March 2005 | Draft (Uncompleted)
Features of Mandelbrot and Julia Sets
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Definitions
Axes: In Figure 1, positive "x" values lie above the horizontal
axis and imaginary "y" values are left and right. These appear to
be widely used axes, common in early mathematics. In fact the "imaginary"
nature of the "y" axis (right and left) is a mathematical device
to enable the representation of complex numbers. The axes are therefore not
of the same kind.
Complex points (numbers): The M-set is a mathematical set, a collection
of numbers. These numbers are different than the real numbers with which people
are familiar and are termed "complex numbers". They have a "real"
part and an "imaginary" part. The real part is an ordinary number,
for example, -2. The imaginary part is a real number times a special number
called i, for example, 3i. An example of a complex number would be -2 + 3i.
The number i was invented because no real number can be squared (multiplied
by itself) and result in a negative number. This means that you can not take
the square root of a negative number and get a real number. When you take the
square root of a number, you find a number that can be squared to get that number.
The number i is defined to be the square root of -1. This means that i squared
is equal to -1. So when you square an imaginary number you can get a negative
number. For example, 3i squared is -9.
xxxxxxx ????
Points: The axes permit various complex numbers to be positioned in
relation to one another in a systematic manner:
Fixed point: The M-set has only one fixed point, the origin. A point
whose value under a mapping function is itself. Many J-sets have fixed points.
"indifferent" fixed points, which are neither attractive or repellent, arise
for values of c right on the cardioid including at the touching points of
buds.
Thus we have three regions associated with f(z)=z*z. Two regions tend to
two of the fixed points of f, namely 0 and infinity. We define a fixed point
to be an "attracting fixed point" if all points near the fixed point have
orbits that have that fixed point as a limit. Thus 0 and infinity are attracting
fixed points, and the regions that tend to them are their "basins of attraction."
The fixed point 1 is not attracting since it has points nearby whose orbits
do not tend to 1. The third region is the unit circle. No point there has
an orbit that has an attracting fixed point as a limit.
Peitgen elaborate that fixed points are repelling when the complex derivative
at that point has absolute value >1, attracting when <1, and "indifferent"
when =1
For the special case where c=0+0i, the Julia set is simply a (nonfractal)
circle with radius 1
Fixed pointy in peoples lives -- home birth placeetc
Origin: This is the point at which the axes cross, defined as (x=0;
y= 0).
- Given a family of complex iterative maps, the set of all parameter values
that produce wholly connected J-sets is determined by the behavior of a
single seed value: the origin.
Birth place, start of argument, etc
Complex points: These are the positions of complex numbers taken up
in terms of the axes.
Points of an argument. Where we have got to in our relationship
Critical point: This is the starting point for the process of generation
of the M-set and is in this case equivalent to the origin.
- Zero is the critical point of z^2+c of the M-set, that is, a point where
d/dz (z^2+c) = 0. If a different function is used, the starting value will
have to be modified. Start with a z of 0+0i, the origin of the complex plane
-- the critical point for this equation.
- critical points (points where the derivative of the function vanishes)
of a function with the behavior of iterations.
- The choice of the point z (x = 0.0, iy = 0.0) is related to a Julia-Fatous
theorem stating that every immediate and connected basin of attraction (the
so-called Fatou set) includes a critical point (where the first order derivative
vanishes).
- The converse is also true: if the critical point belongs to a basin of
attraction then its orbit stays bounded (since it's convergent to the limit
point of the basin) in the same basin, which shows to be both immediate
and connected.
- An interesting corollary is that the number of distinct critical points
is the same as the number of the basins of attraction.
- Critical points are important because every attracting cycle for a polynomial
or rational function attracts at least one critical point. In some cases,
there may be multiple critical values, so they all should be tested. Thus,
testing the critical point shows if there is any stable attractive cycle.
- Mandelbrot was looking for some sort of clue as to which c numbers made
disconnected sets, and which made connected sets. It turns out that the
test is easy. You just start with a z of 0+0i, the origin of the complex
plane. This is called a "critical" point for this equation. If this point
is class (1), the Julia set is of the dust type. If this point is class
(2), the Julia set is of the solid type. And in the rare case where it's
class (3), the Julia set is of the dendritic type. The interesting thing
is that if you plot all these critical points on the screen, coloring based
on whether you get a class (1), (2), or (3) Julia set, you get a different
fractal - sort of a "master" Julia set:
Critical starting point in discussion? Kairos Dramatic moment Critical popint
in a negotiation
Iterative generation of M-set: Chaos occurs in objects like quadratic
equations when they are regarded as dynamical systems by treating simple mathematical
operations like taking the square root, squaring, or cubing and repeating the
same procedure over and over, using the output of the previous operation as
the input for the next (iteration). This procedure generates a list of real
or complex numbers that are changing as the procedure continues -- a dynamic
system. The M-set is a set of points that fail to escape to infinity under an
iterated point process.
It is amazing that the orbit of 0 "knows" the shape of the filled Julia set
for x2 + c. The reason that 0 is so special stems from the fact that 0 is the
critical point of x2 + c. That is, the derivative of x2 + c is 2x, and this
derivative only vanishes at x=0.
Indeed, it is not possible to determine whether certain c-values lie in the
Mandelbrot set. We can only iterate a finite number of times to determine if
a point lies in M . Certain c-values close to the boundary of M have orbits
that escape only after a very large number of iterations. A second question
is: How do we know that the orbit of 0 under x2 + c really does escape to infinity?
Fortunately, there is an easy criterion which helps:
Computation of complex points: The M-set is defined as the set of
points c in the complex plane for which the iteratively defined sequence zn+1=zn2
+ c : does not tend to infinity (where zo=0 and c=x + iy). Each
such number being transformed into its corresponding image point (a pixel
for display purposes)
- To see if a point is part of the M-set, just take a complex number z.
Square it, then add the original number. Then square the result, and add
the original number. Repeat this process, and if the number keeps on going
up to infinity, it is not part of the M-set. In other words, for each complex
point c (displayed as a single pixel), start with z=0. Repeat z=z2+c
up to N iterations, exiting if the magnitude of z gets large. If the iteration
loop finishes, the point is probably inside the M-set.
- If a point, under the generative operation of iterated squaring, gets
more than a distance of 2 from 0+0i then it is not in the M-set. The set
of complex points that are in the Mandelbrot set are thus within 2 units
of the origin.
In other words, our map is z ? z2 + c, where c is the first complex number
in our iteration. So the sequence goes c, c2 + c, (c2 + c)2 + c, ((c2 + c)2
+ c)2 + c, etc. When we use this process, the not-sent-to-infinity set is
called the Mandelbrot set. There are many Julia sets, but only one Mandelbrot
set.
Iterations: The maximum number of iterations (N) used in testing points
in the computation, can be selected as desired, for instance 100. Larger N
will give sharper detail but take longer.
- If the point is beyond 2 units from the origin, the point is therefore
outside the M-set (and z will tend to go to infinity). It can be colored
(see below) according to how many iterations were completed.
