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 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.

Fractals in nature are ubiquitous - clouds, plants, galaxies, shells and more.

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 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
      
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 Oceanic-Scene with Fractal Seaweed and
Coral composed from 3-d rendered IFS fractals.
Below we see more amazing quaternion 4-D 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 cross-section of wood stalk there is a circular Fractal
  

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