The Fractal Tree : Julia Sets

The Fractal Tree : Julia Sets

A fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,”[1] a property called self-similarity. Roots of mathematically rigorous treatment of fractals can be traced back to functions studied by Karl WeierstrassGeorg Cantor and Felix Hausdorff in studying functions that were analyticbut not differentiable; however, the term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractusmeaning “broken” or “fractured.” A mathematical fractal is based on an equation that undergoes iteration, a form of feedbackbased on recursion.[2]

This principle of self-similarity recursion can be seen in the Spiral of the Golden Mean. The Fibonacci Sequence is the iterating pattern that geanerates the Golden mean. SEE ALSO; Newton iteration ; Newton’s method ;

The Glynn Set is a special Julia Set, named after Earl Glynn. It kind of looks like a tree. See also my 3D Glynn Julia set. Here is some Mathematica code:
(* runtime: 48 seconds *)
Julia[z0_] := Module[{z = z0, i = 0}, While[i < 100 && Abs[z] < 2, z = z^1.5 + c; i++]; i];
c = -0.2;
DensityPlot[Julia[-x + I y], {y, -0.2, 0.2}, {x, 0.35, 0.75}, PlotPoints -> 275, Mesh -> False, Frame -> False]

Here is some code to plot this using the Modified Inverse Iteration Method (MIIM). Note that special care must be taken to verify each root’s validity:
(* runtime 7 seconds *)
Pow[z_, n_, k_] := Module[{theta = Arg[z]}, theta = n(theta + 2Pi (k – Floor[(theta/Pi +Abs[1/n])/2])); If[Abs[theta] > Pi, Null, Abs[z]^n Exp[I theta]]];
power = 1.5; nroot = 3; c = -0.2; zlist = {}; dzmax = 25.0; imax =1000; z = Table[-0.61, {imax}]; dz = Table[1, {imax}]; roots = Table[1, {imax}]; i = 2;
While[i > 1, z[[i]] = Pow[z[[i – 1]] – c, 1/power, roots[[i]] – 1]; If[z[[i]] === Null, prune = True, dz[[i]] = Abs[power]Abs[z[[i]]]^(power – 1)dz[[i – 1]]; zlist = Append[zlist, z[[i]]]; prune = (i == imax || dz[[i]] > dzmax)]; If[prune, While[i > 1 && roots[[i]] == nroot, roots[[i]] = 1; i–]; roots[[i]]++, i++; roots[[i]] = 1]];
ListPlot[{Re[#], Im[#]} & /@ zlist, PlotStyle -> PointSize[0.005], AspectRatio ->Automatic]


From Wikipedia, the free encyclopedia

A Julia set.

Julia set 3d slice animation.ogg

Three-dimensional slices through the (four-dimensional) Julia set of a function on thequaternions.

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly underrepeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is ‘regular’, while on the Julia set its behavior is ‘chaotic‘.

The Julia set of a function ƒ is commonly denoted J(ƒ), and the Fatou set is denoted F(ƒ).[1] These sets are named after the French mathematicians Gaston Julia,[2] and Pierre Fatou[3] whose work began the study of complex dynamics during the early 20th century.



[edit]Formal definition

Let f(z) be a complex rational map from the plane into itself, that is, f(z) = p(z) / q(z), where p(z) and q(z) are complex polynomials. Then there are a finite number of open sets F_i, i = 1, \dots, r, that are left invariant by f(z) and are such that:

  1. the union of the Fi‘s is dense in the plane and
  2. f(z) behaves in a regular and equal way on each of the sets Fi.

The last statement means that the termini of the sequences of iterations generated by the points of Fi are either precisely the same set, which is then a finite cycle, or they are finite cycles of finite or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second it is neutral.

These sets Fi are the Fatou domains of f(z), and their union is the Fatou set F(f) of f(z). Each of the Fatou domains contains at least one critical point of f(z), that is, a (finite) point zsatisfying f‘(z) = 0, or z = ∞, if the degree of the numerator p(z) is at least two larger than the degree of the denominator q(z), or if f(z) = 1 / g(z) + c for some c and a rational functiong(z) satisfying this condition.