Human life is characterized by iterative processes. Physiologically these
include breathing and the pumping action of the heart. Vision is based on rapid
eye movement (REM). The circadian rhythm of the waking/sleeping cycle can also
be understood in this way, as can the cycle of consumption/excretion. Many habits
are characteristically iterative, as is engaging in sex. The succession of human
generations, through which society (and the planetary surface) is populated,
may also be considered iterative. A number of religions hold strong convictions
regarding reincarnation, itself an iterative process. Within society
there are many regular processes that can be usefully seen as iterative: rituals,
regular meetings, festivals, etc that provide benchmark points indicative of
its status. The most fundamental debates have an iterative aspect as the same
points are explored are explored again and again.
** dilemma: going round and round without resolution / periodic cycle
Surfaces and volumes: The M-set can be represented on a surface or on
a volume.
Complex plane: A complex plane can be thought of as being the set
of all complex numbers, and then used as a way of visualizing relationships
between those numbers using images. The cardioid characteristic of the M-set
(as discussed here) emerges through the representation of the M-set on a complex
plane.
- Every point in the plane of complex numbers is either outside the M-set,
infinite, or inside of it, finite.
- The M-set fractal thus portrays two-dimensionally the infinity between
the whole numbers zero and one, the potential and the actual.
- Views of the M-set may therefore be thought of as being views of a subset
of the complex plane.
- In general, a M-set marks the set of points in the complex plane such
that the corresponding J-set is connected and not computable.
Complex sphere: The complex numbers may alternatively be represented
as points on a complex sphere. The origin (below) would then be one pole and
infinity the other pole (above) with the unit sphere being the equator. Lit
from "infinity", points on the sphere would leave shadows in unique
positions on the complex plane (as described above). The complex plane is
thus a projection of the complex sphere.
Scope
Size: It can be shown that the entire M-set lies inside a disk of
radius 2, centered at the origin.
Boundedness: A complex number c is in the M-set if the iteration of
z2+c (beginning with zero) remains bounded. If |z| exceeds 2, the
z sequence diverges. The boundary of the set of complex values of c -- such
that z does not escape to infinity -- is very complicated. The M-set therefore
lies within |c|<=2.
Boundary zone: We may take the view that the process of self-organization
takes place at the ‘edge of chaos’, where the system is able to poise itself
at a position of optimum fitness, between (ultimately stultifying) stability
and the chaos and unpredictability (and therefore unmanageability) of some
form of strange attractor. Such a situation is often depicted by the infinite
(fractal) variety of the boundaries of the Mandelbrot or J-sets. Systems in
the edge of chaos position may show a pattern of what is called punctuated
equilibrium in which phases of seeming equilibrium are interspersed with what
appears to be chaotic behaviour; so a strange attractors is the focal point
of a phase of equilibrium.
Dimension: Dimension is a measurement of how complex an object is.
We are used to the idea of a point being zero dimensional, a line being one
dimensional, and a solid square being two dimensional. Present conceptions
of dimensionality and fractals are practical working definitions and are by
no means rigorously defined. In general terms, the M-set has a dimension of
2 because the entire set is contained in a disk which has a dimension of 2.
The topological dimension of the M-set is 1 -- the boundary has an empty interior,
so the dimension must be less than 2. Despite its complexity, the M-Set has
a singular fractal dimension. Whereas other fractals have a non-integer dimension,
the M-set fractal dimension is 2.
For some types of functions, the set of numbers that yield chaotic or unpredictable
behavior in the plane is called the Julia set These Julia sets are complicated
even for quadratic equations. They are examples of fractals - sets which,
when magnified over and over, always resemble the original image. The closer
you look at a fractal, the more you see exactly the same object. Fractals
naturally have a dimension that is not an integer - not 1 or 2, but often
somewhere in between.
Human experience may be understood as lying within the world of polarization
and duality. The coherence and integrity of human experience -- any sense
of unity -- therefore emerge within the framework of that duality.
Sets and connectedness -- J-set and M-set
A dynamical system is generally
defined mathematically on a configuration space consisting of a topological
manifold such as a plane, whether or not it is complex. Non-linear dynamical
systems may be defined on such a plane by quadratic functions (of the form f(x)
= ax2 + bx + c, where a, b, and c are numbers with a not equal to
zero). Under suitable conditions, deviations from linearity result in complex
chaotic behavior (in which the orbits of the system are attracted to a complex
higher-dimensional subset called a strange attractor, or are ergodic). For some
types of functions, the set of numbers that yield chaotic or unpredictable behavior
in the plane is called the Julia set (J-sets). These sets are complicated even
for quadratic equations. They are examples of fractals - sets which, when magnified
over and over, always resemble the original image.
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Dynamic system
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Julia
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Mandelbrot
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Non-linear system
Any curve, system, or set of equations that cannot be differentiated,
namely lines cannot be found to approximate the rate of change of the
system at any given point.
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| basic definitions |
A Julia set is almost the same thing. It is defined to be : the set of
all the complex numbers, z, such that the iteration of f(z) -- > z 2 +
c is bounded for a particular value of c. Again, more simply put it is
the graph of all the complex numbers z, that do not go to infinity when
iterated in f(z) -- > z 2 + c, where c is constant.
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the definition of the Mandelbrot set is : the set of all the
complex numbers, c, such that the iteration of f(z) -- > z 2 + c is bounded
(starting with z =0 + 0i). More simply put, the Mandelbrot set is the graph
of all the complex numbers c, that do not go to infinity when iterated in
f(z) -- > z 2 + c, with a starting value of z =0 + 0i. |
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Generation / Iteration
The results of a procedure whose result is fed back into the same
procedure many times
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Selection of a particular point, then multiplying
every other point by it repeatedly, then adding the original point at each
iteration. |
Multiplication of every point on a complex plane
by itself, repeatedly, adding the oriinal point at each repetition. |
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The set associated with the function z = z2 + c,
where c is an arbitrary constant. The J-set therefore iterates z2
+ c for fixed c and varying starting z values. Any set containing
only those points that remain stable during iteration. |
The set of points that do not escape to infinity when the
function z = z2 + c is iterated, where c is the point itself
and z starts at the orgin. The M-set iterates z2 + c with
z starting at 0 and varying c |
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Julia sets come from iterating a map; that is, applying a function again
and again.
The function that is iterated can be practically anything, as long as
it uses complex numbers.
Quadratic Julia sets are generated by the quadratic mapping (2) for fixed
c.
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| Space |
J-set is in dynamical or variable space (z-plane). |
M-set is in parameter space (c-plane) |
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| Number |
There are an infinite number of different J-sets possible,
each |
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| Specification |
Defined for a given value of c -- a complex number, but for
any given Julia set, it is held constant |
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The Julia set J(c) is made of all points z , which
do not go to an attractor (or infinity) under iterations. |
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Orbits
The trajectory of a point or other object, whether through physical
space (eg a planet) or through mathematical space (eg a complex plane).
