(almost)the
Mother of All Fractals: The Mandelbrot Set
incl: interesting
relationships, occurrence in other fractals, physical
constant, unexplained properties, etc
UPDATE: The man who invented the word FRACTAL & discoverer of the Mandelbrot set,
Benoit Mandelbrot (often called the Father of Fractal Geometry) has died.
 Benoît B. Mandelbrot (19242010)
Mandelbrot Zoom Sequence: Increasing Complexity
The
Mandelbrot set is the black rounded branching circular shape in the center,
it contains infinitely patterns and many copies of itself buried deep in the curls and branchings,
each one unique
and containing equally many submandelbrotsets.
Since it encloses a finite area on the complex plane the whole pattern (with all the curls,
crimps, turns & trillions of branches and spirals) is ALL
one single line enclosing the boudary in black. 






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 XY 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 Set is the SUM of ALL possible Julia Sets in
the Complex plane.
If you start at the needle and move to the inner cusp, it's a map of every possible curve or spiral.

Here's the Wikipedia summary of the Mset.
" the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal.
Mathematically, the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the
complex quadratic polynomial zn+1 = zn2 + c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when
starting with
z0=0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number
(that number
depends on c) however large n gets.
For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.
On the other hand, c = i gives the sequence 0, i, (−1 + i), −i, (−1 + i), −i…, which is bounded, and so i belongs to the Mandelbrot set.
When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an
elaborate boundary, which does not simplify at any given magnification. "  (end Wiki quote)
Below are a few SelfSame JuliaSets with lines
to where they correspond
to Points on the boundary of the SelfSimilar but neverrepeating
Mandelbrot Set.
(
image below by Paul
Bourke, Swinburne University AU)
For each point in the MSet there
is a corresponding Juliaset,
the difference is Jsets repeat themselves perfectly over
and over as you "zoom in"
by Iterating the equation into finer and finer points on
the grid.
The Mset 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 millioncoil spiral to lightning like fragments).
The Mset is the Master pattern to
ALL 2D curves, every possible combination is contained
within it's infinitely thin boundary.
Notice that below right
one jset is nearly a straight line, while on the
left we have a nearly perfect circle
the Mandelbrot set is these and everything inbetween 
a truly amazing discovery.

"The Mandelbrot set was named after the work of mathematician Benoit Mandelbrot in the 1980's, who was one of the early researchers in the field of dynamic complexity. The Mandelbrot set has a fractallike geometry, which means that it exhibits selfsimilarity at multiple scales. However, the smallscale details are not identical to the whole, and in fact, the set is infinitely complex, revealing new geometric surprises at ever increasing magnification. Belying this mindboggling complexity is the extremely simple mathematic process used to produce it. ... , to generate the set, take a complex number, multiply it by itself, and add it to the original number; take that result, multiply it by itself, and add it to the original number; and so on. If the resulting numbers generated during the iteration process grows ever and ever larger, then the original complex number C is not in the Mandelbrot set. If the sequence converges, drifts chaotically, or cycles periodically, then C is in the set. " (text from http://www.visualbots.com/mandelbrot_project.htm)
Counting bulbs & stalks on the Mandelbrot boundary (image:Chris King)

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 or intersect.
At Left we see the Mandelbrot relationship
to the perioddoubling
'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 nonlinear equations (Fractal
made from Newtons method of deriving a Cuberoot being
the most obvious.)
Also Mandelbrot curves have been discovered
in crosssections of magnetic field borders,
implying there is a 3D mandelbrot equivalent that is
closely tied to electromagnetism
and therefore a deep structural and fundamental
aspect
of life, and physical space/time.
(note: I read this in "Turbulent Mirror" can anyone cite a reference for this? email me design[at]miqel[dot]com)
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, yettobeclarified
fundamental importance of the Mandelbrot Set
in many physical process ....
not just visually
pleasant mathematical abstractions.
Here the Mandelbrot Set makes an appearance within the Newton Basin fractal
more views of Mandelbrot Set in Newton Basin fractal
There are zillions of ways to render the mandelbrot set
depending on which mathematical relationships you
desire to highlight.
This Rendering of the internal field relationships
is given a color gradient to almost resembles a Blue Rose.
An Inverse Mandelbrot takes the form of a Fractal TearDrop!

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
__
View of some of the Interior Relationships in the Mandelbrot
Set.
Beautiful organic forms deep within the mandelbrot fractal border
Fractal Humor
Mandelbrot Tattoo!!

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