Naturally
Occurring Fractals
(incl:
plants, rivers, galaxies, clouds, weather, population
patterns, stocks, video feedback, crystal growth, etc.)
 The
geometry of Fractals brings us a new appreciation for
the natural world and the patterns we observe in it.
Many things previously called chaos are now known
to follow subtle subtle fractal laws of behavior. So many
things turned out to be fractal that the word "chaos"
itself (in operational science) had redefined, or actually
for the FIRST time Formally Defined as following
inherently unpredictable yet generally deterministic rules
based on nonlinear iterative equations. Fractals
are unpredictable in specific details yet deterministic
when viewed as a total pattern - in many ways this reflects
what we observe in the small details & total pattern
of life in all it's physical and mental varieties, too
....
FRACTAL FERN: One very simple way to understand fractals
and the meaning of "lteration" is to examine
a simple recursive operation that produces a fractal fern
thru a "chaos game' of generating random numbers
and then placing them on a grid.
After a few dozen repetitions or ITERATIONS the shape
we would recognize as a Perfect Fern appears from the
abstract world of math.
How and Why can this be?
The answer to why is that it Simply IS - and it's quite
surprising too!
Answering How is that nature always follows the simplest
& most efficient path. Fractals are maps of the simplest
paths sliding up the scale of Dimensions (from 2-D to
3-D and so on). So maybe it's simply an artifact of nature's
elegance that we find exact correspondences between these
inherently existing mathematical forms and natural patterns,
and even living creatures of many types.
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| Edible
Fractals: Romanesco (a cross between broccoli
and Cauliflower, which accentuates the great fractal
spiral patterns on the top. Tastes a-ok too) |
Iterative
Desert Landscape: this windswept landscape,
looking fa bit like sandstone is completely generated
by math/fractal equations |
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Lightning Strikes and Electrical Discharge creates fractal
formations
called Lichtenberg figures on rocks, grass,wood or even
people!
From Tennessee's Oak-Ridge Laboratory Museum
"This particular Lichtenberg Figure was created by
exposing a rotating lucite cylinder to the electron beam.
When the cylinder was discharged by striking one end with
a grounded rod, the electrons created a three dimensional
Lichtenberg Figure. Size: 12.5 " long, 2" diameter
Experiments in optical visualizations of magnetic fields reveal fractal shapes
The Ginger-Root-Dragon Looking object
in the large image below is a 4-Dimensional Julia-Set.
The Cross-section of this shape reveals the regular
2-D julia-sets we are familiar with
(if this is not making sense; read the section on Mandelbrot
and Julia sets)
Familiar Forms; looking like waterspouts,
roots, clouds, and dragons.
to the LEFT we see a simulated Oceanic-Scene with Fractal
Seaweed and Coral composed from 3-d rendered IFS fractals.
Below we see more amazing quaternion 4-D julia sets.
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from the Irish / Celtic Book of Kells: Stunning Examples
of (pre-Fractal) Fractal Art
The mind has always had a fixation
with recursive and fractal patterns,
largely because our environment is filled with them. Only
in the last 40 years have we been able
to finally describe this exquisitely subtle math of interacting
patterns mapped as dimensionality.
Interestingly the representations of fractals and therefore
INFINITY
most frequently appear in human art of the Religious and
Spiritual varieties
 
BELOW:
Virus, mold and bacterial aggregate colonies spontaneously
assume Fractal Shapes
(from Science
News)
Below: Snowflakes are naturally occurring Fractal Formations
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