The DAWN of Fractals

[Charleswehner]

We all know that the remarkable images of Fractals arise out of chaotic arithmetic. I decided however, that by hands-on research I might explore the world of less chaotic mathematics to see what might arise.

The original Julia and Mandelbrot sets arose out of Z <- Z * Z in a recursion loop. This leads to Z², then to Z⁴, then to Z¹⁶, then to Z²⁵⁶ and onward.

I decided to explore the situation when Z <- Z * Z₀, where Z₀ is the starting position - after zero iterations. This gives rise to the sequence Z₀ followed by Z₀² followed by Z₀³ and Z₀⁴ and so on.

The first test program is at http://wehner.org/tools/fractals/expo/expo.asm and gives this image:

That is the typical de Moivre circle - not a Fractal image. However, the outer edges are feathered much more gently. Instead of the circle rapidly giving way to a white background, it has a soft gradation of colours around it.

The difference comes about because the Julia superset and Mandelbrot set have a function (how did I type that?) whilst this exponential function is only Zⁿ. If a Fractal function could be built out of this, it would have large areas of soft gradation.

I then added the Mandelbrot vector. There are two ways of doing this. One way is from Z <- (Z+1) * Z₀, which was my first modification to the program, at http://wehner.org/tools/fractals/expo/expoa.asm. The second - at http://wehner.org/tools/fractals/expo/expob.asm takes Z <- (Z * Z₀) + Z₀ . If ever a Fractal system arose out of this, one would use the latter file to replace Z₀ with a fixed vector to get a Julia version.

Here is the result at http://wehner.org/tools/fractals/expo/expoab.gif :

The addition of the Mandelbrot vector (the pixel vector Z_pixel is Z₀) has provided a notch to the right - as in the proper Mandelbrot set. Furthermore, it does not seem to matter whether one adds or subtracts. I tried both.

Inside the notch there is colour. Enlarged 64 times, by means of http://wehner.org/tools/fractals/expo/expoc.asm gives this:

There is nothing Fractal-like there.

And when one adds TWICE Z₀ by means of http://wehner.org/tools/fractals/expo/expod.asm the notch is deeper:

And when FOURFOLD Z₀ is added, deeper still http://wehner.org/tools/fractals/expo/expoe.asm

And EIGHTFOLD (http://wehner.org/tools/fractals/expo/expof.asm ) gives

These are not Fractals - but one can see the dawning of the Mandelbrot shape. All we have at present is a notch.

The simplest function is a linear one. Here, for example with the identity function Y=X, there is no chaos. The next more complicated for our interests is the exponential - Z <- Z * Z₀. After that come the FACTORIALS. These are of the form 1, 1*1, 1*1*2, 1*1*2*3 and so on, whereas exponentials are of the form 1, 1*2, 1*2*2, 1*2*2*2.

In factorials, the last number is rising - it is 1, 2 and 3. In the exponential it is fixed. I demonstrated a base of 2. Yes, that last number is the base.

Because the base of a factorial is always rising, the factorial will always catch up with and overtake an exponential. James Stirling and Leonhard Euler both used this fact in order to find an approximation for the factorials. At infinity, the base is rising to infinity-plus-one and infinity-plus two - so it is not rising at all. For high numbers near infinity, the factorial therefore approximates a high exponential.

However, the Julia-Mandelbrot function gives which is guaranteed ultimately to overtake the factorials. That is because it is exponential in an exponential. The best a factorial can do is to approximate an exponential.

So, although Fractals do not begin in earnest with Z <- Zⁿ, they might begin with Z <- Z! where Z! is the (fractional) factorial of the complex duple X + iY.

Charles

 

[heneganj]

Charles your posts are lecture like in structure and I'm finding them fascinating!

[Charleswehner]

Charles your posts are lecture like in structure and I'm finding them fascinating!

Perhaps they are lecture-like because the subject is technical.

I am just reporting what I find as I feel my way back into Fractals.

There is a lot more on the way. Some will be known, but some (hopefully) will be new.

Glad you enjoy them. They are research results - not opinion.

Charles

[Charleswehner]

I originally described the Julia/Mandelbrot algorithm as throwing a point around in a spin drier.  We have ZZ*Z, which doubles the angle and squares the range from the origin. The de Moivre theorem deals with that. After that comes a shift along a vector (the "tumble" as in a tumble-drier) which I will tackle later.