The complement of F(f) is the Julia set J(f) of f(z). J(f) is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). LikeF(f), J(f) is left invariant by f(z), and on this set the iteration is repelling, meaning that | f(z) − f(w) | > | z − w | for all w in a neighbourhood of z (within J(f)). This means that f(z)behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.

For f(z) = z2 the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the non-rational points). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.

For f(z) = z2 − 2 the Julia set is the line segment between -2 and 2, and the iteration corresponds to x \to 4(x - 0.5)^{2} in the unit interval – a very used method for generation of random numbers. There is one Fatou domain: the points not on the line segment iterate towards ∞.

These two functions are of the form z2c, where c is a complex number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes. See the pictures below.

Julia set (in white) for the rational function associated to Newton’s method for ƒ:zz3−1. Coloring of Fatou set according to attractor (the roots of ƒ)

For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following main theorem on the iterations of a rational function:

Each of the Fatou domains has the same boundary, which consequently is the Julia set

This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if there are more than two Fatou domains, each point of the Julia set must have points of more than two different open sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for instance, when f(z) is the Newton iteration for solving the equationzn = 1(n > 2):  f(z) = z − f(z) / f‘(z) = (1 + (n − 1)zn) / (nzn − 1). The image on the right shows the case n = 3.

There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many components.[4] Each component of the Fatou set of a rational map can be classified into one of four different classes.[5]

[edit]Equivalent descriptions of the Julia set

  • J(f) is the smallest closed set containing at least three points which is completely invariant under f.
  • J(f) is the closure of the set of repelling periodic points.
  • For all but at most two points z\in X, the Julia set is the set of limit points of the full backwards orbit \bigcup_n f^{-n}(z). (This suggests a simple algorithm for plotting Julia sets, see below.)
  • If f is an entire function – in particular, when f is a polynomial, then J(f) is the boundary of the set of points which converge to infinity under iteration.
  • If f is a polynomial, then J(f) is the boundary of the filled Julia set; that is, those points whose orbits under iterations of f remain bounded.

[edit]Properties of the Julia set and Fatou set

The Julia set and the Fatou set of f are both completely invariant under iterations of the holomorphic function f, i.e.

f − 1(J(f)) = f(J(f)) = J(f)


f − 1(F(f)) = f(F(f)) = F(f).[6]

[edit]Quadratic polynomials

A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational maps. The quadratic polynomials can be expressed as

f_c(z) = z^2 + c\,

where c is a complex parameter.

Filled Julia set for fc, c=1−φ where φ is thegolden ratio

Julia set for fc, c=(φ−2)+(φ−1)i =-0.4+0.6i

Julia set for fc, c=0.285+0i

Julia set for fc, c=0.285+0.01i

Julia set for fc, c=0.45+0.1428i

Julia set for fc, c=-0.70176-0.3842i

Julia set for fc, c=-0.835-0.2321i

Julia set for fc, c=-0.8+0.156i

A Julia set plot showing julia sets for different values of c, the plot resembles theMandelbrot set

The parameter plane of quadratic polynomials – that is, the plane of possible c-values – gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all c such that J(fc) is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor set: in this case it is sometimes referred to as Fatou dust.

In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. This is true, in particular, for so-called‘Misiurewicz’ parameters, i.e. parameters c for which the critical point is pre-periodic. For instance:

  • At ci, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
  • At c = − 2, the tip of the long spiky tail, the Julia set is a straight line segment.

In other words the Julia sets J(fc) are locally similar around Misiurewicz points.[7]


The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Epstein’s ‘finite-type maps’.

Julia sets are also commonly defined in the study of dynamics in several complex variables.

