“ Be the Change that you want to see in the world. ” ...is by Mahatma Gandhi...

17 agosto 2006

Simple Fractals

The geometry of Fractals lies somewhere between dimensions. To be totally accurate "fractal" is even not a 'thing' at all but more like a unit of measure or mathematical characteristic. For example each fractal has a 'fractal dimension' which is it's degree of regularity and repetition.

One very simple way to understand fractals and the meaning of "iteration" is to examine a simple recursive operation that produces a fractal pattern known as Cantor Set. you take a line of arbitrary length and remove the middle third. this is the first step or "Iteration", then take the remaining two lines and repeat the clipping procedure. Eventually after 5 or 10 iterations you have dozens of tiny lines which take up only as much room as the two original ones from the first step.

From Wikipedia "The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. The Cantor set is defined by repeatedly removing the middle thirds of line segments. The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 1/3 and translated. Its Hausdorff dimension is equal to ln(2)/ln(3). It can be formed by intersecting a Sierpinski carpet (see below) with any of its lines of reflectional symmetry (such as reading the center scanline).

THE KOCH SNOWFLAKE: example at right is similar, except rather than subtracting the middle of the line in each step or iteration - we add a triangular bulge to each line, and then to each resulting line ... and so forth until the border goes from a triangle to a star, to a wrinkled snowflake.

This also illustrates a fundamental property of fractals .. infinite boundaries.

Examine the illustration again. In step one (the triangle) the top line is length x. lets say it's 100 centimeters in length. In step 2 (the star) this same line is roughly twice as long because of the extra length of the added sub-triangles.

By the sixth iteration the top line has become 1000 times as long as the starting line, but without stretching past the original starting points (a & b).Fractals can enclose a finite area with an infinitely long & intricate line or boundary.

From Wikipedia "The Koch Curve is a mathematical curve, and one of the earliest fractal curves to have been described. It appeared in a 1904 paper entitled "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire" by the Swedish mathematician Helge von Koch. The better known Koch Snowflake (or Koch Star) is the same as the curve, except it starts with an equilateral triangle"

Animation of Koch Curve Iteration

The Opposite of the KOCH CURVE is....

(produced by subtracting triangles from the interior,
instead of adding them to the surface as in the Koch curve)

Perfect Fern
Produced by Chaotic Iterative Process

Pythagorean Trees

In addition to Ferns, Fractal iterative shapes also reveal tree-like structures,
which can take on an infinite variety of formations

A couple of pentagonal fractal possibilities

The Apollony fractal (images by Paul Bourke, Swinburne University AU)
packing Circles or Spheres Together - Another Way of Creating Fractals

(images by Paul Bourke, Swinburne University AU)
packing Circles or Spheres Together - Another Way of Creating Fractals
(image by Paul Bourke, Swinburne University AU)

Side-View of same fractal as above
(image by Paul Bourke, Swinburne University AU)

Queen Anne's Lace is another beautiful fractal in Nature

Almost like a fractal braid or chain-link.
By Roger Bagula, image by Paul Bourke, Swinburne University AU

Realistic Sponge-Like Fractal
made only from Circles:

Dodecahedral Fractal/Harmonic Subdivision of Spherical Surface

the major shape-varieties of Julia Sets
which correspond to regions in the Mandelbrot Set

Crop-Circle Designs are often simple Fractal Iterations such as the
5-way repeating vortex (left), or the fractal pentagonal / golden ratio sub-division (right)