An Introduction to Chaos Theory and Fractal Geometry

"God's Thumbprint"  The "Mandelbrot Set"
"God's Thumbprint" The "Mandelbrot Set"

A Butterfly in Brazil Can Cause a Tornado in Texas?


We just begin to think we understand what's going on around us in the universe. We get comfortable with the "Big Bang," "Black Holes," and "Pulsars." Most of us agree life follows Evolution's directives and that the latest Conservative Government is going to tax us to death. We are copasetic with the fact that the world's nations are skidding all around the planet on Tectonic Plates and that a random rock from space could end life as we know it with no warning; some even believe Cliff Richard when he maintains he’s “not gay!”

All-in-all, mankind is a pretty clued-up lot in 2011.

The, on one hard-to-forget night, we switch BBC 3 or 4 on in the UK and find ourselves in the middle of an erudite discussion about something that has just been words - to me at least - for the last few years; something I though had more the smack of alchemy than that of true scientific doctrine.

I am talking about Chaos Theory, describing a State of Disorder in systems - slowly becoming fact - and the application of Fractals, the discovery that many things we thought to just possess the properties we could see and describe in Euclidean terms, were much more complex and harder to determine.

Chaos Theory has its basic parameters in mathematical theory with some pretty complex formulae to tell the expert what it is. The discipline also takes in philosophy, biology, ecological systems and, of course, physics. Which has allowed the theory to attract the attention of a whole lot of braniacs.

What Chaos Theory does is tell us we cannot - and to an extent, why - predict what many systems will do. That means forecasting like Weather, for example (and explains why so many errors), It also explains much about the movements of the Solar System, Population Growth, Sub-Atomic Particles, Medical Conditions and much more.

Edward Lorenz in 1972 famously described the force which might first edge a system towards chaos as "A butterfly flapping its wings in Brazil causing a tornado in Texas!"

This is why long-term weather prediction of more than a week or so has proved impossible...these are too many "butterfly's wings" at work. And it states simply why predicting many systems accurately is impossible: any tiny force can cause huge changes.

Fractals.

When we look at a mountain, a cloud, a lightning strike, or a coastline we see it as a simple shape: we might say a mountain looks like a cone; a cloud a sphere, or blob; we usually say lightning zigzags until it hits something; a coastline is seen as just that, a line which rises and falls and goes in or out.

When we look at snowflakes without a microscope, we see nothing of the incredible patterns of the always-six-sided crystalline shapes. Same for frost forming on our windowpanes. Even the prosaic cauliflower or broccoli - our bathroom fern - show nothing of the complex Fractals which really make up their forms.

A Fractal, then, is "a whole which can be fragmented and its parts are at least a close copy of the whole." This sometimes called "Self-Similarity." The term "Fractal" was introduced by a great scientist, Benoit Mandelbrot quite recently (1975).

His fractal geometry covers the construction of plant structure, our blood vessels, the above mentioned geological features. Also art, music, architecture and much more. He showed how masses can be broken down into their underlying construction. His famous "Mandelbrot Set" (see pic) amazed the scientific community; one saw it as God's Thumbprint." (This is more amazing as a computer model and has to be seen).

In fact, this hub can only be an introduction to Chaos Theory and Fractals without becoming a scientific work: beyond space here and beyond the author's ability; it might inspire you to read more.


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Comments 14 comments

joanwz profile image

joanwz 5 years ago from Katy, Texas

Makes me want to learn more. THanks.


diogenes profile image

diogenes 5 years ago from UK and Mexico Author

Hi Joan: I didn't realize it at the time, but this subject has been covered by other hubbers. You may find some of their articles worth reading. Bob

PS Thanks for following me; I will do so for you, but later as I just can't keep up with the load I unwisely took on...Bob


writeronline 5 years ago

Hi Bob, that was a challenging way to start my day. But very instructive, thankyou.

Chaos Theory is one of those terms you hear bandied about a lot, but in my case, have no real idea, beyond the implicit, of what it actually is. Now I do.

