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