This website is a gallery of computer-generated fractal art as well as a text that explains what it is and how it is created. Central to the website are a myriad of fractal images shown in "Stories about Fractal Art" augmented by the text as optional references. In addition, there are two galleries of fractal art: Gallery 2D is a collection of two-dimensional (2D) images made recently, while Gallery 3D comprises such 3D objects as fractal mountains (some are realistic and others imaginative), fractal forests and fractals painted on various nonplanar surfaces. Here are examples:

 "Symmetric Rocks in Desert"

 "Pyramid" "Escher-like Fern Mountains" 

 "Newton's Apple" "Dragon's Egg" "Broken Taiko Drum" 

 "Copper Bowl" "Antique Vase" 

 "Tamarack Forest"

"Birdless Island" 

and
"Mandelbrot Moon Over Fractal Mountains"

Copyright © 1997-2025 Junpei Sekino 
The website was last updated on January 14, 2025 

Digital Artist (Author's Profile): When Junpei Sekino was 10 years old he won first prize for the junior division in a national printmaking contest in Japan.
He now combines art and mathematics to create fractal art. ...from MathThematics, Book 3, Houghton Mifflin, 1998, 2008.
Stories about Fractal Art
From Art and Mathematics Married by Computer

Contents   Introduction § 1. Prep Math: Orbits, Dynamical Systems and Topological Ideas
§ 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
§ 5. Julia Sets and the Fundamental Dichotomy
§ 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
.
Fractal Coloring Algorithms Gallery 2D Gallery 3D Sekino's Home Page


Introduction: Speaking loosely without using technical terms such as the Hausdorff-Besicovitch dimension, a fractal is an object that is self-similar, i.e., a large part of it contains smaller parts that resemble the large part in some way; see Figures 0.1-0.5 shown below. Benoit Mandelbrot coined the term "fractal" in 1975 and created a branch of mathematics called fractal geometry seven years later. As an "IBM fellow," he had access to some of the best computers and technical assistants available for his research at the time.
Figure 0.1(A) Figure 0.1(B).  A Julia Set Figure 0.1(C)
The Mandelbrot Set Found on the "Map" of Figure 0.2(A) A Newton Fractal



Figure 0.1(D)
"Heart to Heart" 		Figure 0.1(E)
"Puffer Fish in Love"


Various "Field Guide Maps" for Finding Julia Sets of Quadratic Equations;
The Mandelbrot set is the simplest of all.


The (Generalized) Mandelbrot Sets
Figure 0.2(A) Of Certain Cubic (Left) and Quartic Equations Figure 0.2(B)


They are also "Field Guide Maps" for finding Julia sets; see § 6.


Julia Sets
Figure 0.2(C) "Born" from the "Green Leaves" of Figure 0.2(B) Figure 0.2(D)


Every Julia set from the "green leaves" is "connected" in one piece; see Figure 6.11(B)



Figure 0.2(E). "Line Dancing Seahorses"

A "Disconnected" Julia Set "Born" from an Orange Disk of Figure 0.2(B)



Figure 0.2(F). Julia Sets Found Without "Maps"

Julia sets are defined independently of special "maps" such as Figure 0.2(B).



Our world has fractals everywhere exemplified by trees, mountains, blood vessels, mycelium strands, stock market graphs, weather patterns, seismic rhythms, ECG signals and brain waves. In its article entitled "How Mandelbrot's fractals changed the world," the BBC states that fractal geometry has practical applications in diverse areas including diagnosing some diseases, computer file compression systems and the architecture of the networks that make up the Internet.

The idea of fractal was not particularly new in mathematics for Mandelbrot's time or the computer age, as Georg Cantor introduced the prototypical Cantor set in 1883 and various others appeared as shown in Figure 0.3. In the 1910s, Pierre Fatou and Gaston Julia independently laid the foundations for fractals generated by "dynamical systems" of complex numbers, the idea pioneered by Henri Poincaré a few decades earlier.

The Julia sets, representing their fractals and named after Gaston Julia, were largely ignored, however, because of the lack of technology to show their beauty, before the Mandelbrot set was born on a computer screen. The Cantor sets, the Julia sets and the Mandelbrot set are all subsets of the complex plane.