[edit]The potential function and the real iteration number

The Julia set for f(z) = z2 is the unit circle, and on the outer Fatou domain, the potential function φ(z) is defined by φ(z) = log | z | . The equipotential lines for this function are concentric circles. As | f(z) | = | z2 we have \phi(z) = \lim_{k\to\infty} \log|z_k|/2^{k}, where zk is the sequence of iteration generated by z. For the more general iteration f(z) = z2c, it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle such that | ψ(f(z)) | = | ψ(z) | 2[8]. This means that the potential function on the outer Fatou domain defined by this correspondence is given by:

\phi(z) = \lim_{k\to\infty} \log|z_k|/2^k. \,

This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function f(z) such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n of the denominator, we define the potential function on the Fatou domain containing ∞ by:

\phi(z) = \lim_{k\to\infty} \log|z_k|/d^k, \,

where d = m – n is the degree of the rational function[9].

If N is a very large number (e.g. 10100), and if k is the first iteration number such that | zk | > N, we have that log | zk | / dk = log(N) / dν(z), for some real number ν(z), which should be regarded as the real iteration number, and we have that:

ν(z) = k − log(log | zk | / log(N)) / log(d),

where the last number is in the interval [0, 1).

For iteration towards a finite attracting cycle of order r, we have that if z* is a point of the cycle, then f(f(…f(z * ))) = z * (the r-fold composition), and the number \alpha = 1/|(d(f(f(\cdots f(z))))/dz)_{z=z*}| (> 1) is the attraction of the cycle. If w is a point very near z* and w’ is w iterated r times, we have that \alpha = \lim_{k\to\infty} |w - z*|/|w' - z*|. Therefore the number | zkr − z * | αk is almost independent of k. We define the potential function on the Fatou domain by:

\phi(z) = \lim_{k\to\infty} 1/(|z_{kr} - z*|\alpha^{k}).

If ε is a very small number and k is the first iteration number such that | zk − z * | < ε, we have that \phi(z) = 1/(\varepsilon \alpha^{\nu(z)}) for some real number ν(z), which should be regarded as the real iteration number, and we have that:

\nu(z) = k - \log(\varepsilon/|z_k - z*|)/\log(\alpha).

If the attraction is ∞, meaning that the cycle is super-attracting, meaning again that one of the points of the cycle is a critical point, we must replace α by \alpha = \lim_{k\to\infty} \log|w' - z*|/\log|w - z*| (where w’ is w iterated r times) and the formula for φ(z) by:

\phi(z) = \lim_{k\to\infty} \log(1/|z_{kr} - z*|)/\alpha^k. \,

And now the real iteration number is given by:

\nu(z) = k - \log(\log|z_k - z*|/\log(\varepsilon))/\log(\alpha). \,

For the colouring we must have a cyclic scale of colours (constructed mathematically, for instance) and containing H colours numbered from 0 to H-1 (H = 500, for instance). We multiply the real number ν(z) by a fixed real number determining the density of the colours in the picture, and take the integral part of this number modulo H.

The definition of the potential function and our way of colouring presuppose that the cycle is attracting, that is, not neutral. If the cycle is neutral, we cannot colour the Fatou domain in a natural way. As the terminus of the iteration is a revolving movement, we can, for instance, colour by the minimum distance from the cycle left fixed by the iteration.

[edit]Field lines

The equipotential lines for iteration towards infinity

Field lines for an iteration of the form(1 − z3 / 6) / (z − z2 / 2)2c

In each Fatou domain (that is not neutral) there are two systems of lines orthogonal to each other: the equipotential lines (for the potential function or the real iteration number) and the field lines.

If we colour the Fatou domain according to the iteration number (and not the real iteration number), the bands of iteration show the course of the equipotential lines. If the iteration is towards ∞ (as is the case with the outer Fatou domain for the usual iteration z2c), we can easily show the course of the field lines, namely by altering the colour according as the last point in the sequence of iteration is above or below the x-axis (first picture), but in this case (more precisely: when the Fatou domain is super-attracting) we cannot draw the field lines coherently – at least not by the method we describe here. In this case a field line is also called an external ray.

