Fractals are SMART: Science, Math and Art!

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.

Don't blame me!

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

-Albert Einstein


  1. Jennifer Schuetz Jennifer Schuetz
    March 5, 2016    

    Please reference Dr. Jonathan Wolfe, one of the founders of the Fractal Foundation and current board president, Albuquerque, NM.

  2. Deborah Theodore Deborah Theodore
    February 29, 2016    

    How do I cite your website in research paper. Who authored or what is the source of “What is Chaos Theory?”
    Thank you!
    Deb Theodore

  3. Manisha Manisha
    December 8, 2015    

    I simply LOVE THIS WEBSITE. I am very new to the concepts of chaos theory , Fractal world of realities. Beautifully explained !

  4. jj jj
    November 19, 2015    

    Nice article. Thanks.

  5. Anna Anna
    November 17, 2015    

    LOVE THIS WEBSITE. New to chaos theory and you, Fractal Folk, really rock my chaotic socks off! Way to go! See you at Fractal Fest!

  6. kirk barrett kirk barrett
    April 18, 2015    

    “The Butterfly Effect” stated that a butterfly flapping its wings _CAUSES_ a hurricane far away in time and space is obviously credible only as a metaphor. If a flap could have _any_ noticeable effect on the weather,even near in time and space seems like an actual hypothesis. But there doesn’t seem to be a way to test it, cause we cannot repeat the exact conditions of an experiment, one time with the flap and one time without. The only way approach to testing the hypothesis is with a mathematical model — but a model is, by definition, a simplified representation of the real system. And if you are looking for an effect of a very small variation, using a _simplified_ representation to detect it is not very credible. As such, going back to origin of the “butterfly affect” (that a small change in the initial conditions in a mathematical model of the weather leads to large changes in the models predictions), I would say its more likely that the effect is an artifact of the need to simplify the system to be able to model it, rather than a real property of the weather system itself. This is what meteorologicalengineer was getting at in the post above.

    Whether the effect is real in other real-world systems seems another question, as is whether there is anyway to test if its real.

  7. April 17, 2015    

    Great theory for expanding one’s thinking and musing. I am a Lutheran pastor and find that much of Chaos explains the church. The dynamics fit well with my process explanations of church life and the graphics I use. I find my thoughts going in directions that might provide all human organizations some needed insights on themselves.

  8. March 31, 2015    

    Fascinating, I’m a nurse, not a scholar. I confess I lack a basic understanding but this gives me a small insight…and a start! Thank you

  9. Ed Hodges Ed Hodges
    March 29, 2015    

    I like what Marina has shared. It makes simple the human thought of chaos.

    To me chaos theory explains the relationship that one particle, molecule, insect, animal, person, world, asteroid or any other simple thing broken down to the smallest or built into the largest object can and will have an effect on every other object.

    To those confused about the “Butterfly Effect”, remember this was one mans attempt to explain what I have just stated above, but he used a metaphor. It can be that butterfly that moved the air or it could have been the child that scared the butterfly into flight or maybe the mother that sent the child out to play.
    It is not just the wind the butterfly expelled with its wings it is the fact that if it had not moved it might not have affected the world at all.
    Just my opinion. I am not a scholar, but I do try to think of myself as being able to grasp ideas that a lot of people may not be able to.

    In point,
    I have a though and please let me know if there is more on this.
    Chaos theory as has been stated as “One Particle will affect another and so on for eternity with varying degrees. Thus the determination of an outcome is ever changing.” So if you add String theory into this concept and show that that particle is actually a string and therefore one wiggle of that string will cause another wiggle within that string as well as along other parts of that string. Would the two theories support each other.

  10. March 2, 2015    

    Here’s a passing thought I had about Chaos Theory:
    There is order in chaos.
    There is no chaos in the theory of chaos. Chaos is a human construct to explain the difference between control and not controlled.
    It is the substance between the here and now and then and there. There is no dis order in the universe for that would mean the universe has a plan of what should be and should not be.
    There is no plan to the universal mind. It is capable of creating whatever is wanted whenever it is wanted by whoever would want to do it.
    The chaos you speak of is the human minds ability to substantiate and bring to the forefront an explanation of how things should be so and what should not be so.
    This theory of chaos does not exist as such but will exist as long as the human construct deems it to be so.

  11. Ed Ed
    February 15, 2015    

    Paul Koszarny says:
    April 12, 2013 at 1:28 pm

    “I have my objections to what meteorologicalengineer wrote. He is wrong on what mathematically “initial conditions” are and as a result he claims that chaos deals with a lot of approximations. That is entirely untrue because chaos is remarkably precise in terms of chaotic behavior. Turbulences, triggers, fractal expansions, borders of chaos possess the quality of exactness…”

    I have heard regarding fractals a demand that physical phenomena are not fractal unless they conform to mathematical exactness. What often is forgotten in such demands is that mathematics merely describe physical phenomena. They are not in themselves a physical phenomenon nor do they necessarily explain physical phenomena. Yet as a mode of description, the fractal concept describes many physical phenomena far better than do other alternatives. One should challenge people who make such demands for perfection to show truly physically perfect equivalents of Euclidean forms such as rectangles of circles. At some scale of measure imperfections will appear that depart from the theoretical. Nevertheless, the theoretical equations of Euclidian rectangles and circles describe these imperfect forms better than any other alternatives. The frustrating problem is that narrow-minded critics do not apply the standard they demand in fractals to the Euclidian physical forms with which they are familiar.

