Fractal Mathematics

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Fractal Mathematics
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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
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• 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.
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• 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
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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.
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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.
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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.
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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.
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I asked other fractal artists to create a fractal
that resembled bacteria growing in a petri dish.
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Everything in the cosmos, large and small
reflects the beauty of fractal mathematics.
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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.

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Exploring the Infinite

• How fractal mathematics can be used as a form of
  self-expression in art

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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
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Art Deco Circle
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Branching Out
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Eccleston
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Prisma
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Its Electric
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Steam Punk
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Dreamcatcher
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State of Medicinal Meditation
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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.
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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
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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
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Realities Crashing
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Erecting the New Zion
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Profonda
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Scarlet
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Cold Breeze
Death is an Angel
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Art Deco Collage
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Black Bird Fly
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Contessa
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Oceanic Opera Omnia
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Efflorescence
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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.
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Rhiannon
Ever and Anon
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Your Possible Pasts
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Storyteller
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Mermaid
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A Winter Fractal
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Return of the Swirly Queen
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Opus 5
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Totality of Physical Entities
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Liquid Dynamics
Legacy
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My love of fractals and patterns carries over
into my traditional artwork as well.
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Infinite Mask - Etching
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The Shore - Chameleon Lovers of the Land and Sea
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Seed