Figure 0.3. Classical Fractals in Mathematics (Refurbished by a Computer)


Koch Snowflake (1904) Sierpinski Rectangle (1916) Pythagorean Tree (1942)


Sierpinski Triangle (1915) Pascal's Triangles with Modular Arithmetic Devil's Staircase (1884)



It was 1980 when Mandelbrot published his computer-generated image of a novel object, now called the Mandelbrot set, and completely altered the fate of fractals in art and mathematics. Mandelbrot, who once studied under Gaston Julia and later became an "IBM fellow," used the computer to visualize the fundamental dichotomy of the Julia sets, which is a special case of the Fatou-Julia Theorem and divides up the complex plane into two regions.

Thus born was an object that turned out to be astoundingly complex as well as beautiful, unimaginable from the dichotomy. It invigorated the interests in fractals by numerous mathematicians including Adrien Douady and John H. Hubbard, who named the object the "Mandelbrot set" and established many of its properties. Because attractive images of the Mandelbrot set can be generated by simple computer algorithms, it also found a way out of mathematical communities and into popular culture.

The NOVA program on the subject was broadcast by PBS in 2008 and stated, "Largely because of its haunting beauty, the Mandelbrot set has become the most famous object in modern mathematics. It is also the breeding ground for the world's most famous fractals. Since 1980, the set has provided an inspiration for artists, a source of wonder for schoolchildren, and a fertile testing ground for the science of linear dynamics."  (Note: "linear" was probably a typo for "nonlinear.")



Figure 0.4(A). A Subset of the Mandelbrot Set in Color




Figure 0.4(B). Another Subset of the Mandelbrot Set in Color




Figure 0.4(C). "Nighttime View" of the Mandelbrot Set in Figure 0.4(B)

It is given by lighting up the invisible filaments of the boundary of the Mandelbrot set.



The Mandelbrot set conceals infinitely many subsets with exquisite details and captivating compositions. Moreover, the detailed patterns are dictated by the boundary of the Mandelbrot set comprising razor-thin "filaments" in the manner illustrated in Figure 0.6(A).  Figure 0.4(C) shows a portion of the boundary of the Mandelbrot set, which is a monochromatic line art and is astoundingly complex reflecting the infinitely recursive self-similarity.

Mandelbrot defined fractals using the Hausdorff dimension (aka Hausdorff-Besicovitch dimension), which is a generalization of the familiar dimensions and quantifies the complexities caused by the self-similarity, roughness and irregularities not found in traditional Euclidean geometry. For example, the Hausdorff dimensions of a line and the boundary (perimeter) of the Koch snowflake shown in Figure 0.3 are 1 and approximately 1.26, respectively, showing that the latter is more complex than the line.

Like the Julia sets, the Hausdorff dimension was created in the 1910s and remained rather obscure before 1980 when the Mandelbrot set emerged and fractals became popular. In 1998, Mitsuhiro Shishikura proved that the Hausdorff dimension of the boundary of the Mandelbrot set is the same as the dimension of the plane, which is 2 and maximal for any objects lying in the plane. This effectively proved that the Mandelbrot set containing its boundary is one of the most complex objects ever plotted on a plane.



Figure 0.4(D). A Julia Set in Color Found on the "Map" of Figure 0.4(A)

Fatou and Julia were born too early to see computer-generated Julia sets.



Figure 0.4(E). Another Julia Set from Figure 0.4(A) in Different Coloring

The last two images are from near the "mini Mandelbrot set" visible in Figure 0.4(A).



Computer users attracted to fractal art soon came to realize that there are infinitely many varieties of Julia sets providing some of the most beautiful fractal art images. Meanwhile, the Mandelbrot set, initially created to show the whereabouts of the Julia sets classified into the dichotomy's two types, grew to be a well designed map on the complex plane to show how the wider and more detailed varieties of the Julia sets are distributed.

Any image of a subset of the Mandelbrot set featuring a "mini Mandelbrot set" functions as such a map as well. A mini Mandelbrot set is a smaller copy of the Mandelbrot set, and it is known amazingly that any segment of the boundary of the Mandelbrot set tangles with infinitely many mini Mandelbrot sets. Most of them are so small they are invisible in a computer image, but when a computer zooms in on them and they become visible, they may reveal unique, intricate and dazzling patterns surrounding them. The Julia sets "born" from such an object naturally inherit the intricate patterns.



