A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.
For a simple description of fractals, please download our “One Pager” (380Kb).
For more detailed info, please download our 20 page “Educators’ Guide” (7.5Mb).
Explore the Mandelbrot fractal yourself, with the amazing real-time fractal zoomer, XaoS:
What is Chaos Theory?
Chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected. While most traditional science deals with supposedly predictable phenomena like gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on. These phenomena are often described by fractal mathematics, which captures the infinite complexity of nature. Many natural objects exhibit fractal properties, including landscapes, clouds, trees, organs, rivers etc, and many of the systems in which we live exhibit complex, chaotic behavior. Recognizing the chaotic, fractal nature of our world can give us new insight, power, and wisdom. For example, by understanding the complex, chaotic dynamics of the atmosphere, a balloon pilot can “steer” a balloon to a desired location. By understanding that our ecosystems, our social systems, and our economic systems are interconnected, we can hope to avoid actions which may end up being detrimental to our long-term well-being.
Principles of Chaos
- The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. It may take a very long time, but the connection is real. If the butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. A more rigorous way to express this is that small changes in the initial conditions lead to drastic changes in the results. Our lives are an ongoing demonstration of this principle. Who knows what the long-term effects of teaching millions of kids about chaos and fractals will be?
- Unpredictability: Because we can never know all the initial conditions of a complex system in sufficient (i.e. perfect) detail, we cannot hope to predict the ultimate fate of a complex system. Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless. Since it is impossible to measure the effects of all the butterflies (etc) in the World, accurate long-range weather prediction will always remain impossible.
- Order / Disorder Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.
- Mixing: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed. Examples: Two neighboring water molecules may end up in different parts of the ocean or even in different oceans. A group of helium balloons that launch together will eventually land in drastically different places. Mixing is thorough because turbulence occurs at all scales. It is also nonlinear: fluids cannot be unmixed.
- Feedback: Systems often become chaotic when there is feedback present. A good example is the behavior of the stock market. As the value of a stock rises or falls, people are inclined to buy or sell that stock. This in turn further affects the price of the stock, causing it to rise or fall chaotically.
- Fractals: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.
“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”
-Albert Einstein
It explains every thing soo simply that every one can understand I love this website?
Fractals changed my life forever. I wanted to commit suicide but then I learned about fractals. I do fractals everyday. I see fractals everyday. I fack fractals everyday. This website is amazing. I’ve sold my soul to fractals. Love them. Bye
Once I started looking further into Fractals, I started to like them.Fractals are very interesting and artistic.
When I first found this website I didn’t know what Fractals were. Once I started looking further into them, I started to like doing fractals. They are really cool! Fractals are very interesting and artistic.
When I first found this website I didn’t know what Fractals were. Once I started looking further into them, I started to like doing fractals. They are really cool!
The Sierpinski Triangle Fractal can serve as a model in Management, it appears.
Presently, a centralised organization functions at some place with subdelegated units at other locations.
The trickle down effect is varied, from not satisfactory to excellent.
If the last unit also has the characteristics of the central organization, the trickle down effect in most cases would be excellent.
Be it a production unit, sales unit, service unit, administrative unit, if like in the fractal, they imbibe the characteristcs of the basic unit, the result very well expected to be the best!
SV Anbazhagan
Absolutely awesome!!! I was helping a graduating geophysical engineer to translate his curriculum from Spanish to English and came upon the term “chaos and fractals geophysical applications.” I decided to google the word fractals and lo and behold!! I’m at the gate of knowledge. I had been told years ago that we could be part of a hair on the eyebrow of a collosal being but wasn’t able to fully grasp the meaning alltogether,although I did have an idea. Habichuelas M has really helped me out here. Thank you guys.
google Frax for a wonderful iPad-only app that is visually stunning in its beauty.
Now I understand that line in Frozen’s “Let It Go”. Thanks for the explanation. You are reliable.
that is a really good definition! whoever made this is really smart!
Great question, Barbara! It’s not a formal term, but we like the word “Fracticle” to describe the repeating subunit that makes up a fractal. Feel free to adopt that if it works for you 🙂
Help! I’ve been trying to find the term for the basic pattern that is repeated at every scale in the fractal–as might be used the sentence, “Fractals are composed of an endless repetition of a single geometric pattern, called the [fractalus? fractalette? fractoid?] of the fractal.” Surely there is a word for this, and I am desperate to know it! Thanks so much–
i love this its awesome
I am using your wonderful site to teach my elementary art students the science AND the creative process of mono-prints! They are having a BLAST!! Thank you!
Thank you Tom. Your response to Habichuelas Magicas was beautifully put and very clear. And of course your quote from a great mind. I can only add for Magicas, keep your feet firmly on tree ground as you reach into the stars….
Habichuelas Magicas – I think your idea is not stupid at all, the holographic principle, and other real life theories include your ideas – and in my Scientific American, there is a new theory emerging based on the beautiful and repeating geometry of nature. Then of course you are correct that many eastern religions are huge on geometry as meaning.. I think the holographic principle and fractals have a lot in common in that in every square inch or square mile of the “film” or membrane, there is the whole contained within that. So if you Cut a piece from it, the whole is preserved… in the holograms we make, the fidelity gets less and less, like loss of resolution, but its not hard to imagine that if every inch of space is infinitely dense, the a celestial hologram would not loose fidelity. “Every particular in nature, a leaf, a drop, a crystal, a moment of time is related to the whole, and partakes of the perfection of the whole. Each particle is a microcosm, and faithfully renders the likeness of the world.” ~Ralph Waldo Emerson~ Emerson
After a long time loving fractals, and thinking about them I got a theory that maybe sounds stupid or difficult to understand about existence in a Universal level and meaning. Considerating Fractals are the model of how naturally LIFE is created and putting it on in the macro and microcosmos would be a chance to find our planet, would be just an electron, of an atom, where the core is the sun. This atom would be part of a molecule, that is our galaxy, and with all those molecules you will find the shape of an organism, like any kind of LIFE, like the Universe.
So if we could see the Universe from a real external perspective, maybe we would find out we live inside other human being who lives in other Universe, that is other human being, Or could be any Lifeform made up with particles, and molecules like an animal, a tree, or an apple…This could fit with the multiverse and multidimenional THEORIES, where there are infinite number of universes…
And about the microcosmos,if we would be able to see an electron, as close as possible,maybe we would find humans on his surface, like in a planet…
We find metaphisic and spiritul texts that say there’s a universe inside every of us, or that our planet or the universe is other kind of organism.
The problem is we are not able to see or perceive it but in that case, we could use fractal to explain time and space like fractals, so what is 1sec for us, may be 1 year from other point of view in other place of that fractalic time, or what is 1 meter for us might be 100000km or light years from other perspective.
I would love if somebody know about some scientis working in this kind of theory, please let me know.
I LOVE FRACTALS SO MUCH THAT I COULD MAKE THEM ALL DAY. SOMETIMES I DO MAKE THEM ALL DAY.
Awesome!!!! Love this.. 😀 Thanks! 😉