straight forward.
To multiply them, you simply multiply the rows of one matrix with the collumns of the other and place the result into the corresponding position of the matrix. So in the general 2x2-case, that means:
For our general algebra matrix, that means:
Note, how for the ns i,j or epsilon, the real part of the multiplication would come out true.
Also note how the n²Im still is intact - the n can be taken out - and how also the Im in the lower left corner still is intact.
That means, any Mbrot, based on an algebra defined by that, has the form:
Where c is the constant for each dimension, that's altered for the Mset and indeed constant for the Jset.
So, you just take the upper left and the lower left results and use those to define your iteration
One of the nice properties of this class of matrices is, that all of them have a general inverse:
- which if you just look at the left part of the matrix and compare with the inverse of complex numbers, can easily be tested to turn out true
*/ to be continued with even more general but maybe (maybe as not yet tried) less interesting extensions.