Figure 0.5(A). A Mini Mandelbrot Set in Color
The Mandelbrot set is about 1026 times as big as the mini Mandelbrot set; see § 3.
By comparison, the number of all stars in the observable universe is about 2 × 1023.



Figure 0.5(B). A Julia Set in Color

A Julia set "born" from the Mini Mandelbrot set of Figure 0.5(A)



Figure 0.5(C). Another Mini Mandelbrot Set in Color




Figure 0.5(D). "Ghost Dragons"

A Julia set in color "born" from the mini-Mandelbrot set of Figure 0.5(C)



The late 1970s and the early 1980s were an exciting period for computer lovers as portable desktop computers such as Apple II not only became widely available but also grew into major entertainment devices boosting their popularity, thanks to the arrivals of "Space Invaders" and "Pacman" from Nintendo.

Since then, a large part of mathematics became experimental like chemistry and physics as younger mathematicians began to utilize computers as their research tools. They find clues and solutions by conducting simulations and numerical and graphical experiments on computers. The trend began mainly because they witnessed the debuts of the subjects where computers played an essential role, namely, "fractal geometry"―and "chaos theory."



Figure 0.6(A). Classical Fractal in Art: Hokusai's "Great Wave off Kanagawa" (1831)

Key Line Art Block Depicting
"Chaotic" Wave of the Sea 		Final Woodblock Print
= Key Block + Color Blocks


Figure 0.6(B). Similarities
Between Hokusai's Woodblock Print and Computer-Generated Fractal Art
Chaotic Julia Set Julia Set in Color


in Fractal Geometry in Fractal Art


Chaotic Julia Set Julia Set in Color


in Fractal Geometry in Fractal Art



Figure 0.6(C). "Congregating Owls"

A Julia set (in color) born from the "Mandelbrot Moon" shown below



Figure 0.6(D). "Mandelbrot Moon"

A Mini Mandelbrot set painted on a sphere for a change; see Gallery 3D.



Figure 0.6(E). "Running Corolla"

A chaotic Julia set born from the "Mandelbrot Moon" shown above; cf. Figure 1.2(G).



Figure 0.6(F). "Running Corolla"

A highly chaotic Julia set born from the Mini Mandelbrot set of Figure 0.5(A)



Figure 0.6(G). "Running Corolla"

A Julia set Found in the Next Image as a Map



Chaos was basically born as a brand new subject in 1974 from biologist Robert May's computer simulations of population dynamics through the dynamical system called the logistic equation. He then discovered an extremely complicated sequence of numbers that was unlike any of the population changes he had observed earlier and that reacted "sensitively" to a minuscule perturbation on its number to drastically alter its behavior. He called these sequences and the source of the sequences "chaotic".

Like "fractal," the word "chaos" was used as a mathematical term for the first time in 1975 when the American Mathematical Monthly published "Period Three Implies Chaos" by T.Y. Li and James Yorke. The paper inspired by May's discovery received a great sensation especially because there appeared very little difference between chaotic and random outcomes even though the former resulted from deterministic processes.



Figure 0.7(A). "Spring Reflection"

A fractal from the area where chaos was discovered.
Note the bifurcation pattern on the "leaves."



Figure 0.7(B). "Autumn Leaves"

Another fractal from the area where chaos was discovered.



The idea of chaos quickly evolved into comprehensive chaos theory in science and at the same time its affinity with fractal art became clearer. As we'll see, it is the aforementioned sensitivity of chaos that causes drastic and unpredictable changes of colors and patterns in the image, and the boundary of the Mandelbrot set turned out to be exactly where the set is chaotic.

As we have seen, the boundary is razor-thin and borders between the two regions in the complex plane comprising totally opposite characters defined by the dichotomy. It is not surprising that such a border becomes "chaotic" like in human interactions. A Julia set, which is also defined as the boundary of another object, became known to be chaotic as well and its chaotic tendency gets more pronounced if it is born from nearer the boundary of the Mandelbrot set.

Chaos contributes to fractal art by adding randomness, unpredictability and irregularities, and, while it makes the artwork more exciting and creative, excessive chaos may generate overwhelming noise and disorder. Thus, balanced chaos is a key to success in fractal art and one way of accomplishing it is to capitalize on the property of the boundary of the Mandelbrot set by tweaking the parameters in the dynamical system when we engage in fractal plotting.



