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Fractal Mathematics

Fractal Logic

- One of the most recognizable fractals by far is

the Mandelbrot Set, discovered by Benoit

Mandelbrot when he programmed this nonlinear

equation into a computer. - Real numbers can be represented on a one

dimensional line called the real number line.

Negative numbers like -2 are plotted to the left

of zero and positive numbers like 2 are plotted

to the right of zero. - Complex numbers have two parts, a real one and an

imaginary one. They need a second dimension to

graph them so we simply add a vertical dimension

to the real number line for the imaginary part.

Since our graph is now two-dimensional, it is a

plane. We can graph any complex number on the

complex number plane.

Z Z 2 C

The Mandelbrot Set

- The Mandelbrot set is made of complex numbers, so

it needs to be graphed on the complex number

plane. However, first we have to find many

numbers that are part of the set. To do this we

need a test that will determine if a given number

is inside the set or outside the set. - C represents a constant number, meaning that it

does not change during the testing process. C is

the number we are testing, the point on the

complex plane that will be plotted when testing

is complete. Z starts out as zero, but it changes

as we repeatedly iterate this equation. With each

iteration we create a new Z that is equal to the

old Z squared plus the constant C . So the number

Z keeps changing throughout the test. - As we iterate our equation, the magnitude of Z

will do one of two things. It will either stay

equal to or below 2 forever, or it will

eventually surpass two. Once the magnitude of Z

surpasses 2, it will increase forever. In the

first case, where the magnitude of Z stays small,

the number we are testing is part of the

Mandelbrot set. If the magnitude of Z eventually

surpasses 2, the number is not part of the

Mandelbrot set.

Z Z 2 C

There are two major characteristics of the

Mandelbrot Set. First, it occupies a finite

space. You can see it in its entirety above.

Secondly, and paradoxically, its border is

infinite.

- If you look at the border of the set through

successively more powerful magnifications, you

will see parts of the fractal that are similar,

but not identical to those in the previous image. - This is called self-similarity.

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Here are two snapshots taken from different

portions of the same fractal. Besides the

coloring, we can see other similarities that

prove they are from the same equation.

Speak To Me

Fractals in Nature

Fractals do not only exist as computations they

can be found in nature as well. It seems

everything in the cosmos has been created with

the same basic principles found in fractal

geometry.

High voltage dielectric breakdown within a block

of Plexiglas creates a beautiful fractal pattern.

The branching discharges ultimately become

hair-like, but are thought to extend down to the

molecular level.

Real Fern Fractal Generated

Most plants show some form of branching. This

happens when the main stem (the trunk) splits

into a number of branches. Each of those branches

splits into smaller branches and so forth. A tree

branch usually looks similar to the entire tree

and a fern leaf looks almost identical to the

entire fern. This is another example of

self-similarity, one of the most important

properties of fractals.

This is a picture of Norway taken by satellite.

It shows the fractal nature of the rivers along

the coast which are very visible against the

snow. Scientists say that even our weather

patterns are fractal in nature.

I asked other fractal artists to create a fractal

that resembled bacteria growing in a petri dish.

Everything in the cosmos, large and small

reflects the beauty of fractal mathematics.

Application of Fractal Concepts

- Musicians and composers are experimenting with

fractal coordinates to define pitch. - Physicists use fractals to analyze the gait

patterns in Parkinsons sufferers which helps

monitor the progress of the disease so doctors

can and tailor their therapy and drug regime. - New camouflage patterns have been made from

fractal designs that prove more effective than

previous patterns. New technology allows the

fractal patterns to be adhered to everything from

weapons to vehicles. - Fractal compression is a lossy compression method

used to compress digital images using fractals.

The method is best suited for photographs of

natural scenes. The fractal compression technique

relies on the fact that in certain images, parts

of the image resemble other parts of the same

image.

- Benoit Mandelbrot discovered fractals 40 years

ago. Born in Poland, he was raised in France and

has done most of his work in the U.S. at

institutions such as Yale and IBM's research

laboratories. His mathematical models have been

applied to the stock market, to air currents, and

to the British coast.

Exploring the Infinite

- How fractal mathematics can be used as a form of

self-expression in art

Often times, the software is capable of

rendering magnificent images that already contain

all of the elements (like depth, color, shape and

intrigue) to create a good piece of art. These

images do not not require much manipulation to

look their best because the beauty lies in the

plotted mathematical equation alone.

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America

Art Deco Circle

Branching Out

Eccleston

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Prisma

Its Electric

Steam Punk

Dreamcatcher

State of Medicinal Meditation

Though a lot of work goes into creating and

exploring the fractal, sometimes its just the

beginning of the creative process.

This was the original image that was created in

Apophysis.

Using a photo editing program like Photoshop,

many filters, layers and color corrections can be

applied in order to alter the way a fractal looks.

Fractal purists are artists who feel that the

beauty lies in the perfection of the plotted

mathematics and will not alter the rendered

fractal. On the other hand, many traditional

and digital artists alike feel they should be

able to use whatever tools are available to them

in order to express themselves.

The Sound of Your Voice

The human eye is used to seeing the chaotic

patterns that occur in nature. Because of this,

I think sometimes fractals tend to appear

generic, lifeless and unmoving.

The Certainty of Chance

I try to avoid this downfall in my work by

incorporating items that subtly bring chaos,

chance and randomness to the image.

Heads With Ears

Realities Crashing

Erecting the New Zion

Profonda

Scarlet

Cold Breeze

Death is an Angel

Art Deco Collage

Black Bird Fly

Contessa

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Oceanic Opera Omnia

Efflorescence

Sometimes I choose to make the image look as

though it has been painted on canvas by applying

texture and stock images as layers in Photoshop.

Rhiannon

Ever and Anon

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Your Possible Pasts

Storyteller

Mermaid

A Winter Fractal

Return of the Swirly Queen

Opus 5

Totality of Physical Entities

Liquid Dynamics

Legacy

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My love of fractals and patterns carries over

into my traditional artwork as well.

Infinite Mask - Etching

The Shore - Chameleon Lovers of the Land and Sea

Seed

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