My first test was to see what would happen if the angle simply grew by a fixed amount at each iteration. I explored that above - and there were no Fractals.

Then I questioned what might happen if the angle had the angle added to it the first time, then twice the angle the second time, then three times the angle the third time. The result would be 1 + 2 + 3 + 4 ..... n . That is SUMMA N.

For a start, I created an image with no tumble. It should be a circle, and it is. That tests the program.

There is a tinge of colour around the edges, but nothing like as much as in the circle I showed previously.

Now I add the tumble as a Mandelbrot vector. It tumbles at each iteration to an extent equal to the range of the pixel from the origin, and parallel with it. The Mandelbrot vector is, after all, the pixel vector.

There are slight tufts appearing. The Fractals are dawning. I decided to choose the tuft on the left for enlargement:

A disappointment. The pattern has a slight curve to it, but there are no exciting shapes.

The program had given one multiplication by the pixel Z, two multiplications, three multiplications and so on. This led to (n*(n+1))/2 multiplications. It is an n-squared exponentiation of Z.

That same program can easily be converted to use the previous result, instead of the pixel Z. So we get Z² followed by two multiplications of Z² by Z² giving Z⁶, followed by three multiplications of Z⁶ by Z⁶ giving Z²⁴, followed by four multiplications of Z²⁴ by Z²⁴ giving Z¹²⁰.

The numbers 2, 6, 24 and 120 are factorials.

Numbers less than one shrink instantly towards the attractor at the origin. Numbers above 1 fly rapidly away. The picture is in colour, but we perceive only black-and-white because the transition from many iteration to few is so abrupt.

With the Mandelbrot tumble added, the whole set appears. But the bits of black in the air on the left suggest that the pattern is breaking up. I explored the area on the left.

The fierce arithmetic of a factorial exponentiation of Z leads to the points being flung around the complex plane and hitting the traps too soon. The fine, feathery detail has been "singed" off. Where with Julia/Mandelbrot we had "sea-horses", here we only have Ss.

This is what happens to the arithmetic.
Consider ZZ*Z+Z₀. We get Z⁴+2Z⁴+Z₀²+Z₀, and at each iteration more binomials of this kind as can be discerned from Pascal's triangle. I am interested in just the first term, the Z⁴.

With linear exponentiation, beginning at 4, we get:
4
8
16
32

With n-squared (specifically summa-n) exponentiation:
4
16
128
2048

With exponential exponentiation (Julia/Mandelbrot):
4
16
256
65536

With factorial exponentiation:

4
64
16777216
1.3292279 times 10³⁶

So Fractals seem to be dawning somewhere near the n-squared region and reach their perfection, high noon, somewhere near the exponential exponentiation of Julia and Mandelbrot. By the time the exponentiation becomes factorial the chaos mathematics is too extreme, and the sun is setting on the Fractals.

It is a pity. I had hoped to have found something new and glorious. Nevertheless, there are other approaches to consider......

Until later.

Charles

[rloldershaw]

Hi Charles,

This type of intuitive/empirical/experimental approach to understanding things is not always guaranteed to result in glorious discoveries.  But it is a lot of fun, and when glorious discoveries *are* made, it is usually the result of this approach.

I have enjoyed following your progress and experimentation.  Long live intuition!

Rob

[Charleswehner]

I have enjoyed following your progress and experimentation.  Long live intuition!

Rob

MISTAKES can be fun, too:

Charles

[Charleswehner]

Just for completeness, here are some addresses for the source-code of the above:

http://wehner.org/tools/fractals/summa/summa.asm - also summa2.asm and summa3.asm .

http://wehner.org/tools/fractals/expo/expo.asm - also expoa.asm, expob.asm expoc.asm, expod.asm, expoe.asm, expof.asm, expog.asm and expoh.asm .

http://wehner.org/tools/fractals/fact/fact.asm - also fact2.asm and fact3.asm .

Research reports must show how the results were obtained - so I have completed this section of study.

Charles

[Nahee_Enterprises]

Charles Wehner wrote:
>
>    Just for completeness, here are some addresses for the source-code of the above:
>       ............
>    Research reports must show how the results were obtained - so I have completed
>    this section of study.

Thank you for your postings of this area of study and research, and especially in supplying the source code used in these explorations.  It is always interesting to see such detailed progression.