Let z be a point in the attracting Fatou domain. If we iterate z a large number of times, the terminus of the sequence of iteration is a finite cycleC, and the Fatou domain is (by definition) the set of points whose sequence of iteration converges towards C. The field lines issue from the points of C and from the (infinite number of) points that iterate into a point of C. And they end on the Julia set in points that are non-chaotic (that is, generating a finite cycle). Let r be the order of the cycle C (its number of points) and let z* be a point in C. We have f(f(\dots f(z*))) = z* (the r-fold composition), and we define the complex number α by

\alpha = (d(f(f(\dots f(z))))/dz)_{z=z*}. \,

If the points of C are z_i, i = 1, \dots, r (z_1 = z*), α is the product of the r numbers f‘(zi). The real number 1/ | α | is the attraction of the cycle, and our assumption that the cycle is neither neutral nor super-attracting, means that 1 < 1/|α| < ∞. The point z* is a fixed point for f(f(\dots f(z))), and near this point the map f(f(\dots f(z))) has (in connection with field lines) character of a rotation with the argument β of α (that is, α = | α | eβi).

In order to colour the Fatou domain, we have chosen a small number ε and set the sequences of iteration z_k (k = 0, 1, 2, \dots, z_0 = z)to stop when | zk − z * | < ε, and we colour the point z according to the number k (or the real iteration number, if we prefer a smooth colouring). If we choose a direction from z* given by an angle θ, the field line issuing from z* in this direction consists of the points z such that the argument ψ of the number zk − z * satisfies the condition that

\psi - k\beta = \theta \mod \pi. \,

For if we pass an iteration band in the direction of the field lines (and away from the cycle), the iteration number k is increased by 1 and the number ψ is increased by β, therefore the number \psi - k\beta \mod \pi is constant along the field line.

Pictures in the field lines for an iteration of the form z2c

A colouring of the field lines of the Fatou domain means that we colour the spaces between pairs of field lines: we choose a number of regularly situated directions issuing from z*, and in each of these directions we choose two directions around this direction. As it can happen that the two field lines of a pair do not end in the same point of the Julia set, our coloured field lines can ramify (endlessly) in their way towards the Julia set. We can colour on the basis of the distance to the centre line of the field line, and we can mix this colouring with the usual colouring. Such pictures can be very decorative (second picture).

A coloured field line (the domain between two field lines) is divided up by the iteration bands, and such a part can be put into a one-to-one correspondence with the unit square: the one coordinate is (calculated from) the distance from one of the bounding field lines, the other is (calculated from) the distance from the inner of the bounding iteration bands (this number is the non-integral part of the real iteration number). Therefore we can put pictures into the field lines (third picture).

[edit]Distance estimation

Julia set drawn by distance estimation, the iteration is of the form1 − z2z5 / (2 + 4z) + c

Three-dimensional rendering of Julia set using distance estimation.

As a Julia set is infinitely thin we cannot draw it effectively by backwards iteration from the pixels. It will appear fragmented because of the impracticality of examining infinitely many startpoints. Since the iteration count changes vigorously near the Julia set, a partial solution is to imply the outline of the set from the nearest color contours, but the set will tend to look muddy.

A better way to draw the Julia set in black and white is to estimate the distance of pixels from the set and to color every pixel whose center is close to the set. The formula for the distance estimation is derived from the formula for the potential function φ(z). When the equipotential lines for φ(z) lie close, the number | φ(z) | is large, and conversely, therefore the equipotential lines for the function δ(z) = φ(z) / | φ(z) | should lie approximately regularly. It has been proven that the value found by this formula (up to a constant factor) converges towards the true distance for z converging towards the Julia set [10].

We assume that f(z) is rational, that is, f(z) = p(z) / q(z) where p(z) and q(z) are complex polynomials of degrees m and n, respectively, and we have to find the derivative of the above expressions for φ(z). And as it is only zk that varies, we must calculate the derivative zk of zk with respect to z. But as z_k = f(f(\cdots f(z))) (the k-fold composition), zk is the product of the numbers f‘(zk), and this sequence can be calculated recursively by zk + 1f‘(zk)zk, starting with z0 = 1 (before the calculation of the next iteration zk + 1f(zk)).

