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Hey guys

I was hoping people could help me understand and tell me if this idea has been done

Basically I was thinking about putting the mandelbrot into 3d,
My understanding is that the problem is complex numbers have 2 parts which translates well into 2 dimensions but not 3. So to try and replicate it we've tried to map two of the dimensions to either the real part or the imaginary, presumably trying every combination, adding, multimpling modulus average etc of two axis to make one part of the complex number.

So my question is why did spherical coordinates work, didn't they have to do a similar mapping of two parts to one and one to the other. Also which part is imaginary is it the radius or one/both of the angles, or one/two of the axis, what happens if you change these around? I know it only really works in higher dimensions so what about higher dimension versions of the methods I mentioned earlier?


Idea:

Anyway my idea was to try to represent two dimensions with one number, if you have a sphere start at a pole and wrap it up in a line in a spiral. The line will be infinitely long but we map it to the limits of the real or imaginary part of the fractal, then when we take some degree of accuracy the line has finite length and any point on the sphere can be represented by one number.

Similarly in cartesian you could have a 2d spiral on one plane for the real or imaginary. Or for simple maths do a grid shape so if we could do 10 d.p of accuracy each line on the grid could be 0.00001 so work out the volume covered first then the position down each line. Going from 0-1 a point in the middle 3/4 across would be 0.7500050000

Sorry my maths isn't good enough for the spirals unless you took each circumference to be the same but then motion around it wouldn't be linear, well my brain is hurting thinking about it.


Has this idea been done or does it just produce the same result as other methods? Sorry I'm not that good at maths hopefully I made myself clear.

Thanks

What do people think, I'm liking the idea the more i think about it

The problem with conventional methods is you end up multiplying or adding axis which maps the same number the multiple joining points, here each value is only present in one location. If we think about the grid version in cartesian each point on the plane will be a point on a line section of the normal version. I was worried moving in one direction changes the value a lot more that the other but it will be like moving alone the line at different scales. If every line in each direction works then the composite should look ok right?

Then the other part of the complex number can be taken as z or you could even do the same thing and give it as a function of z and x or z and y


Could someone test this out for me I did the maths for vertical lines running in the same direction
I was thinking we would map the imaginary part as it has symmetry

a + b i

b = ( ( ( (((Ymax-Ymin) x ((X-ΔX)-Xmin)) / ((Ymax-Ymin)x(Xmax-Xmin))) + (ΔX((Y-Ymin) / (Ymax-Ymix))) ) x (imax – imin) ) -imax )

imax is how much of the imaginary part we want to plot , it should be at the boundary for the Mandelbrot set or just above it, around 1.12?
The other max and min values are arbitrary, we should choose values that will correspond to the z values

I was thinking of doing it between 1 and -1 on both axis as it's similar to the limit of the imaginary and it makes the formula easy

It becomes:

b = ( ( ( ((X-ΔX)/2 +0.5) + (ΔX(Y/2 + 0.5)) ) x (imax-imin) ) -imax )