To study this equation as a dynamic system, we use an iterative process
whereby we input the initial condition, compute the output, and then feed
the output back into the original equation. This list of successive iterations
is called the orbit of the given initial condition. Orbits about an attractor
can be super-stable, periodic, or chaotic. Given a specific choice of
c and z0, the iterative recursion leads to a sequence of complex
numbers z1, z2, z3 -- called the orbit
of z0. Depending on the exact choice of c and z0,
a large range of orbit patterns are possible. All sequences of z computed
through the iterative equation will fall into one of three classes of
behaviour:
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| Behaviour 1 |
Convergence: the sequence of points {xk} converges to a limit
Values increase without bound (towards infinity), the J-set is then known
as of the dust type. For a given fixed c, most choices of z0
yield orbits that tend towards infinity.
Some Julia sets may consist of many disconnected points (called "dust
sets"). The further from the origin, the quicker the J-sets break up and
fall into Cantor dust. either f(z) can continue to grow without bounds
or it will stay bounded.
Points z0 in the complex plane that do not stay bounded with successive
iterations of f(z) are said to be in the escape set Ec.
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The M-set graphically depicts for which c-values
the orbit of fc (0) will have an attracting fixed point (main cardioid),
an attracting periodic orbit (primary bulbs), or will diverge to infinity
(colored region). |
| Behaviour 2 |
Periodic cycle: for some p>0 x0=xp so that the sequence repeats itself
Values collapse (to zero), the J-set is then of the solid type. others
from larger "solid" areas that seem all connected.
For some values of c certain choices of z0 yield orbits that
eventually go into a periodic loop.
All other points in the complex plane stay bounded as n is taken to infinity
-- they are termed prisoners and are said to be in the prisoner set Pc
defined for a given c.
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Since for c inside M, the iterations remain bounded, pixels
corresponding to the Mandelbrot set consume the greatest amount of the computational
time. However, iterations inside M evolve differently depending on the value
of c. (Behavior of the iterations is related to the appearance of the Julia
sets Jc.) For example, for c inside the big cardioid, the iterations converge.
For c inside the big circle to the left of the cardioid, the iterations
converge to a cycle of period 2. For c inside each wart attached to the
cardioid, the iterations converge to a periodic cycle whose period is determined
by the corresponding wart. |
| Behaviour 3 |
Chaos: none of the above. The points {xk} go from one place to another
in apparently chaotic manner. The set of points with chaotic orbits is
called the Julia set for a given function f.
Values change, but do not seem be (1) or (2). J-sets are strictly defined
as class (3) points, when they drift around to other class (3) points
and do not tend towards zero or infinity. Finally, some starting values
yield orbits that appear to dance around the complex plane, apparently
at random. Points in the Julia set, however, are said to be chaotic, meaning
that very small differences in points tend to show wildly different results.Julia
sets are all points where the sequence of z values change, often drastically,
but do not approach infinity nor zero.
The J-set is of the dendritic type. some form thin, wiggly lines that
are all connected but do not outline any shapes ("dendritic" types).
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Attractor
A value, or set of values, to which an iterated mathematical function
converges, no matter the initial value. While infinity is a point attractor,
depending on the choice of c, there may also exist one other attractor
in the system. For two dimesional functions, a region of points with attractors
is termed a basin of attraction.
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Infinity as a point attractor. If there is no second attractor
(i.e., infinity is the only attractor) then the Julia set is a disconnected
Cantor dust set. |
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If this second attractor does exist for a particular c, then
the Julia set is topologically connected, and is in fact the boundary between
the basin of attraction to infinity and the basin of attraction to the finite
attractor. |
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, the second attractor may be either a point attractor or
a periodic cycle. (A point attractor is essentially a periodic cycle of
period 1.) The exact shape of the basin of attraction to this second attractor
depends on c. |
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Connectedness / (Boundedness?)
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There are an infinite number of different J-sets possible.
But unlike the M-set grounded in zero where the black portions are all connected
with each other in the complex plane, the different J-sets are disconnected
with each other. |
An M-set is the set of all parameter values whose J-sets are
wholly connected.The M-set is a wholly connected archipelago of self-similar
islets linked by an array of extremely twisty, ever branching fibers. |
| Connected 1 |
Topologically equivalent to a severely deformed circle. Origin
trapped inside the set, the set is topologically equivalent to a circle
and thus is wholly connected. |
Mandelbrot has discovered the set M of parameter values c
for which Julia sets are connected. This set that now bears his name may
also be defined as the set of c's for which iterations {zk} starting with
z0 = 0 remain bounded. ????? |
| Connected 2 |
Topologically equivalent to a curve (or line) with an infinite
series of branches and sub-branches called a dendrite (e.g., the Julia set
for c=0+i)" (Elert 22.shtml). Origin is a part of the set, the set is dendritic. |
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| Disconnected |
Orbit of the origin eventually escapes to infinity. Disconnected
sets are completely disconnected into a countably infinite assembly of isolated
points. In addition, these points are arranged in dense groups such that
any finite disk surrounding a point contains at least one other point in
the set. Such sets are said to be dustlike. As they can be shown to be similar
to the Cantor middle thirds set, they are often called Cantor dusts. |
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We do not have to calculate the whole Julia set, we only have
to examine the orbits of specific points: These specific points are the
critical points, i.e. all points where the first derivation vanishes (f'(z)=0),
as defined above. The orbits of the critical points define the type of the
Julia set: If all orbits stay limited, then the Julia set is connected.
If at least one orbit tends to converge to infinity, then the Julia set
is dust like. |
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Graphic representation
J-sets and M-sets have a close relation to each other.
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The image around a selected point in an M-set resembles the
image of the associated J-set. |
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Julia set (J-set)
Quadratic J-sets are the family of sets generated
by the special quadratic case form f(z) = z2 + c. Here z represents a variable
of the form x+iy (x and y real numbers) which can take on all values in the
complex plane. This is sometimes referred to as a quadratic map, and is a
type of dynamical system. Each point in the complex plane corresponds to
a different J-set derived from a function z representing a variable (of the
form a+ib) which can take on all values in the complex plane.. *** Points
in the Fatou set tend to stick together; that is, points close to each other
will follow similar paths, drawing closer to either infinity or zero. http://www.mcgoodwin.net/julia/juliajewels.html
- It is the ones with smaller values of c (i.e., |c| < ~ 2) are particularly
interesting graphically. For almost every c, the function generates a fractal
(for c = -2 and c = 0 ).
- There are several types of J-sets. The broadest distinction, though, is
whether there is an "inside" to the J-set or not.
- ** For any given starting value of z, say z0, there are two possibilities
for what will happen to the iterated values of f(z) as n increases toward
infinity:
- ** All points must either be in one or the other set. The common boundary
between the escape set and the prisoner set is called the Julia set Jc,
defined for a particular value of c.
- ** In fractalspeak, infinity and zero are called "attractors" because
lots of points end up heading towards (are attracted by) these places. All
the points that fall into class (3) are parts of the "Julia Set".
- ** The values that fit in the third category are said to be in the Julia
set.
