fractal theory

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. 

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. 

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. 

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.

Illustration of Pi Expanding Forever Closer to a Circle.

This is an illustration of an n-sided polygon with n=360 (or 360 right triangles that when you draw secant lines around the edge gives you an area equal to an n sided polygon with n=360). As n gets larger and approaches infinity the value approaches Pi forever because you are getting closer and closer to a circle for ever and as you fill in the edge of the circle (or it gets smoother as n gets larger). The area gets a little larger and the circumference get larger also as you add sides (as n grows larger), but the diameter stays the same. When you use secant lines (a line through two points on the edge of the ‘circle’ every one degree in this drawing) you are approaching Pi from the inside of the circle. This is the inner boundary of Pi. If you use tangent lines around the drawing (a line through only one point around the 'circle’) then as you add sides the value you get is larger than Pi but begins to get smaller and it approaches a Pi from the outside of the perimeter. This is the outer boundary of Pi. Then as the secant lines and tangent lines from the inner and outer boundary of Pi approach each other they trap Pi, or a shape forever getting smoother and smoother (a circle), forever between them.

Most interesting part is that perfect circles don’t exist. Illustration of Pi with 180 sides has big empty spaces on the edge of the circle, then when you look at this drawing with 360 sides you see that some of that empty space has been filled in so it is closer to a circle and then look at the drawing of Pi with 720 sides and you see that it fills in a little more of the space as it is even closer to a circle. So as you keep adding and adding sides and you get closer and closer to a circle forever but you never get all the way there. Just closer and closer forever. That is the beauty of Pi. 

The area of Pi with 180 sides is 3.141433159… When you have 360 sides like this drawing the area is 3.141552779… The area of the drawing of Pi with 720 sides is 3.141582685… So a reason Pi can never repeat itself is that each time you add sides to the 'circle’ you get a new and unique area and circumference. The can never find the 'end’ to Pi mathematically because you can add sides to a circle forever and get a larger and unique value as you forever approach an infinite number of sides.

They way Pi is calculated now is that they say let the number of sides to a n-sided polygon forever approach infinity and it is that diameter divided by its circumference that we will call Pi. The reason Pi can never end is because you can mathematically makes the sides to a 'circle’ smaller and smaller to infinity and the smaller the sides get the further the circumference gets. Your calculator says that x goes to infinity so no matter how many side the polygon has, Pi will always give you a value that is slightly too large. The only way you can avoid this problem with infinity is to apply the Planck length. The Planck length is the smallest observable distance. Once you have a circle where the sides are one Planck length the that may be the closest you can get to observing a perfect circle in our Universe.

Anton Stankowski

Anton Stankowski was a German graphic designer, photographer and painter. He developed an original Theory of Design and pioneered Constructive Graphic Art. Typical Stankowski designs attempt to illustrate processes or behaviours rather than objects. Such experiments resulted in the use of fractal-like structures long before their popularization. Enjoy: Pure Stankowski!

Giuseppe Peano, born on 27th August 1858, was an Italian mathematician. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation ( in his first book dealing with mathematical logic, the modern symbols for the union and intersection of sets appeared for the first time.). The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction.

Moreover, Peano’s famous space-filling curve appeared in 1890 as a counterexample. He used it to show that a continuous curve cannot always be enclosed in an arbitrarily small region. This was an early example of what came to be known as a fractal.

He has done so many things ^_^ 

Whether we’re gazing up through a telescope or down through a microscope, either way we’re looking at infinity unfolding in much the same patterns. This elegance and consistency seems to me rather more impressive than the petty myths of some desert-dwelling tribesmen a couple of thousand years ago.