For iteration towards ∞ (more precisely when m ≥ n + 2, so that ∞ is a super-attracting fixed point), we have

|\phi^'(z)| = \lim_{k\to\infty} |z'_k|/|z_k|d^k, \,

(dm − n) and consequently:

\delta(z) = \phi(z)/|\phi^'(z)| = \lim_{k\to\infty} \log|z_k||z_k|/|z'_k|. \,

For iteration towards a finite attracting cycle (that is not super-attracting) containing the point z* and having order r, we have

|\phi^'(z)| = \lim_{k\to\infty} |z'_{kr}|/(|z_{kr} - z*|^{2}\alpha^{k}), \,

and consequently:

\delta(z) = \phi(z)/|\phi^'(z)| = \lim_{k\to\infty} |z_{kr} - z*|/|z'_{kr}|. \,

For a super-attracting cycle, the formula is:

\delta(z) = \lim_{k\to\infty} \log|z_{kr} - z*||z_{kr} - z*|/|z'_{kr}|. \,

We calculate this number when the iteration stops. Note that the distance estimation is independent of the attraction of the cycle. This means that it has meaning for transcendental functions of “degree infinity” (e.g. sin(z) and tan(z)).

Besides drawing of the boundary, the distance function can be introduced as a 3rd dimension to create a solid fractal landscape.

[edit]Plotting the Julia set

[edit]Using backwards (inverse) iteration (IIM)

A Julia set plot, generated using random IIM

A Julia set plot, generated using MIIM

As mentioned above, the Julia set can be found as the set of limit points of the set of pre-images of (essentially) any given point. So we can try to plot the Julia set of a given function as follows. Start with any point z we know to be in the Julia set, such as a repelling periodic point, and compute all pre-images of z under some high iterate fn of f.

Unfortunately, as the number of iterated pre-images grows exponentially, this is not feasible computationally. However, we can adjust this method, in a similar way as the “random game” method for iterated function systems. That is, in each step, we choose at random one of the inverse images of f\,.

For example, for the quadratic polynomial f_c \,, the backwards iteration is described by

z_{n-1} = \sqrt{z_n - c} .

At each step, one of the two square roots is selected at random.

Note that certain parts of the Julia set are quite difficult to access with the reverse Julia algorithm. For this reason, one must modify IIM/J ( it is called MIIM/J) or use other methods to produce better images.

[edit]Using DEM/J

Julia set : image with C source code using DEM/J

[edit]See also

Search Wikimedia Commons Wikimedia Commons has media related to: Julia set
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  1. ^ Note that for other areas of mathematics the notation J(f)\, can also represent the Jacobian matrix of a real valued mapping f\, between smooth manifolds.
  2. ^ Gaston Julia (1918) “Mémoire sur l’iteration des fonctions rationnelles,” Journal de Mathématiques Pures et Appliquées, vol. 8, pages 47–245.
  3. ^ Pierre Fatou (1917) “Sur les substitutions rationnelles,” Comptes Rendus de l’Académie des Sciences de Paris, vol. 164, pages 806-808 and vol. 165, pages 992–995.
  4. ^ Beardon, Iteration of Rational Functions, Theorem 5.6.2
  5. ^ Beardon, Theorem 7.1.1
  6. ^ Beardon, Iteration of Rational Functions, Theorem 3.2.4
  7. ^ Lei.pdf Tan Lei, “Similarity between the Mandelbrot set and Julia Sets”, Communications in Mathematical Physics 134 (1990), pp. 587–617.
  8. ^ Adrien Douady and John H. Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d’Orsay 2/4 (1984 / 1985)
  9. ^ Peitgen, Heinz-Otto; Richter Peter (1986). The Beauty of Fractals. Heidelberg: Springer-Verlag. ISBN 0-387-15851-0.
  10. ^ Peitgen, Heinz-Otto; Richter Peter (1986). The Beauty of Fractals. Heidelberg: Springer-Verlag. ISBN 0-387-15851-0.