- *** Our functions are often polynomials or rational functions and are
all defined on the Reimann sphere, which is the plane of complex numbers
along with a point at infinity,
- Points in the Julia set tend to drift to other such points, and their
graphs may connect.
- A J-set is the boundary of all the attractor basins. It may also be described
as the closure of all the repellors.
- A J-set is effectively an event horizon within a phase-state description
of a discrete non-linear dynamic process.
- Julia sets can also be formed from higher degree and more complex expressions.
The following are Julia sets for the iterated functions f(z) = z4 + c and
f(z) = z5 + c, respectively:
- The black points in graphic representations of these sets are the non-chaotic
points, representing values that under iteration eventually tend to cycle
between three different points in the plane so that their dynamical behavior
is predictable. Other points are points that "escape," tending to infinity
under iteration. The boundary between these two points of behavior - the
interface between the escaping and the cycling points - is the Julia set.
- The equation for the quadratic Julia set is a conformal mapping, so angles
are presented.
- For the special case where c=0+0i, the Julia set is simply a (nonfractal)
circle with radius 1
- (That is, the modulus | zn | grows without limit as n increases.) (This
is an example of chaos.) These starting values make up the Julia set of
the map, denoted Jc. Some authors also define the filled-in Julia set, denoted
Kc, which is the set of all z0 with yield orbits which do not tend towards
infinity. The "normal" Julia set Jc is the edge of the filled-in Julia set.
- For a function f, its filled-in Julia set Kf is defined as the set of
starting points z0 for which the iterations {zk} remain bounded. The boundary
of Kf is known as the Julia set, Jf. The Julia set of f(z) = z2 is the unit
circle, its filled-in Julia set is the unit disk (the unit circle plus its
interior.)
- http://www.cut-the-knot.org/ctk/Mandel.shtml
With respect to human behaviour and understanding, a J-set might be usefully
described as a "pattern". A distinction can then be made between
three kinds of pattern:
- Essentially unstable patterns that persist only briefly, if at all,
and may only be briefly assumed to have any existence. These are the behaviours
which seem to be part of an enduring pattern but more or less quickly
prove not to be. Equally they are the modes of thought which may breiefly
appear to be consistent, but quickly prove not to be.
- These are patterns which are essentially habitual and unvarying,
consistent with a single general pattern of behaviour of which they are
an exemplification.
Connectedness
- Depending on the value of c selected, the resultant Julia set may be connected
or disconnected--in fact, either totally connected or totally disconnected.
- The J-set is either a connected set or a Cantor set. A connected set consists
of one piece whereas a Cantor set consists of an uncountably infinite set
of disjoint points. If the J-set is a Cantor set, then any arbitrarily small
neighborhood around any point contains a scattered cloud of infinitely many
points which do not touch
- A set of points is connected if, for any two points in the set, there
is at least one path consisting entirely of points in the set, which leads
from one point to the other.
- The studies about Mandelbrot and Julia Set demonstrate that Mandelbrot
and some Julia Sets are pathwise-connected, so that each pair of points
belonging to a Julia/Mandelbrot set can be connected by a path, i.e. a subset
of points belonging to the set.
- There are several types of mathematical connectedness and I have not taken
the opportunity to explore this topic in detail. In the case of Julia sets,
I understand the type of connectedness referred to is "pathwise" connectedness,
meaning that one can trace a path from a point in the set to other points
in the set without leaving the set (Gagliardo).
- Connected Julia sets are "completely connected" as opposed to being merely
"locally connected"
- Note that graphical displays of connected Julia sets often appear to demonstrate
separate subsets even though they are in fact connected.
Mandelbrot set (M-set)
The M-set is an answer to any query regarding
the existence of an organizing principle for the infinite number of possible
J-sets -- namely an organizing principle that classifies these J-sets.
- The M-set consists of all complex numbers c so that the J-set of z^2 +
c is a connected set. The M-set corresponds to the points c whose J-set
include the origin 0. These J-sets are those which are fully connected.
Each point c in the M-set specifies the geometric structure of the corresponding
J-set. If c is in the M-set, the J-set will be connected. If c is not in
the M-set, the J-set will be a Cantor dust.
- This set that now bears his name may also be defined as the set of c's
for which iterations {zk} starting with z0 = 0 remain bounded.
- A Julia Set depends on an iteration exactly like that for the Mandelbrot
Set except that the initial value of z is the complex number representing
the point whose membership is to be tested, and c is a parameter of the
set.
- The totality of all possible Julia sets for quadratic functions is called
the Mandelbrot set
- The points within the M-set correspond precisely to the connected J-sets,
and the points outside correspond to disconnected ones. In general, a M-set
marks the set of points in the complex plane such that the corresponding
J-set is connected and not computable. Unlike those in the M-set grounded
in zero, and therefore connected, the other J-sets are disconnected with
each other. These Cantor Sets are fragmented into infinitely many pieces.
The further from the edge of the M-set, the quicker the J-sets break up
and fall into Cantor dust.
- The main purpose of the M-set is to index J-sets corresponding to various
values of the parameter c. When c belongs to the M-set, Jc is connected.
For c outside M, Jc is totally disconnected and known as the fractal dust.
- The M-set (as a totally distinct fractal) becomes apparent as a form of
"master" J-set:
- if the critical points giving rise to the three different types of
J-set are coloured differently.
- if a colour value to a pixel depending on how fast it was found out
whether iterations for that pixel escape to infinity.
- Since the M-set is an amalgamation of all J-sets, the detail that becomes
apparent on zooming is based on the precise location of the zoom. Different
locations will therefore give rise to different detail -- often similar
in shape to Julia sets taken from that area.
- In the iteration of the complex quadratic map, there is a unique trapping
set Tc and a corresponding escape set Ec. The J-set (Jc) is the boundary
between the set Tc and the set Ec.
- Those with a value of c just on the outer border of the M-set are the
most complex and beautifully ornate of all.
- Alternatively, it can be defined as the set of values of c for which the
orbits (successive iterations) of z0 = 0+0i remain bounded
- The boundary of the Mandelbrot set acts as a catalog for the shapes of
Julia sets. That is, the Julia set corresponding to a point c in the boundary
of the Mandelbrot set will have as part or most of its shape an infinite
repetition of the shape of the Mandelbrot set near c.
- Each point in a Mandelbrot set shows us the type of the Julia set: If
we take an arbitrary point 'c' and it lies 'inside' the Mandelbrot set (i.e.
normally the black region), then this tells us, that the Julia set J(z^2+c)
is connected. If we take another point 'd' from 'outside' the Mandelbrot
set, then this tells us, that the Julia set J(z^2+d) is dust like.
- The family of functions f(z)=z*z+c as c varies over the complex plane
has its dynamic behavior crudely classified by the Mandelbrot set: the function
f(z)=z*z+c has a connected Julia set if and only if c is in the Mandelbrot
set.
- The Mandelbrot set gives a bit more information since the shape of the
Mandelbrot set near c gives hints as to the appearance of the Julia set
for f(z)=z*z+c.