Figure 0.7(C). A Mini Mandelbrot Set with its Chaotic Boundary

Is chaos too little, too much or balanced in this image ?



Googling we can find a host of websites displaying numerous computer-generated fractal art images that are often stunningly beautiful. It indicates that a large population not only appreciates the digital art forms but also participates in the eye-opening creative activities. Since computer use is essential for such aspirations, a fairly large part of this article is devoted to show how to program a computer and plot popular types of fractals generated by simple dynamical systems. It is not a text on computer programming and instead tells the general principles in everyday language easily translatable to a computer language.

Central to our programming are iterations comprising repetitions of thousands (or occasionally millions) of a simple process per "pixel" defined by a dynamical system. Through iterations our computer zooms in and miraculously reaches a "picoscopic" object like the mini Mandelbrot set of Figure 0.5(A) within a reasonable duration and generates amazing self-similarity like in the fractal output of Figure 0.4(C). Chaos may also ensue after the simple process is iterated so many times. So, it is through iterations that an output image tends to be enormously complex and with unpredictable details. If all goes well, the image may become a breathtaking masterpiece of art.

Particularly exciting in fractal plotting is, therefore, the moment the fractal image generated by our personal program emerges on our computer screen. Even if it turned out not to be an artistic masterpiece, it may still stir our imaginations in the part of mathematics that is in fact quite deep and still filled with unknowns. One might even discover a thing or two in the relatively young field of fractal geometry in which computer experiments often lead the way. It is plain fun.



Figure 0.8(A). "The Lion from the Mandelbrot Moon"

A Julia set born from the Mini Mandelbrot set of Figure 0.6(D)
"Lions," "Elephants" and "Seahorses," etc. are star actors in the Julia set scenes.



Figure 0.8(B). "Ghost Elephant"

A Julia set born from the Mini Mandelbrot set of Figure 0.5(C)




Go to   Top of the Page Introduction § 1. Prep Math
§ 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
§ 5. Julia Sets and the Fundamental Dichotomy
§ 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
Fractal Coloring Algorithms Gallery 2D Gallery 3D


§ 1.  Preparatory Mathematics

As mentioned at the outset, the text in this article is meant as an optional reading and is not needed in appreciating the fractal art images. On the other hand, people who wish to understand fractals in some depth by following the article need to know (a) elementary algebra and geometry of complex numbers and (b) beginning calculus. In addition, people interested in learning fractal plotting should have (c) some basic computer programming experience. Courses covering (a), (b) and (c) are available in high school.

(a) includes the practice of writing a complex number z as a point (x, y) in the xy-plane as well as the standard algebraic expression z = x + yi and ability to do basic arithmetic of complex numbers such as addition and multiplication. The complex plane means the set of all complex numbers z = (x, y) which coincides with the Cartesian xy-plane.

For each complex number z = (x, y), the absolute value of z means |z| = √(x2 + y2) and it represents, through the Pythagorean theorem, the distance of z from the origin of the complex plane. More generally, if u and v are complex numbers, |u - v| represents the distance between u and v, which satisfies the triangle inequality  |u - v| ≤ |u| + |v|. Setting w = u - v, we get |w| ≤ |w + v| + |v|, or equivalently, |w + v| ≥ |w| - |v|. The last form of the triangle inequality will be used a few times in the upcoming sections to justify certain key propositions.

The only ideas we need from (b) are a sequence of numbers that may converge or diverge and the derivative and a critical point of a function where the derivative vanishes.


Figure 1.0(A).  "Dancing Seahorses"

See Fractal Coloring for the Step-by-Step Coloring Algorithm for the Image
Generated by the Mandelbrot Equation


We now introduce several preliminary ideas.