[edit]External links

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From Wikipedia, the free encyclopedia

The Mandelbrot set is a famous example of a fractal

Frost crystals formed naturally on cold glass illustrate fractal process development in a purely physical system

fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,”[1] a property called self-similarity. Roots of mathematically rigorous treatment of fractals can be traced back to functions studied by Karl WeierstrassGeorg Cantor and Felix Hausdorff in studying functions that were analyticbut not differentiable; however, the term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractusmeaning “broken” or “fractured.” A mathematical fractal is based on an equation that undergoes iteration, a form of feedbackbased on recursion.[2]

A fractal often has the following features:[3]

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.

Images of fractals can be created using fractal-generating software. Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, such as when it is possible to zoom into a region of the fractal that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals.




To create a Koch snowflake, one begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral “bump.” One then performs the same replacement on every line segment of the resulting shape, ad infinitum. With every iteration, the perimeter of this shape increases by one third of the previous length. The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains finite. For this reason, the Koch snowflake and similar constructions were sometimes called “monster curves.”

The mathematics behind fractals began to take shape in the 17th century when mathematician and philosopher Gottfried Leibnizconsidered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).

It was not until 1872 that a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with Weierstrass’s abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch curve. (The image at right is three Koch curves put together to form what is commonly called the Koch snowflake.) Waclaw Sierpinskiconstructed his triangle in 1915 and, one year later, his carpet. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. The idea of self-similar curves was taken further by Paul Pierre Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, theLévy C curveGeorg Cantor also gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals.

Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri PoincaréFelix KleinPierre Fatou and Gaston Julia. Without the aid of modern computer graphics, however, they lacked the means to visualize the beauty of many of the objects that they had discovered.

In the 1960s, Benoît Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word “fractal” to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term “fractal”.


Julia set, a fractal related to the Mandelbrot set

A class of examples is given by the Cantor setsSierpinski triangle and carpetMenger spongedragon curvespace-filling curve, andKoch curve. Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals can bedeterministic (all the above) or stochastic (that is, non-deterministic). For example, the trajectories of the Brownian motion in the plane have a Hausdorff dimension of 2.

Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (seeattractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension of 2—but what is truly surprising is that theboundary of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by Mitsuhiro Shishikura in 1991. A closely related fractal is the Julia set.

[edit]Generating fractals

The whole Mandelbrot set
Mandelbrot zoomed 6x
Mandelbrot Zoomed 100x
Mandelbrot Zoomed 2000xEven 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set.

Four common techniques for generating fractals are:


Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:

  • Exact self-similarity – This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity. For example, the Sierpinski triangle and Koch snowflake exhibit exact self-similarity.
  • Quasi-self-similarity – This is a looser form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly self-similar. The Mandelbrot set is quasi-self-similar, as the satellites are approximations of the entire set, but not exact copies.
  • Statistical self-similarity – This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of “fractal” trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar. The coastline of Britain is another example; one cannot expect to find microscopic Britains (even distorted ones) by looking at a small section of the coast with a magnifying glass.

Possessing self-similarity is not the sole criterion for an object to be termed a fractal. Examples of self-similar objects that are not fractals include the logarithmic spiral and straight lines, which do contain copies of themselves at increasingly small scales. These do not qualify, since they have the same Hausdorff dimension as topological dimension.

[edit]In nature

Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakescrystals,mountain rangeslightningriver networkscauliflower or broccoli, and systems of blood vessels and pulmonary vesselsCoastlines may be loosely considered fractal in nature.

Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples—a branch from a tree or afrond from a fern is a miniature replica of the whole: not identical, but similar in nature. The connection between fractals and leaves are currently being used to determine how much carbon is contained in trees.[5]

In 1999, certain self similar fractal shapes were shown to have a property of “frequency invariance”—the same electromagnetic properties no matter what the frequency—from Maxwell’s equations (see fractal antenna).[6]

A fractal that models the surface of a mountain (animation)

Photograph of a romanesco broccoli, showing a naturally occurring fractal

Fractal pentagram drawn with a vectoriteration program

[edit]In creative works

Further information: Fractal art

Fractal patterns have been found in the paintings of American artist Jackson Pollock. While Pollock’s paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work.[7]

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns.[8] It involves pressing paint between two surfaces and pulling them apart.