- The totality of all possible Julia sets for quadratic functions is called
the Mandelbrot set: a dictionary or picture book of all possible quadratic
Julia sets.
- The Mandelbrot set completely characterizes the Julia sets of quadratic
functions, and has been called one of the most intricate and beautiful objects
in mathematics.
- The main purpose of the Mandelbrot set is to index Julia sets corresponding
to various values of the parameter c. When c belongs to the Mandelbrot set,
Jc is connected. For c outside M, Jc is totally disconnected and known as
the fractal dust.
- The boundary of the Mandelbrot set acts as a catalog for the shapes of
Julia sets.
- That is, the Julia set corresponding to a point c in the boundary of the
Mandelbrot set will have as part or most of its shape an infinite repetition
of the shape of the Mandelbrot set near c.
- Each point in a Mandelbrot set shows us the type of the Julia set:
- If we take an arbitrary point 'c' and it lies 'inside' the Mandelbrot
set (i.e. normally the black region), then this tells us, that the Julia
set J(z^2+c) is connected. If we take another point 'd' from 'outside' the
Mandelbrot set, then this tells us, that the Julia set J(z^2+d) is dust
like.
applet:
- http://www.cut-the-knot.org/Curriculum/Algebra/MandelbrotIterations.shtml
- http://www.cut-the-knot.org/ctk/Mandel.shtml
- http://www.cut-the-knot.org/blue/julia.shtml
Form and features
The Natural definition of a fractal: A geometric figure or natural object that
combines the following characteristics: (a) its parts have the same form or
structure as the whole, except that they are at a different scale and may be
slightly deformed; (b) its form is extremely irregular or fragmented, and remains
so, whatever the scale of examination; (c) it contains "distinct elements" whose
scales are very varied and cover a large range. This looks a little more within
our grasp to prove. The first part is what we like to call self-similarity,
which the Julia set demonstrates easily (no matter how far in we zoom, we see
the same structures), but the Mandelbrot set is harder to see. However, on closer
examination, we see the master Mandelbrot set repeated over and over leading
us to believe that the first part is true. Part (b) is highly obvious for our
sets, as is part (c).
Self-similarity:
- The most important characteristic of a fractal for the purposes of our
metaphor is that the patterns on the border of the image generated in the
M-set recur at different levels; that is, one can see the same pattern recurring
as one magnifies the image to see finer detail.
- Note that the M-set in general is _not_ strictly self-similar; the tiny
copies of the M-set are all slightly different, mainly because of the thin
threads connecting them to the main body of the M-set. However, the M-set
is quasi-self-similar.
- If a fractal is self-similar, you can specify mappings that map the whole
onto the parts. Iteration of these mappings will result in convergence to
the fractal attractor.
- "[Suitably] selected fragments of a Julia set are strictly [self]-similar
to the set as a whole." (Elert 23.shtml). In contrast, "fragments of the
Mandelbrot [set] are only quasi-similar to the set as a whole. Furthermore,
the motif of this quasi-self-similarity varies from one region to another
and from one level of magnification to another." (Elert 23.shtml).
- The Mandelbrot set serves as a roadmap to or table of contents for the
Julia sets (Peitgen et al. 855-895). To varying degrees, but in some cases
quite striking, there is a correspondence or quasi-similarity between the
appearance of portions of the Mandelbrot set and the Julia sets corresponding
to the c values in that region of M.
Symmetry: The bilateral symmetry results from the imaginary components
of the complex numbers while the vertical, head and tail, directions result
from the real number components.
Other M-sets: Any iterated function can be used to build a M-set.
The original M-set (as discussed here) uses iterated squaring. M-sets with
iterated cubing, third power, fourth power, and fifth power can also be produced.
Such sets exhibit an n-fold rotational symmetry where n is one smaller than
the iterated power used to generate the set.
Cardioid: This is the main body of the set as represented in Figure
1. Attached to it are "bulbs".
- "Vertical x-axis": Positive values of real numbers lie above the
horizontal axis (negative below).
- "Horizontal y-axis": Positive imaginary "y" values are
*** left and right.
- Single-factor / Integral
"Bulbs" (or "circles", or "disks") around the
cardioid:
- The bulb-like regions directly attached radially to the main cardioid are
called primary bulbs.
- There is a (countable) infinity of these which are in direct (tangential)
contact with the cardioid, but they vary in size, tending asymptotically to
zero diameter.
- Each such primary has in turn its own (countable) infinite set of smaller
circles which branch out from it, and this set of surrounding circles also
tends asymptotically in size to zero. The branching out process can be repeated
indefinitely, producing a fractal.
- Bulb period: At the tip of each primary bulb, is a spoke-like structure
emanating from a central junction point. The period of any such bulb is equal
to the number of spokes attached to that bulb.
- triadic, quadrilemma, multi-set
"Head": This region corresponds to the large bulb directly attached
above the main cardioid. This has an attracting cycle of period 2. As mentioned
above, the orbit of fc (0) has an attracting 2-period at c = -1. In fact, it can
be easily proved that the region of 2-period orbits is bounded by a circle given
by | c + 1 | = ¼, which is a circle of radius ¼ centered at z = -1. In other words,
all c-values within this bulb will generate an orbit of fc (0) that approaches
a cycle of period 2.
- Along the vertical "x-axis" -- in the negative direction, the
cardioid has a series of successively smaller circles attached to it running
in a chain.
- Each circle on the x-axis corresponds to a region of differing periodicity.
- The ratio of the diameters of successive circles approaches Feigenbaum's
constant "delta". ****
- A universal constant in mathematics (like pi=3.1415926... and e=2.7182818...)
that applies to nearly any parametrized iteration function, such as that used
for the Mandelbrot Set. It gives the limit of the ratio between the parameter
values at successive period doubling bifurcations in a parameter space. =
4.66920
- The interpretation of the delta constant is as you approach chaos, each
periodic region is smaller than the previous by a factor approaching 4.669...
- The long tail-like region (corresponding to the chaotic regime) is punctuated
with little islands.
- The islands correspond to odd period windows which are miniature mutated
copies of the whole M-set.
- binary thinking, dualistic, polarization
Filaments / Tendrils:
- These are associated with the whole structure, each with its own array of
window-like, mutated copies of the whole set. In these some new cardoids appear,
not attached to "bulbs" (or "circles") of lower period.
- The filaments contain all the variety in the Mandelbrot Set. While the islands
are extremely alike, the filaments attached to them are remarkably different.
Any two filaments which appear alike actually contain many subtle differences.
- These small copies are connected with the main cardioid by filaments which
are formed by other tiny cardioids. These strucrures are called the "Mandelbrot
hair" or filaments.
Orbits: Super-stable orbits quickly converge to a fixed point. In other
words, each primary bulb corresponds to a different period, and all c-values
in the same primary bulb generate orbits which approach a cycle of the same
period p.
- Inside the cardioid, all orbits of f c (0) are attracted to an attracting
fixed point.
