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In the search for another dimension, with which to introduce topology into a fractal landscape, I explored random-access plotting. My plotting routine allows any pixel in the 512 by 512 array to be accessed.

I then described what happens to the entire Mandelbrot set when the starting position (bounded within its grid), the first position to which the dots were "flung", the second and so on, are all used for plotting. It led to a series of images which I showed under "programming" as "Moir Effects".

Here is what happens when it is not the entire set, but just a half-pixel by half-pixel area that has been enlarged:


That image came from:


Sadly, there is no discernable structure.

Charles

Continuing the story:

Given that I have now managed to capture the last location of the dot after it left its starting pixel, it becomes possible to use either the X or the Y value of that location as a substitute for the iteration number. That is to say, to use the X or the Y as a palette pointer.

I modified my machine-code programs as follows:

mov  ax,[juliax2+offset+2]  ; Upper 2 bytes describe pixel X
add  ax,two/65536               ; when no longer negative
rcr  eax,1                               ; and halved (128 represents 1)
clc
rcr  eax,1                               ; halved again (0 to 255)
and  ax,000ffh                       ; and masked.
mov  cl,al


The result, created by http://wehner.org/tools/fractals/position/lastpal.asm is http://wehner.org/tools/fractals/position/lastpal.gif as follows:


Much the same can be done with Y as the palette.

The source is http://wehner.org/tools/fractals/position/lastpaly.asm the result is http://wehner.org/tools/fractals/position/lastpaly.gif and it looks like this:


It is time to dig deeper into the fractals to see what fine structure might be there. The standard I have fixed upon is http://wehner.org/tools/fractals/man/man8.gif , modified by the various experiments. Here it is with a "last fling" X palette:

The source is http://wehner.org/tools/fractals/position/lastpal8.asm the image is http://wehner.org/tools/fractals/position/lastpal8.jpg


And here is man8.gif with a Y palette:

The source is http://wehner.org/tools/fractals/position/lastpl8y.asm the image is http://wehner.org/tools/fractals/position/lastpl8y.jpg


The specular image of the "last fling", with which I started this report, shows that each time the Mandelbrot iteration is carried out, the dots are flung into a more and more random pattern. Strictly, it is pseudorandom, because everything is deterministic. However, it is so close to total randomness that the GIF compression process breaks down. The 250,000 bytes "compress" to 344,000 bytes - and I had to switch to JPEG.

Even with JPEG, compression is poor. I had to use quite a high level of compression just to reduce the file to half-size. This brightened the colours a little (the pink is a little more pink, the blue a little more blue) and sharpened the "grain". However, the images are a reasonable representation of what has happened.

Where there was just one, or a few like two or three, iterations, a patch of the smooth-palette background was swung round and pasted on top of that background. This produced interesting shapes. However, already at ten iterations or so the pattern has become pseudo-random. The "last-fling" analysis of an image is therefore unsuitable as a means of finding a fourth dimension for the topology of fractal "landscapes".

Charles

There is, of course, another well-known way of showing where a point was "flung" to before it left the map.

That method is to count how many times a point lands on a particular pixel just before escape, and use the count as an index into the palette.

The outcome is the famous "Buddhabrot" image (terrible pun on "Butterbrot"), of which an example was made by http://wehner.org/tools/fractals/buddha/buddha.asm


There is a lot more about Buddhabrot here:

http://www.superliminal.com/fractals/bbrot/bbrot.htm

Charles

Of course, if one oversampled by 16 by 16, instead of eight by eight, one would get a denser image - with the shadows clogged up:


That is why those who take the "Buddhabrot" seriously tend to reserve at least two bytes for each pixel, count out the "arrivals" at each pixel, and finally convert the count into a sigmoid curve - with low contrast in the light and dark regions, and normal contrast in the middle.

That way, far more detail can be preserved.

Another trick is "acutancing". Here, one would exaggerate the edge-contrast of the features, in order to draw attention to them.

I am not delving too deeply into these fields, which have been adequately explored.

Charles