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

Mandelbrot passes on - Oct 15 2010UPDATE: 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 (1924-2010)

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 sub-mandelbrot-sets. 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.
Mandelbrot sequence 1 Mandelbrot sequence 1 Mandelbrot sequence 1
Mandelbrot sequence 1 Mandelbrot sequence 1 Mandelbrot sequence 1

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 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.

rainbow mandelbrot

complax plane & mandelbrot set

Here's the Wikipedia summary of the M-set.

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 Self-Same Julia-Sets
with lines to where they correspond
to Points on the boundary of the Self-Similar but never-repeating Mandelbrot Set.
( image below 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).

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.

"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 fractal-like geometry, which means that it exhibits self-similarity at multiple scales. However, the small-scale 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 mind-boggling 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 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.
(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, yet-to-be-clarified
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
Mandelbrot set in Newton fractal
more views of Mandelbrot Set in Newton Basin fractal
Mandelbrot set in Newton fractal
Mandelbrot set in Newton 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 Tear-Drop!
inverse mandelbrot teardrop

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
organic fractal mandelbrot

Fractal Humor

Mandelbrot Tattoo!!
fractal mandelbrot tattoo

An Introduction to the Fascinating Patterns of Chaos & Visual Math
Platonic & Vectoral Geometry Simple Iterative Fractals Natural fractals
Multi Dimensional Forms Mandelbrot Set   |  Visual Mandelbrot Guide Fractal Types & Categories
PHI - the Golden Ratio Mandelbulb & 3 Dimensional Fractals Fractal "Wada" Basin Reflections
Minimal Surfaces Fractal Technology & Historical Fractals Other Fractals & Cool Patterns

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