- ***In the primary bulbs, all orbits of f c (0) are attracted to an attracting
p-period cycle. As the c-value crosses from the cardioid into one of
the primary bulbs the attracting fixed point switches stability and becomes
a repelling fixed point. As this happens, an attracting cycle of some period
is born. Before this bifurcation, the p-period cycle was repelling.
- ??? These orbits can settle on to attracting fixed points, be periodic,
or ergodic. A small set of fixed points, the repelling fixed points, do not
generate orbits in the traditional sense. They neither roam nor run off to
infinity and one need not wait for them to exhibit "characteristic" behavior.
They are permanently and immutably fixed and nearby points avoid them. They
lie on the frontier between those seeds with bounded orbits and those with
unbounded orbits.
- Under iterates of f(z)=z*z, each complex number z follows a "path" called
the forward orbit of z. This consists of the sequence z, f(z), f(f(z)), f(f(f(z))),
f(f(f(f(z)))), etc. We would like to understand the orbits to whatever extent
possible. This turns out to be easy to describe for most complex numbers,
and more complicated to describe for the rest. For all z inside the unit circle
centered at 0, the orbits all tend to 0. For all z outside the unit circle,
the orbits all tend to infinity. On the unit circle, the orbits are more complicated.
Some of the orbits on the unit circle end at 1. These are the orbits of fractional
powers of 1 with the denominator of the fraction an integral power of 2. Some
of the orbits are finite and cyclically repeating. We such an orbit a "periodic
orbit" and the number of points in it the "period" of the orbit. The two non-real
cube roots of 1 form a finite cyclic orbit of two points. Some orbits end
in a periodic orbit, and lastly, some orbits never repeat, never stop and
never reach a limit. These orbits are orbits of points on the unit circle
that are an irrational multiple of 2*pi from 1. Each of these last orbits
gets arbitrarily close to any point on the unit circle.
- Then the list of successive iterates of a point or number is called the
orbit of that point. In a common sense the Julia set (named after Gaston Julia)
consists of all starting points z, whose orbits behave abnormal. Normally
the orbit leads to something, the orbit leads to an attracting point, or,
to be more general, to an attracting set. But this is not a must. If we use
another function we will see that some orbits lead to nothing, they simply
jump around, not knowing where they should go. The Julia set consists of all
numbers whose orbits don't know where to go. That's a rather strange definition,
isn't it? Well, how does one actually calculate a Julia set? What we have
to do is the following: We have to search for all points whose orbits lead
to something. All those points don't belong to the Julia set, the Julia set
consists of all other (remaining) points.
- This is sometimes referred to as a quadratic map, and is a type of dynamical
system. Given a specific choice of c and z0, the above recursion leads to
a sequence of complex numbers z1, z2, z3... called the orbit of z0. Depending
on the exact choice of c and z0, a large range of orbit patterns are possible.
For a given fixed c, most choices of z0 yield orbits that tend towards infinity.
(That is, the modulus | zn | grows without limit as n increases.) For some
values of c certain choices of z0 yield orbits that eventually go into a periodic
loop. Finally, some starting values yield orbits that appear to dance around
the complex plane, apparently at random. (This is an example of chaos.) These
starting values make up the Julia set of the map, denoted Jc. Some authors
also define the filled-in Julia set, denoted Kc, which is the set of all z0
with yield orbits which do not tend towards infinity. The "normal" Julia set
Jc is the edge of the filled-in Julia set.
- We would like to understand the orbits to whatever extent possible. This
turns out to be easy to describe for most complex numbers, and more complicated
to describe for the rest. For all z inside the unit circle centered at 0,
the orbits all tend to 0. For all z outside the unit circle, the orbits all
tend to infinity. On the unit circle, the orbits are more complicated. Some
of the orbits on the unit circle end at 1. These are the orbits of fractional
powers of 1 with the denominator of the fraction an integral power of 2. Some
of the orbits are finite and cyclically repeating. We such an orbit a "periodic
orbit" and the number of points in it the "period" of the orbit.
Period of attracting cyle: A complex number c is in the M-set if the
iteration of z2+c (beginning with zero) remains bounded. It is in
a (hyperbolic) component of period n if the iteration attracts to a peroidic
orbit of period n. These components are the disks budding off of the M-set or
cardioids for mini-M-sets.
- The main cardioid can be termed the “period-1 bulb”.
- The period-2 bulb, namely the largest primary bulb, is above the cardioid
in Figure 1.
- Moreover, each primary bulb consists of c-values for which fc (0) has an
attracting cycle of some period p, where p is some integer. In other words,
each primary bulb corresponds to a different period, and all c-values in the
same primary bulb generate orbits which approach a cycle of the same period
p.
- The largest primary bulb between them is the period-3 bulb, on either side
of Figure 1. Similarly the largest between between period-2 and period-3 is
period-5. The increasing perioid numbers (and decreasing sizes) form a Fibonacci
sequence ( 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . ).
- Also, there is another amazing fact about the arrangement of the buds. Two
given buds of periods p and q at the cardoid detemine the period of the largest
bud in between them as p+q. (This is illustrated for the case of p = 2 and
q = 3 in figure(3) below). Similar rules are true for buds on buds.
- The bulb-like regions directly attached to the main cardioid are called
primary bulbs, and there are an infinite number of them.
- At the tip of each primary bulb, is a spoke-like structure emanating from
a central junction point. The period of any such bulb is equal to the number
of spokes attached to that bulb.
- It is known that if c lies in the interior of a bulb, then the orbit of
z0=0 is attracted to a cycle of a period n. It is a multiple of n for c inside
the other smaller bulbs attached to the primary bulb.
- One can count rotation number of a bulb by its periodic orbit star. An attracting
period n cycle z1 -> z2 ->...-> zn -> z1 hops among zi as fc is iterated.
If we observe this motion, the cycle jumps exactly m points in the counterclockwise
direction at each iteration. Another way to say this is the cycle rotates
by a m/ n revolution in the counterclockwise direction under iteration. (Robert
L. Devaney. Rotation
Numbers and Internal angles of the Mandelbrot bulbs, 2000)
- Rotation numbers: The "rotation number" of a bulb is the fraction of the
number of "ears" of the J-set the critical orbit jumps between at each iteration.
Bifurcation: Time-dependent systems are capable of abrupt changes in
their topological form called bifurcations as the underlying parameters cross
critical values. Bifurcations result in abrupt catastrophic change in the topology
of the flow under continuous variation of the time-dependent parameters.
- devil , satan
- conversion, temptation
- nature as evil
Attractor / Repellor: Strange attractors have orbits which chaotically
trip from one basin to another. The above period 3 set of attractors can be
modeled physically with an iron pendulum suspended between three magnets. It
is a strange attractor and the final resting position is indeterminate even
if the bob starts close to one particular magnet.
- attractor point / attractor cycle / limit cycle
- As the c-value crosses from the cardioid into one the primary bulbs the
attracting fixed point switches stability and becomes a repelling fixed point.
As this happens, an attracting cycle of some period is born. Before this bifurcation,
the p-period cycle was repelling.
