### Three dimensional Lyapunov space

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#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #30 on: March 17, 2019, 09:43:54 AM »
@claude: Thanks!

I was examining the speed of convergence in 2D images. I did a 2-pass computation, first to calculate the final Lyapunov value, then a second round to find (if existent) an iteration number, after which the intermediate Lyapunov value at later counts only varies +-5% of the final value.

I drew that information in a height diagram under the 2D image to get an impression on how fast the pixels converge. Here are some examples (image at top, speed of convergence at bottom, the higher the peak, the faster the convergence).

There's quite a variety of outcomes here: Regions of fast convergence, spiky parts, whole images at one height and some shapes that are quite regular.

Has anyone experience in the convergence speed of the Lyapunov type? Is order faster converging than chaos - or does it not correlate? (I have to check that next).

In the overview picture, the first one showed a very weird shape. It almost looks like the picture above it (the double torus), was just cut in half and then shifted. That always makes me suspicious of a glitch in my computation routines, but so far I haven't found one.

The reason behind all this was: I plan to impose that speed information onto the 2D images to get a new way of generating 3D images, so that at the end it will (hopefully) look like one of these images of space time that are shown in physics documentaries when they talk about black hole, gravity and space curvature.

For the spiky parts I think I will try vasyans suggestion of supersampling to reduce graining. That might work here as well to somehow smoothen the speed peaks and then not only draw one pixel at the speed height but rather a polygon of a couple.

But I am currently not sure how to connect polygons at different heights to one another. Using octagonal shapes?

Has anyone experience in this regard?

#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #31 on: March 21, 2019, 11:05:42 AM »
I found some interesting cube shapes. They nicely show what lies within a simple concept as the Lyapunov sequence (simple in definition, that is) - pure mathematics with the easiest coloring possible (linear interpolation), especially in the cases where objects seem to float around randomly in space - direction, shape, distance, everything stems from the formula. It'd be great to have some kind of interactive visualization there where one can turn around and move into the objects.

However from an aesthetic artist point of view, those cubes from the ouside are most of the time not very appealing. So I might try something different in the future: Jumping randomly into the cube, taking those color values found (or some sort of average), define them as transparent and take a look around.

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