Naturally
Occurring Fractals
(including
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 2D to
3D 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.
Fractals in nature are ubiquitous  clouds, plants, galaxies, shells and more.

Fractal Mountains In Tibet Fractal Rivers in Savannah, Georgia

Mandelbrot set area displaying dendritic branching like river erosion patterns
Fractal Growth Pattern of a Leaf Chaotic Fractal Branching Pattern from Electrical Discharge

Natural Fractals are visible in these landscapes painted by Maxfield Parish (18701966)

Below: Fractal Electric Discharge Patterns Generated by
Placing a CD into a Microwave for approximately 4 seconds!!
Lightning Strikes and Electrical Discharge creates fractal
formations
called Lichtenberg figures on rocks, grass,wood or even
people!
From Tennessee's OakRidge 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
Lichtenberg Figures on a Patient Due to a Lightning Strike
by Yves Domart, MD, and Emmanuel Garet, MD,
3D FRACTALS
Familiar Forms; looking like waterspouts,
roots, clouds, and dragons.
to the LEFT we see a simulated OceanicScene with Fractal
Seaweed and
Coral composed from 3d rendered IFS fractals.
Below we see more amazing quaternion 4D julia sets.
Ernst Haeckel drawings of Fractal Life Forms from the sea
Cut into a Cabbage and you'll find a Fractal  Inside of a thin crosssection of wood stalk there is a circular Fractal

BELOW:
Virus, mold and bacterial aggregate colonies spontaneously
assume Fractal Shapes
(from Science
News)

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