  12. Justin Justin
    December 16, 2014    

    Note that the butterfly does not power or directly create the tornado. The Butterfly effect does not convey the notion – as is often misconstrued – that the flap of the butterfly’s wings causes the tornado. The flap of the wings is a part of the initial conditions; one set of conditions leads to a tornado while the other set of conditions doesn’t. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale alterations of events. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different

  13. Poincaré Poincaré
    December 16, 2014    

    The n-body problem (gravity) is chaotic (might want to update your intro). Julia set fractals and the like are predictable in the same context as gravity, through iterative and rounded calculation. There is a difference between systems such as weather and stock markets which have many unknown variables, and systems like the purely mathematical n-body problem and the Mandelbrot set, with which we know all variables, but for which solutions require exhaustive calculation. These type of chaotic systems are chaotic in the sense that given some set of parameters, there exists no known “shortcut” equation to calculate the final result. Instead, the calculation expands into a massive Taylor series, which in practice we don’t/can’t expand fully and must truncate. Those n-body simulations of galaxies you see on youtube use non-relativistic gravity by the way, the relativist n-body problem is a whole other cat.

  14. Senex Senex
    December 14, 2014    

    Very intrigued by this “chaos theory”. However as Stephen has so graciously pointed out. The concept of a butterfly flapping its wings, thus causing a hurricane to occur in a country miles away seems a bit too far-fetched in my opinion. Now I do not entirely dismiss the butterfly effect as simple hogwash. I thoroughly believe that with time and effort one would be able to calculate the effects of butterflies around the world to prove whether or not that a simple butterfly flapping it’s wings can cause such strife and chaos in a country miles away. Simply fascinating. How even the most slightest of events can cause such chaos in other parts of the world. Even still, why does the whole idea of the butterfly effect apply only to butterflies. Is there not any other creature out there that can cause chaos in a country miles away? Similarly, like a butterfly, a lion roaring in sub-Saharan Africa, can cause a tsunami to occur in Indonesia. Or a Gorilla pounding its chest in a New York zoo, could cause an earthquake to occur somewhere in Spain? Can someone enlighten me further on this theory? Of what I say could be possible?
    Furthermore, if what Emily said was true, and that universes are in some way interconnected with everything. Think of the possibilities!

  15. Martha Martha
    November 17, 2014    

    Very interesting reading the comments posted and this very well put together site. The comments ranging from Mathematical sciences to Eastern Philosophy show that theories of Chaos have been studied for years. Energy can not be created nor destroyed, some people will even argue this fact, however the idea of a butterflies wings flapping causing exponential expressions until it causes a tropical storm may seem hard to imagine. Let’s try an easier example, I apologize in advance for the morbidity of the topic, but negative effects are easier to notice and more obvious for Chaos Theory. (See mythology for Chaos) Anyway, imagine the equation for nuclear reactions, let’s say an explosion, the amount of elemental particulates increases after the intial explosion, however the reaction continues, with the same formula’s until things are back in balance with our current environments of land, water, air to the molecular level sometimes within thousands of miles from the atomic explosion. You would think that as the environment reaches homeostasis, the Chaos Therory ceases to multiply, in fact it has and will continue until every exponential possibility has expired. (Never)

  16. emily emily
    November 17, 2014    

    i believe that everything in the universe is somehow interconnected. i mean, it makes sense how a butterfly could have so much force. even the smallest of animals can have so much power. by the way, this website explained everything so clearly to me. i have a school project, and i only went to this website, then i had all the information i needed.

  17. Stephen Garramone Stephen Garramone
    November 2, 2014    

    I don’t get it. In no way does a butterfly in China cause a hurricane in the Carribean. The physics concept of entropy explains the ability to mix gases but not unmix them. Entropy (randomness) in the Universe is always increasing. Gaseous expansion of a pure gas from a cylinder to a bigger cylinder can be done without the loss of energy but cannot be compressed back to the original state without the expenditure of energy. Theoretically, if the Universe is finite (albeit large) there will be an eventual run down of intrinsic energy to a steady state of equal energy throughout. Local zones in which there is an increase in energy such as a water hammer rely on more energy being expended by the surrounding environment. This is classic entropy. In a sense, these local zones of hyper energy will occur in ever decreasing amplitudes “forever” asymptotically approaching the eventual uniform mean. Predicting magnitude and location of these local zones of hyper energy. Maybe that is what chaos theory is all about.

  18. November 2, 2014    

    I agree with Molson (above). Chaotic systems may very well be deterministic, but not modelisable due to unstable coefficients, feedbacks, and phase changes. However,the patterns thrown-off therein are identifiable; that may be more useful than attempting to predict in non ergodicity. Great site, bravo.

  19. Doone Doone
    August 13, 2014    

    As I read all the content and comments in this article, I realize that: “Chaos theory is our key to understand every system in the universe no matter how complex it is.” ~Sir Doone, Philippines

  20. Raghavan Raghavan
    June 28, 2014    

    It sure appears that way, but as we move on, chaotic systems of today will be very predictable things as our data gets more robust and we are clear about what problems need accuracy (in initial conditions) with the accuracy we can provide.

  21. jiyoon jiyoon
    February 12, 2014    

    thanks!! it’s a wonderful site!! It helped me so much!!fantastic!! North Korea jiyoon

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