Orbits and Dynamical Systems: When we solve a mathematical problem using a computer, we frequently do it by exploiting what the machine does best, namely an iteration. It means repeating a certain process over and over, often for thousands or even millions of times, at a blinding speed. To see how it works, consider the best-known equation in fractal plotting, which we call the Mandelbrot equation for convenience:

(1.1)     zn+1 = zn2 + p ,  with   n = 0, 1, 2, · · · , 

where zn+1, zn and p are complex numbers and p is called a parameter. The iteration index n is especially important for fractal plotting and it is there for us to iterate the equation to generate a sequence of complex numbers once the value of p and initial value z0 are given. For instance, let p = -2 and z0 = 0. Then setting the index n = 0, 1, 2, · · ·  in (1.1), our properly programmed computer iterates (1.1) and calculates the sequence of numbers

   z0 = 0,  z1 = z02 + p = 02 - 2 = -2 ,  z2 = z12 + p = (-2)2 - 2 = 2 ,  z3 = z22 + p = 22 - 2 = 2 , · · · ,

i.e.,  z0 = 0,  z1 = -2 ,  z2 = 2 ,  z3 = 2 , · · · ,  z30 = 2,  z31 = 2, · · · ,

which is called the orbit of p = -2 with the initial value z0 = 0 or the orbit of z0 = 0 with the parameter value p = -2. If we hold the value of z0 at z0 = 0 and change the value of p from p = -2 to p = -1.9 in (1.1) then the computer again iterates (1.1) and quickly calculates thousands of terms in the sequence

   z0 = 0,  z1 = -1.9,  z2 = 1.71,  z3 = 1.0241, · · · ,  z30 = -1.1626,  z31 = -0.5483, · · · ,

which is now called the orbit of p = -1.9 with the initial value z0 = 0 as well as the orbit of z0 = 0 with the parameter value p = -1.9. It is important to note that an orbit may change its behavior drastically if the parameter value p changes slightly. For instance, unlike the orbit of p = -2 which becomes static after time n = 2, the orbit of p = -1.9 keeps moving along the real axis of the complex plane in a seemingly unpredictable way as time n progresses. Note that we refer to the index n as time or instant.



Figure 1.0(B). "Turquoise Lion"

Technical Description: The Julia Set of p = (0.281215625, 0.0113825)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation


Figure 1.0(C).  "Esmeralda Lion"

Technical Description: The Julia Set of p = (0.281150625, 0.011546875)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation


We have used only real numbers for simplicity, but the orbits used in fractal plotting are sequences of complex numbers in the complex plane. Because most of the orbits dance around in the complex plane with time n, it is appropriate to call a collection of orbits a dynamic mathematical system or dynamical system. For example, the Mandelbrot equation (1.1) is a dynamical system consisting of infinitely many orbits of complex numbers, one orbit zn for each choice of values of p and z0. As we shall see, there are infinitely many dynamical systems including the Mandelbrot equation (1.1) and the logistic equation (7.1), each of which generates infinitely many fractals.



Figure 1.1(A). "Circus Seahorses"

Technical Description: The Julia Set of p = (0.03697296, 0.55091235)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Dynamical System zn+1 = zn3 + zn + p


Figure 1.1(B).  "Circus Elephants"

Technical Description: The Julia Set of p = (0.0641826, 0.5406694)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Dynamical System zn+1 = zn3 + zn + p


Canvases: We begin with a simple example. Let R be the rectangle in the complex plane defined by -2 ≤ x ≤ 2 and -1.28 ≤ y ≤ 1.28 and suppose we wish to plot the graph of the inequality x2 + y2 ≤ 1 on R using a computer. We first decompose R into, say, 50 × 32 miniature rectangles of equal size called picture elements or pixels and then represent the pixels by pixel coordinates (i, j) in such a way that the upper left and lower right pixels are (0, 0) and (49, 31), respectively. Thus, the i- and j-axes of the pixel coordinate system are the rays emanating from the upper left corner of R and pointing east and south, respectively; see the diagram in Figure 1.2(A) on the left.

Let imax = 50, jmax = 32, xmin = -2, xmax = 2, ymin = -1.28 and ymax = 1.28. Then for each i = 0, 1, 2, · · ·, imax-1 and j = 0, 1, 2, · · ·, jmax-1, the pixel (i, j), which is a rectangle, contains infinitely many complex numbers (x, y). For our computational purpose, we choose exactly one representative complex number (x, y) in the pixel (i, j) by setting

(1.2)    Δx = (xmax - xmin) / imax; Δy = (ymax - ymin) / jmax,

(1.3)    x = xmin + i Δx;  y = ymax - j Δy.

