Introduction

By some reason I entered the wrong way round the world of Fractals; first I searched "beautiful" pictures in the margins of the square Mandelbrot fractal, then came the higher potencies; to my surprise they did not show much new stuff.

Next came fractals according to Julia and since the functions for Mandelbrot images from the 2. to the 21. potency existed it was easy to do the same for Julia's; anyway, for potencies above 10 there came pictures that may be used as patterns for coasters or dessert plates.

Then somewhere I stumbled over Mandelbrot resp. Julia pictures showing a three-dimensional view - so I was tempted to write corresponding programs. However case of Mandelbrot only potencies 2-6 and for Julia potencies 2-11 were studied in details; higher potencies did not raise much - except for exaggerating computer times.

Strictly speaking I could have squeezed these four goals into one program but to write four independent-ones seemed to be easier to build.

Well, now I might have leaned back, but then I realized that many of my pictures not only presented well on my computer screen but they also were (including some fantasy) to be found in Nature.

   

So I bought the essay THE FRACTAL GEOMETRIE OF NATURE (Mandelbrot 1983) - but this also was not yet the real thing.

Mandelbrot on his paper touched on the problem concerning the length of the coastline of Great Britain. In practice it is not possible to define its length. On examples that are easier to handle in mathematics he shows this coastline being a line but not an one-dimensional one. Dimension of this thing lies between the integer values 1 and 2; Mandelbrot estimates 1.26, a so called broken figure. Therefrom the name fractal, since fractum in latin means broken.

Such a "coastline" also shows the well known AppleMan (by the way: Apple might be the computer he used and Man might be Mandelbrot). To illustrate the fact I assembled a little animation:

Starting at picture x 1 (original size) a detail always is enlarged up to x 1000. One might continue like this but then the black/white border becomes the more and more fuzzy. This is correct, since nature knows no sharp edges too.

Strangely Mandelbrot seemed not to have realized that the border around his figure hides an immense lot of pleasing pictures. However, it might be the computer he disposed of did not allow coloured presentations. But anyway, if I am not wrong, Mandelbrot was one of the first having graphics on his display.

Fractametry

Geometry, i.e. surveying the earth, this was the art of ancient egyptians to be able to allocate the property to the owners after the annual Nil flood. Later in old Greece this developed, specially under Euclid, to the classical geometry using but circle and ruler.

Circle and ruler, they allow drawing designs for buildings, machines and so on. Only for nature this geometry does not help much: Mountains are no cones, islands no areas of a circle, a flash does not follow a straight line and clouds are no spheres. And the same is for maths, although there are scientists insisting that an observation is ample proof if a mathematical expression for its behaviour could be found. Well, mathematics and with her geometry lived in a kind of ivory tower.

Besides cartography was developed, i.e. presentation of the earths surface either on two-dimensional maps or three-dimensional globes.

Geos (earth) and metry leads to geometry, fractal and metry give fractametry. Usually this is called fractal geometry; I dislike this fractal-earth-metry, so I stay with fractametry.

Fractametry is much like cartography, since every spot on the object is displayed in X, Y and Z; however, for Z often colours are used.

Chaos

Initially CHAOS is greek and stands for yawning chasm, precipice, gaping void.

Its meaning changed under the philosophers Anaxagoras and Plato to primary matter, amorphous, shapeless.

Nowadays Chaos most of the time has a negative meaning: mess, confusion, muddle.

In science chaos first time was used 1975 by J. Yorke in his paper "Period three implies Chaos", but maybe it was not too seriously meant. The writer treated effects when copying intervals to themselves and then noticed aperiodity.

A characteristic feature of chaotic systems is their sensitiveness to smallest modifications of starting or marginal conditions; sometimes regular behaviour suddenly switches to irregular. E. Lorenz (1965) noticed such behaviour on mathematical weather models, so that he started talking of the butterfly in the gulf of Mexico influencing weather in Europe while flapping his wings.

1989 Poincare studied stability of planetary orbits in our solar system. He found smallest orbit disturbances might amplify over time. For fear of consequences he then stopped his research; anyway, nowadays one talks of Poincare-scenarios if a regular system suddenly changes to irregular.

Much easier to understand (relative to the solar system) is a double pendulum: at the end of a first one is fixed a second one. The first one being pushed gently the whole system swings regularly. Stronger pushing leads to irregularities. Despite of simple rules (still being valid) calculating of behaviour no longer is feasible. Then we have "deterministic" chaos.

Chaos = doomsday?

How to define deterministic chaos: It concerns systems with an attractor, i.e. a state the system aims to reach. Our double pendulum has such an attractor, after a time it will hang straight down.

Natural sciences, especially physical science, intend forecasting events. An aeroplane flies, a billiard ball takes aim at the goal. If an event is repeated (same conditions of course), the result will be the same; this is what the principle of causality says.

Strictly speaking it never is possible to repeat starting conditions with absolute precision; but hopefully similar conditions (I beg you, please) shall give similar results.

But Poincare and the meteorologist Lorenz found processes that give completely different results for very similar conditions. For quite a time such stuff was declared an oddity and swept under the carpet. But chaos theory found such events being normality; occasionally it'll be a long time before chaotic behaviour is detected.

In what concerns our solar system, for coming millions or even milliards of years probably it might behave - and then? This question has not be answered yet - Poincare only found, chaos might come, but it needs not. And what means philosopher Murphy? Anything that might go wrong sometimes, will go wrong. Remember Tchernobyl.

Chaos = starting new things?

Chaos is not absolute death and disaster; deterministic chaos is the cause for our own existence.

Look at the picture above, it shows part of a flat roof. Where Chaos is written all is present, Earth, Water, Air and Fire from the sun; but look at the arrowhead, there came as troublemaker some time ago a seed and this started a process: that little shoot took minerals and water from the earth, carbon dioxide from the air and all this got built into stalks and leaves with the help of the sun fire. Chaos has found an attractor and became deterministic chaos. An attractor, that is a fractal, yes, even that is possible.

One of the most interesting natural fractals are the clouds. Go and watch some scattered clouds - and you will find a lot of things.

On the web I found some strange cloud formations and since I no more can run after such pictures, so I raid the net, but I hope, you will understand.

  

  

The shot at top left shows multiple parallel clouds, on top right multiple lenticular clouds are developing. Such formations appear on wavy motions of the air. The formation bottom left appeared over Hawaii. About the shot bottom right it is said it to be a lenticular cloud and also that it really is a genuine photography.

Links

chaos theory

chaos