Simple
Iterative 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.
CANTOR SET:
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 selfsimilar, 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
subtriangles.
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"



Realistic SpongeLike Fractal
made only from Circles:
BELOW:
Dodecahedral
Fractal/Harmonic Subdivision of Spherical Surface

the major shapevarieties of Julia Sets
which correspond to regions in the Mandelbrot Set

CropCircle
Designs are often simple Fractal Iterations such
as the
5way repeating vortex (left), or the fractal pentagonal
/ golden ratio subdivision (right)


NEW!! HYPERDIMENSIONAL
Paul Laffoley Posters
: DIMENSIONALITY, THANATON III & KALIYUGA are Available Now at Miqel.com!
CLICK TO VIEW LARGE PICS
My KPT Fraxplorer Gallery
