(almost)the Mother of All Fractals: The Mandelbrot Set
incl: interesting relationships, occurrence in other fractals, physical constant, unexplained properties, etc.

the Mandelbrot set, perhaps the most famous Fractal. Stunning, enigmatic and potentially useful in future technological applications such as data storage, information analysis, even in fractal antennas.

To begin any introduction to the mandelbrot set we need to first mention Julia Sets.

The mandelbrot Set is a fractal
mapped on an X-Y Coordinate grid.
The mandelbrot is the fractal across the whole 'complex plane' or grid.
For EACH POINT on the grid
there is an infinitely repeating fractal shape called a julia set.
The Mandelbrot is the SUM of ALL possible Julia Sets in the Complex plane.

The Mandelbrot set is the rounded branching image in the center, it contains infinitely many copies of itself, each one unique
and containing equally many sub-mandelbrot-sets. All as one single line enclosing the Boudary in Black.

Below are a few Self-Same Julia-Sets with lines to where they correspond
to Points on the boundary of the Self-Similar but never-repeating Mandelbrot Set.
(this image by Paul Bourke, Swinburne University AU)


For each point in the M-Set there is a corresponding Julia-set,
the difference is J-sets repeat themselves perfectly over and over as you "zoom in"
by Iterating the equation into finer and finer points on the grid.
The M-set however changes constantly as you zoom in, and is a single continuous line that maps the transition
between Every possible julia set (from a straight line to a million-coil spiral to lightning like fragments).

In this way, the M-set is the Master pattern to ALL 2-D curves, every possible combination is contained within it's infinitely thin boundary.
Notice that below right one j-set is nearly a straight line, while on the left we have a nearly perfect circle
the Mandelbrot set is these and everything in-between - a truly amazing discovery.

Flower or Sea-Urchin-like Shapes in the M-set

The Mandelbrot set has a few unique properties Among Fractals:

It was proven to be the absolute maximum Space Filling Curve possible in 2 + Dimensions.
if the boundary region was one 'quanta' more curved inward on itself
it would HAVE TO overlap, popping into 3 dimensions.

At Left we see the Mandelbrot relationship to the period-doubling 'Chaos' equation
which is used to describe population expansion, plant growth, weather instability and a host of other physical processes. Also has a habit of "popping up" unexpectedly in other dynamic non-linear equations (Fractal made from Newtons method of deriving a Cube-root being the most obvious.)

Also Mandelbrot curves have been discovered
in cross-sections of magnetic field borders,
implying there is a 3-D mandelbrot equivalent that is closely tied to electromagnetism
and therefore a deep structural and fundamental aspect of life, and physical space/time.






Think about that for a moment, Visualize it - Taking a slice of the magnetic field of the earth, sun, a plant, the data on audio or video tape, and there's our old familiar buddha looking mandelblob -
ALL THIS DATA IS STORED AS THE MANDELBROT SET! Holy Kraap!
That's weeeeird, and beautiful too.


This suggests an unknown, yet-to-be-clarified
fundamental importance of the Mandelbrot Set
in many physical process ....
not just visually pleasant mathematical abstractions.

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BELOW:
A series of Mandelbrot images I made along the axis of the feature called the 'needle'
to highlight the pattern diversity in relatively featureless looking areas





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View of some of the Interior Relationships in the Mandelbrot Set.