- The simplest example of this type of bifurcation occurs as the c-value
moves from the large cardioid into the 2-period bulb. If we travel along the
real axis, this transition occurs at c = -3/4. As the c-value crosses this
point, the attracting fixed point becomes a repellor and an attracting 2-period
cycle is created. For this reason, c = -3/4, is a period-doubling bifurcation
point. A similar bifurcation occurs as the c-value crosses into any of the
primary bulbs. The only difference is that different primary bulbs are characterized
by different periods.
- The general area around an attractor is called an attractor basin. The boundary
of an attractor basin is a curious thing. In general, points inside the boundary
are convergent and trapped by the attractor. Points outside the boundary are
divergent and escape the attractor. The attractor then acts as a repellor.
Points on the boundary, though, can go either way. The boundary is not equicontinuous.
- Inside the cardioid, all orbits of f c (0) are attracted to an attracting
fixed point. In the primary bulbs, all orbits of f c (0) are attracted to
an attracting p-period cycle.
- In higher dimensions, however, attraction and repulsion are not limited
to points. An iterative map can collapse on to any structure possible in that
dimension. Attractors and repellers can form paths, surfaces, volumes, and
their higher dimensional analogs.
- A Julia set is an attractor in the sense that values of z belonging to Jc
when further iterated continue to produce other values lying in Jc. That is,
the set seems to attract orbits beginning in the set.
- Basins of attraction Thus we have three regions associated with f(z)=z*z.
Two regions tend to two of the fixed points of f, namely 0 and infinity. We
define a fixed point to be an "attracting fixed point" if all points near
the fixed point have orbits that have that fixed point as a limit. Thus 0
and infinity are attracting fixed points, and the regions that tend to them
are their "basins of attraction." The fixed point 1 is not attracting since
it has points nearby whose orbits do not tend to 1. The third region is the
unit circle. No point there has an orbit that has an attracting fixed point
as a limit. One can also have "attracting periodic orbits." For example, consider
f(z)=z*z+c where c is a complex constant. Now f(z)=z or z*z-z+c=0 also has
two solutions giving two finite fixed points, and the usual infinite fixed
point. The equation f(f(z))=z or f(f(z))-z=0 is a quartic with 4 finite solutions.
Two are the solutions to f(z)=z since a fixed point for f is a fixed point
for f(f). Thus z*z-z+c is a factor of f(f(z))-z and long division gives z*z+z+(c+1)
as the other factor. The equation z*z+z+(c+1)=0 has one root if c=-3/4. For
other values of c, we get a periodic orbit of period two. It turns out that
the orbit will be attracting if the derivative of f(f(z)) at one of the points
in the orbit is inside the unit circle. This is easy to calculate since the
derivative of f(f(z)) is 4*z*f(z) when f(z)=z*z+c. Now if z is in a periodic
orbit of period 2, then f(z) is the other point in the orbit. Thus the derivative
of f(f(z)) at such a point is 4 times the product of the two points or 4 times
the product of the two roots of z*z+z+(c+1)=0. This calculation eventually
leads to the conclusion that 4(c+1) must lie inside the unit circle. Equivalently,
c must lie inside the circle of radius 1/4 centered at -1. The points inside
this circle will show up later.
Rotation:
Features: Many of the detailed features of the M-set have been given
colloquial names, usually descriptive in relation to natural phenomena [more].
Colloquial names for the entire M-set R2 and for various parts of it are
derived from the resemblance of the continent to a sitting Buddha (when rotated
90 degrees so that west is up). The term buddhabrot in particular is used
to refer to plots in which all iterates, as well as the parameter value, are
plotted in the same image; with suitable color mappings such plots are said
to resemble a sitting Buddha even more than the normal-style plot. See also
topknot. From the Mandelbrot Set Glossary and Encyclopedia, 2004 Robert
P. Munafo.
Colour:
- The colors are added to the points that are not inside the set, according
to how many iterations were required before the magnitude of Z surpassed two.
Not only do colors enhance the image aesthetically, they help to highlight
parts of the M-set that are too small to show up in the graph.
- One of the best ways to color the Mandelbrot Set uses the HSV color space.
Use the Distance Estimator function for V and Escape-Iterations for H, and
make S alternate between odd and even values of Escape-Iterations. The result
is stunning.
- The simplest form of rendering uses escape times. Pixels are coloured according
to the number of iterations it takes for a pixel to _blow-up_ or escape the
loop.
- Instead of converging to a root they escape. The angle of the vector connecting
the point before escape with the point after escape or the last to points
before giving up on checking for an escape can be used to acieve rather nice
effects.
- Color coding the rate at which different values of c cause z to shoot off
to infinity, stabilize in the realm of finite numbers, or go to zero, creates
the visual embodiment of the “M-world”.
- However, the other points, which are not part of the set are the ones that
result in the beautiful colours. The way the colours are computed is by seeing
how many iterations it takes for the points that are not part of the set to
reach infinity (this is determined by how many iterations it takes them to
move a distance further than two units from the origin, in this case). For
example, if a point were to move a distance further than two units from the
origin after only 10 iterations, it could be coloured blue. Likewise, if it
moved further than two units from the origin after 20 iterations is could
be coloured red.
- Coloured versions of the representation are formed by assigning colours
other than white to points that do not belong to the set, the choice of colour
being a function (usually following a spectral sequence) of the number of
iterations performed before divergence becomes apparent.
Mapping in higher dimensions: The classic M-set is a map in the complex
plane. The M-set can also be mapped in higher dimensions.
- Each coordinate in the plane is essentially 2 dimensional because a complex
number, z, has two parts and is written z = a + ib. Complex numbers have one
part which is real and a second part which we call imaginary. To introduce
M-sets in higher dimensions we must use quaternions. Quaternion numbers are
an extension of complex numbers, which have four parts instead of two. A quaternion
Q equals a + ib + jc + kd, where the coefficient a, b, c, and d are real numbers.
- In higher dimensions, however, attraction and repulsion are not limited
to points. An iterative map can collapse on to any structure possible in that
dimension. Attractors and repellers can form paths, surfaces, volumes, and
their higher dimensional analogs.
- For (Glenn Elert, Strange
& Complex, 2003): "J-sets are slices parallel to the z-axis while
the M-set is a slice along the c-axis through the origin. As the coordinate
system is complex, however, these "axes" are actually planes. The Mandelbrot
and J-sets are therefore two-dimensional cross sections through a four-dimensional
parent set; the mother of all iterated quadratic mappings so to speak".
- Traditionally the familiar pictures are drawn by considering complex arithmetic,
which is 2 dimensional in nature. However, the complex number system is a
subset of a higher number system, known as the quaternions. Quaternions are
based in 4 dimensions as the name may suggest. It follows that drawings of
fractal sets using complex numbers are showing you 2d slices of higher dimensional
objects, and that by using quaternion maths we can view these higher dimensional
objects. That is what the animations on this page are all about.
