Fractals, Chaos, and Religion in a Nutshell

by Tom Seddon

(I hereby promise that the following explanation has only three numbers and absolutely NO MATH. tes)

What is a Fractal?

A fractal is a new kind of geometric shape which no matter how greatly magnified always looks surprisingly like the original shape, that is to say, it is “self-similar at all scales.” Curious, but “so what?” Well, first, nature employs fractals to build all sorts of things, and second, fractals are closely related to “deterministic Chaos.” And some would say, Fractals and Chaos together cast a new light on dusty old religion.

First things first:

How to Make a Fractal.

Fractals are created by starting with a simple shape and then changing the shape by a simple rule and then repeating that rule infinitely.

Here are the steps in creating a classic fractal--the Koch Snowflake:

Start with an equilateral triangle,

Insert smaller copies of the triangle at the center of each side,

Insert still smaller copies of the triangle in each of the now 12 sides,

Keep
repeating the process, *ad infinitum*...

Now here’s where things get curious-er. If you carry the process to infinity (and obviously you can’t draw it anymore, but you can keep track of it mathematically), one finds that the area of the snowflake, say the amount of yellow paint needed to paint the inside, ends up being exactly 1.6 times the area of the original triangle BUT the perimeter or distance around the outside is now infinitely great, so an ant walking around the border lines would take forever to make it around. Curious, eh?

Fractal Landscape Generation

The same process can be applied to a surface. In the picture below, the large original starting triangle is easily seen. The “fractal generator” not only subdivided the original into little triangles, it pushed them up random amounts. Starting to look realistic, isn’t it?

Screenshot from Jon Shemitz’s article, “3D Fractal Landscapes,” at http://www.midnightbeach.com/jon/pubs/3D_Fractal_Landscapes.html

Shown below is a complex fractal landscape computer generated and posted on Wikipedia by an artist calling himself “The Ostrich.”

This is not a real photo—it is mathematically generated! This obviously has had enormous application in modern digital movie making.

Fractals in Nature

Mother nature employs fractals to build things by taking a simple design and then adding some randomness and repeating the design on smaller and smaller scales. We just never saw them before. These photos are of real things that reveal an underlying fractal structure.

Fractal Tree by Samuel Judge

Lighting Fractal by HowStuffWorks

Broccoli floret by tinG

The internet is full of other photos like these; just Google “Fractals in Nature.”

One of the most famous fractal images is the Mandelbrot set. It uses a quite complex mathematical formula to eventually generate the pattern below. By fiddling around, the pattern can be displayed in amazingly kaleidoscope colors. This image is rather tame.

Created by Wolfgang Beyer with the program Ultra Fractal 3.

The Mandelbrot Set in general is not precisely self-similar but is quasi-self-similar, as smaller slightly different versions of itself keep appearing at arbitrarily large magnifications. In the sequence below, each picture to the right is a huge enlargement of the piece inside the little box in the picture that came before.

Illustrations from Wikipedia’s article on the Mandelbrot Set.

Deterministic or Creative Chaos

Chaos theory relates to fractals since it was discovered that the progression of many natural systems through time couldn’t be predicted with the certainty everyone hoped would come with more powerful computers. Tiny differences in the starting conditions of the computer’s multiple repetitions will eventually lead to enormous differences in the results. And this will always be true no matter how powerful the computer you can imagine.

This is popularly referred to as the “butterfly effect.” That is, “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” (Although everyone starts with a butterfly, the places and consequences change each time the story is told.) The butterfly effect was invoked by fictional chaotician Ian Malcolm in both the novel Jurassic Park and in the film (played by Jeff Goldblum). He used it to explain the inherent instability of (among other things) an amusement park with dinosaurs as the attraction.

How is such a thing possible? Well, the butterfly does not “cause” the tornado in the classical sense of cause and effect, but by flapping its wings, the butterfly has changed its environment, and this tiny change, infinitely multiplied with random modifications, just like a fractal is computed, sets into motion the sequence of events that creates the tornado. Don’t mistake the metaphor; it’s not that butterflies cause tornados, it’s that we can never know what triggered the situation that led to the tornado.

This stuff is real.
The “butterfly effect” was discovered by a programmer developing
computer software to make weather predictions.
He made a tiny change in one of the numbers that represented an actual
weather observation. Thinking it could
hardly make a difference; he used the number 0.506 instead of the complete
0.506127. This tiny change resulted in a
__completely different weather forecast__.
This discovery renders long-term weather prediction impossible in
general.

Now even though the process involved repeating the same calculations over and over, that is to say, the process was completely “deterministic,” the final results were unpredictable! This kind of behavior has come to be known as “deterministic Chaos,” or simply “Chaos.” It is not to be confused with statistical randomness.

Indeed, chagrined meteorologists now admit that for practical purposes of weather forecasting, predicting any individual tornado is, and always will be, totally impossible. Even the seemingly mechanical laws of Newton are not immune to Chaos. Astronomers know that because of Chaos, it is impossible to perfectly predict the path of an asteroid years in advance. A new discovery proves that the seemingly perpetual motions of planetary orbits are actually Chaotic.

Chaos problems have been explored in biology, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, robotics, and religion!

Wake turbulence behind a landing jet aircraft starts as a small airflow disruption at the wing tip.

Studies of the critical point beyond which a system changes over from smooth flow to turbulence involve applications of Chaos theory,

Chaos theory is also currently being applied to medical studies of epilepsy, specifically to attempts at the prediction of seemingly random seizures by closely observing the patient for critical initial conditions. This is an example of working the problem backwards. If we always see the same outcome, can we possibly narrow down the small differences that triggered it? It is a colossal challenge; something akin to watching tornados in an attempt to find those fluttering butterflies.

Fractals, Chaos, and Religion

Philosophers and theologians have gotten involved. We now have a solid science that implies
(according to some) that even given a specific Creation and a set of definite
rules for acting out that Creation (a kind of Intelligent Design by a Deist,
using the “Supreme Architect” notion of God); that even then the end results
today could well be unpredictable. **Chaos may have done its own Creation!**

For an excellent example of the fervor this has caused browse the website: http://www.fractalwisdom.com/ whose banner headline is “FRACTAL CHAOS Crashes the Wall between Science and Religion.”

The web site does begin with this intriguing quote from Aldous Huxley,

“At any given moment, life is completely senseless. But viewed over a period, it seems to reveal itself as an organism existing in time, having a purpose, trending in a certain direction.” (Hmmm? Is this Chaos theory inside out or upside down?)

And thus you have it, Chaos, Fractals, and Religion, each helping to re-generate and change the other with unpredictable results!

Tom Seddon, Salt Lake City, 29 November 2009.