All this stuff does serve to remind us (me) that, in the words of one Ian Dury (a reference not all will get, I know) "There ain't arf been some clever bast*rds."

Cheers


diogenes profile image

diogenes 5 years ago from UK and Mexico Author

So true. Scientists continue to hold me in awe. But I wonder sometimes if any of it is really true, or they manufacture all the stuff to entertain us! Bob


kimh039 profile image

kimh039 5 years ago

Hello Bob! I believe chaos theory applies to humans, human systems and groups as well as any other system, but the human systems are "my thing." I wish I could wrap my head around it a little better, but the bite size piece you provide here is enough for now. I never heard of fractal geometry. That seems interesting though. Thanks for sharing:)


Christopher Price profile image

Christopher Price 5 years ago from Vermont, USA

Earlier tonight I stumbled upon a program on the Discovery channel (here in the US) about fractals. Though I missed some of the program it was fascinating to see the many ways fractal sets can be used, from fabric design to determining the amount of CO2 a rain forest processes. (Something about how the pattern and ratio of twigs to branches on one tree has a direct correlation to the number and dispersion of trees in the immediate vicinity.)

As far as the chaos thing goes...it could be the story of my life.

CP


diogenes profile image

diogenes 5 years ago from UK and Mexico Author

Hi Kim. Yes, I believe you are right. Many systems have the element of chaos in them. Thanks for visit.

Christopher: Hahaha...Mine too! I read some of that about fractal sets and how they can be applied to nature.

Bob


rjsadowski profile image

rjsadowski 5 years ago

I haven't thought about fractals for quite some time. Thanks for reminding me. As I recall you can generate these images on your computer.


Hello, hello, profile image

Hello, hello, 5 years ago from London, UK

Wow, Bob, you made my brain work overtime. That was interesting and I never had a clue of it. Thank hyou for sharing it. I knew about the snowflake and the frost on window panes because I saw plenty of them. I miss them because they were so beautiful. The kids don't know what they are missing sitting there all day long and clicking away on games. Oh well, I better get off my soapbox.


MartieCoetser profile image

MartieCoetser 5 years ago from South Africa

diogenes – Thank you so much for explaining Chaos Theory and Fractal Geometry in such a way that I finally got the picture. It looks kind of like a scientific edition of Fate.


Trish_M profile image

Trish_M 5 years ago from The English Midlands

Hi Bob :)

I remember reading about 'fractal geometry' a while ago ~ amazing, fascinating and beyond my mathematical and scientific knowledge.

'Chaos Theory'? ~ Beyond me, too.

We are learning so much, and know so little.

Very interesting! :)


Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man

For an author who says it is beyond his ability you have done a mighty fine job here not just in explaining the basics of what many people would otherwise find very daunting your hub does inspire the reader to read more. Thank you.


Alladream74 profile image

Alladream74 5 years ago from Oakland, California

An interesting and complex subject.Goes to show how intricate this world is.It is amazing how much order appears in things under the microscope that other wise look chaotic and unrelated.Great intro to a complex subject


Sophie's soap box profile image

Sophie's soap box 4 years ago from Australia

Thanks for the great hub. Ever since I was forced to study Chaos and Compexity Theory at univsersity, I have been in awe of this potentially unfathomable subject!! I loved your examples of broccoli and coastlines too. Every time I cut off a little 'tree' of broccoli, I realise that there are many more little trees that make it up.

In addition, this theory suggests that the whole is worth more than the sum of its parts and that the more 'information rich' a system is, the stronger it is against internal or external stressors. If you have an apple orchard (a monoculture of cloned species) and one apple tree is vulnerable to a specific pathogen, all the apple trees in that orchard could be wiped out if exposed to it.

On the other hand, if you introduced that pathogen to a climax community (e.g. a rainforest) a few species may be affected however most would not be, resulting in a resilient, healthy ecosystem which has experienced minimal damage.

I really love this topic as it has such far reaching implications for our understanding of planetary health, human health, the way small businesses function, our bodies (micro villi and villi in the small intestine) and so much more! Thanks again.

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