>>>>>>>>>>>>>>>>>
Attraction to Shape http://home.earthlink.net/~sroof/Abraxas/sar/shape/shape.htm
The Mandelbrot set is then the set of complex c values for which the z orbit
remains bounded. From Julia and Fatou , it is known that the basin of attraction
of any finite attractor will contain the critical point (see, for example, Devaney
devaney), so the Mandelbrot set catalogues the parameter values for which a
finite attractor exists. Other initial conditions may not fall in the basin
of attraction of a finite attractor even if one exists; thus the Mandelbrot
set is the maximum region in parameter space for which orbits can remain bounded.
[more]
>>>>>>>>>>>
Ben Moore. Fractls. http://krone.physik.unizh.ch/~moore/complex/fractals2.html
Has a: A Mandelbrot Set with 8 Accompanying Julia Sets in a Constellation Diagram
The Mandelbrot set for the quadratic mapping f: z --> z2 = c is shown below
for all parameters c = x = iy in the range x = [-2, 1/2] y = [-2, 2]. Some wholly
connected Julia sets were also added and their approximate location in parameter
space indicated. This type of arrangement is known as a constellation diagram.
>>>>>>>>>>>>>>
Glenn Elert. The Chaos Hypertextbook. 1995-2003 http://hypertextbook.com/chaos/
Strange & Complex ™ © by Glenn Elert
http://hypertextbook.com/chaos/23.shtml
The factor that determines whether a Julia set is wholly connected or wholly
disconnected is the parameter value. Thus it would be instructive to plot the
behavior of the Julia sets for all parameter values. The resulting construction
would be the complex analog of a bifurcation drawing. At first glance, this
seems a daunting task. Plotting every possible Julia set and then examining
it to determine whether it was connected or not would take an eternity. Luckily
for us, however, we need only study the behavior of one point in the complex
plane. This trick was discovered by the Polish-American mathematician Benoit
Mandelbrot and in his honor the set of all parameter values whose Julia sets
are wholly connected is called a Mandelbrot set. The Mandelbrot set for the
quadratic mapping f: z --> z2 = c is shown below for all parameters c = x =
iy in the range x = [-2, 1/2] y = [-2, 2]. Some wholly connected Julia sets
were also added and their approximate location in parameter space indicated.
This type of arrangement is known as a constellation diagram.
In higher dimensions, however, attraction and repulsion are not limited
to points. An iterative map can collapse on to any structure possible in that
dimension. Attractors and repellers can form paths, surfaces, volumes, and their
higher dimensional analogs. For example, the two-dimensional map f: (x,
y) --> (x, y/2) attracts all points asymptotically to the x-axis. Likewise,
a two-dimensional object can act as a repeller. Such is the case for the map
f: (x, y) --> (x2 - y2, 2xy). Points inside the unit circle head for the origin
while those outside fly off to infinity. Points on the circle remain there and
thus for this map the unit circle can be considered a fixed repeller.
>>>>>>>>>>>>
Complex Quadradic Dynamics: A Study of the Mandelbrot and Julia Sets [text]
The Mandelbrot set is a map in the parameter plane that graphically demonstrates
the behavior of fc (0) for all c-values.
The Mandelbrot set, M, includes all c-values for which the orbit of z o = 0,
under the mapping of fc , remains bounded.
Using the equations above, one can easily see that c = z - z2 . Equivalently,
c = x(q) = ½eqi - ¼ e2qi . This parameterized curve traces out the large central
region of the Mandelbrot set. The interior of this region contains all c-values
for which fc has an attracting fixed point. This region is often referred to
as the main cardioid of the Mandelbrot set.
However, any given Julia set falls into one of two categories. The discovery
of this fundamental dichotomy dates back to 1919 when G. Julia and P. Fatou
proved that for each c-value,
In this way, it is easy to see that the location of the c-value in the Mandelbrot
set immediately gives one an idea of what the corresponding Julia set will look
like.
7, 5, 8, 3, 7, 4, 5, 6, 7 --- 3, 4, 5
2 -------------------------------- 7
7, 5, 8, 3, 7, 4, 5, 6, 7 -- 3, 4, 5
What if our life is an animation of 3D slices taken from a 4D continuum?
The maps of this quadratic equation are not necessarily chaotic attractors
in the normal sense of the term. In general, a chaotic attractor is the limit
set of an aperiodic trajectory. This region remains bounded, but the pattern
never repeats so the attractor will always have a fractal structure. The dynamic
mapping of f c (z) = z2 + c, for a fixed c, is the Julia set. The filled black
region represents the z-values that behave orderly, either as periodic cycles
or fixed points. The colored region represents the z-values that will escape
to infinity. The Julia set, J c , is the boundary between these two competing
basins of attraction.
In closing, let us make sure we clearly understand that the Mandelbrot and
Julia sets are different ways of looking at the same thing. For the Mandelbrot,
we hold z constant at 0 and check all c-values. For the Julia set, we hold c
constant and check all z-values. Perhaps with more sophisticated computer graphics
technology we will be able to analyze the mapping of f c (z) by modulating both
the parameter values and the initial condition values at the same time. In any
case, it is certain that the frontier of Chaos Theory is a field of infinite
fractal possibilities.
Chaotic processes are characterised by a critical dependence on initial conditions
and other external influences. The state of a system can be represented by a
point in a multidimensional 'phase space', and within this there are 'basins
of attraction' towards different types of behaviour. A characteristic of chaotic
systems is that the basins of attraction have a complicated interface like that
between the dark (member) and light (non-member) areas of the Mandelbrot Set.
As with the Mandelbrot Set, the interface has similar form when viewed at any
level of resolution, so the basins of attraction are intimately mixed no matter
how precisely the co-ordinates are specified. This means there is no limit to
the smallness of the disturbance, or deviation of an initial setting, that can
drastically change the system behaviour. The behaviour is unpredictable in principle,
irrespective of the precision of the methods of measurement and manipulation.
Bohm introduces a new concept in which he describes the Implicate Order as
a kind of *generative order.* He notes that "This order is primarily concerned
not with the outward side of development, and evolution in a sequence of successions,
but with a deeper and more inward order out of which the manifest form of things
can emerge *creatively.*" Bohm believes that the generative order "proceeds
from an origin in free play which then unfolds into ever more crystallized forms."
Generative order can be seen in the work of an artist. Bohm uses the example
of Mandelbrot's mathematically-derived fractals to illustrate more scientifically
this cosmic generativity. "Fractals involve an order of similar differences
which include changes of scale as well as other possible changes." Bohm notes
that "By choosing different base figures and generators, but each time applying
the generator on a smaller and smaller scale, Mandelbrot is able to produce
a great variety of shapes and figures--All are filled with infinitesimal detail
and are evocative of the types of complexity found in natural forms."
The Mandelbrot set of degree d 2, denoted by Md, is defined as the set of
parameters c for which any of the following equivalent conditions holds:
• Origin’s orbit under iteration of the polynomial Pc (i.e., the sequence
{0, c, cd + c, (cd + c)d + c, . . . }) is bounded.
• Julia set of Pc is connected.
• Julia set of Pc contains the origin (the only critical point of Pc).
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| Figure 5: Illustration of different colouring conventions
inside the M-set |
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