Fractals are also prevalent in African art and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.[9]

In a 1996 interview David Foster Wallace admitted that the structure of his novel Infinite Jest was inspired by fractals, specifically the Sierpinski triangle.[10]

The song “Hilarious Movie of the 90’s” from Pause (album) by the artist Four Tet employs the use of fractals.[11]


A fractal is formed when pulling apart two glue-covered acrylic sheets.

High voltage breakdown within a 4″ block of acrylic creates a fractal Lichtenberg figure.

Fractal branching occurs in a fractured surface such as a microwave-irradiatedDVD.[12]

DLA cluster grown from a copper(II) sulfate solution in an electrodepositioncell

A “woodburn” fractal

A magnification of the phoenix set

fractal flame created with the programApophysis

Fractal made by the program Sterling

A fractal created using the programApophysis and a julian transform

A double fractal found in nature. Ice on a vine. Two fractals naturally occurring at once


Main article: Fractal analysis

As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications of fractals include:[13]

[edit]See also


  1. ^ Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company.. ISBN 0-7167-1186-9.
  2. ^ Briggs, John (1992). Fractals:The Patterns of Chaos. London : Thames and Hudson, 1992.. pp. 148. ISBN 0500276935, 0500276935.
  3. ^ Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd.. xxv. ISBN 0-470-84862-6.
  4. ^ The Hilbert curve map is not a homeomorhpism, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert map in R2are both 2. Note, however, that the topological dimension of the graph of the Hilbert map (a set in R3) is 1.
  5. ^ “Hunting the Hidden Dimension.” Nova. PBS. WPMB-Maryland. 28 October 2008.
  6. ^ Hohlfeld R, Cohen N (1999). “Self-similarity and the geometric requirements for frequency independence in Antennae”. Fractals 7 (1): 79–84. doi:10.1142/S0218348X99000098.
  7. ^ Richard Taylor, Adam P. Micolich and David Jonas. Fractal Expressionism : Can Science Be Used To Further Our Understanding Of Art?
  8. ^ A Panorama of Fractals and Their Uses by Michael Frame and Benoît B. Mandelbrot
  9. ^ Ron Eglash. African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press 1999.
  10. ^
  11. ^
  12. ^ Peng, Gongwen; Decheng Tian (21 July 1990). “The fractal nature of a fracture surface”Journal of Physics A 23 (14): 3257–3261. doi:10.1088/0305-4470/23/14/022. Retrieved 2007-06-02.
  13. ^ “Applications”. Retrieved 2007-10-21.

[edit]Further reading

[edit]External links

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From Wikipedia, the free encyclopedia

ray-traced image of the 3D Mandelbulb
for the iteration z ↦ z8c.

Daniel White and Paul Nylander constructed the Mandelbulb, a 3-dimensional analog of the Mandelbrot set, using a hypercomplex algebra based on spherical coordinates.[1]

White and Nylander’s formula for the nth power of the 3d hypercomplex number \langle x, y, z\rangle is:

\langle x, y, z\rangle^n = r^n\langle\cos(n\theta)\cos(n\phi),\sin(n\theta)\cos(n\phi),\sin(n\phi)\rangle


\begin{align}r&=\sqrt{x^2+y^2+z^2} \\  \theta&=\arctan(y/x) \\  {\rm and\ } \phi&=\arctan(z/\sqrt{x^2+y^2})=\arcsin(z/r).\end{align}

They use the iteration z\mapsto z^n+c where z and c are 3-dimensional hypercomplex numbers with the power map z\mapsto z^n defined as above.[2] For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of “lobes” controlled by the parameter n. Many of their graphic renderings use n = 8.


[edit]External links

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