Consequently, we may view R as the rectangle comprising imax × jmax = 50 × 32 pixels, each of which has a unique representative complex number. The rectangle R with the pixel structure is called a canvas for plotting the output image with the image resolution of 50 × 32 pixels.

Plotting the graph of the inequality x2 + y2 ≤ 1 on the canvas is now easy. For each pixel (i, j), we examine its representative complex number (x, y) on the canvas R. If it satisfies the inequality, color the pixel red, and otherwise, color it white. Since the coloring process uses only finitely many pixels of the canvas R, the output image that resembles the Japanese flag is an approximation of the true graph. The greater the number of pixels, the higher the image resolution and the more accurate the output image.

Figure 1.2(A) shows two approximations of a fractal called "Goldfish in Love." The one on the left is painted on a canvas with 50 × 32 pixels and the other on a canvas with 500 × 320 pixels.


Figure 1.2(A). "Goldfish in Love" with Different Image Resolutions


Technically, a canvas can be defined by any positive integers imax and jmax and any real numbers xmin, xmax, ymin and ymax with  xmin < xmax and ymin < ymax, but we normally impose the ratio equality

(1.4)    (ymax - ymin)/(xmax - xmin) = jmax/imax

on the input values xmin, xmax, ymin and ymax. Then (1.4) implies that Δx = Δy in (1.2) so each pixel is a square as shown in Figure 1.2(A). This way the red circle in the aforementioned output would not look oval.


Figure 1.2(B). "Jellyfish Queue"

Technical Description: A Subset of the Mandelbrot Set on a p-Canvas
Centered at p = (0.28212284496875, 0.0110092373125)
Generated by the Mandelbrot Equation with z0 = 0


Figure 1.2(C). Mini-Mandelbrot Sets

Technical Description: A Mandelbrot Fractal of z0 = i/√3 on a p-Canvas
Centered at p = (0.00401324, -1.98544205)
Generated by the Dynamical System zn+1 = zn3 + zn + p


p-Canvases and z-Canvases:  Consider a dynamical system, say, the Mandelbrot Equation comprising infinitely many orbits. Each orbit is determined by values of p and z0 and interchangeably called the orbit of p with the initial value z0 or the orbit of z0 with the parameter value p. We have also seen that a canvas is a rectangle in the complex plane consisting of pixels (i, j), each of which has a representative complex number (x, y). In fractal plotting, we view the complex numbers representing the pixels as values of p and call the canvas a p-canvas or view these complex numbers as values of z0 and call the canvas a z-canvas.

In Introduction we mainly discussed two types of fractals, the Mandelbrot set (and its subsets) and Julia sets (and their subsets), both generated by the Mandelbrot equation. In § 2 and § 4 we plot the former on p-canvases and in § 5 the latter on z-canvases. On a p-canvas, the value of p varies and and the value of z0 is fixed at a constant, and that's where we use orbits of p with fixed z0. On a z-canvas, it's the other way around, and that's where we use orbits of z0 with fixed p.

For example, at each parameter p on the p-canvas (with fixed z0), there corresponds the orbit of p (with fixed z0), and as we'll see, its behavior determines the color of the pixel containing p. As we have seen, the orbits of p belonging to two adjacent pixels on the p-canvas may have totally different behaviors, in which case the colors of the pixels may be drastically different. These ideas will appear over and over with examples and become clear in the upcoming sections.

Because a fractal image plotted on a p-canvas is typically the Mandelbrot set, we often call such a fractal a Mandelbrot fractal. This is especially convenient when we plot a fractal on a p-canvas generated by a dynamical system other than the Mandelbrot equation. Similarly, we often call a fractal plotted on a z-canvas a Julia fractal.


Figure 1.2(D). "Jady Unicorns"

Technical Description: A Julia Fractal of p = (-0.3959, 0.0312)
On a z-Canvas Centered at z0 = (-1.254375, 0.052756)
Generated by the Dynamical System zn+1 = zn4 + zn + p



Figure 1.2(E). "Metallic Unicorns"

Technical Description: A Julia Fractal of p = (-0.39985, 0.02025)
On a z-Canvas Centered at z0 = (-1.254375, 0.052756)
Generated by the Dynamical System zn+1 = zn4 + zn + p



Figure 1.2(F). "Gold Elephants"

A Subset of the Julia Set in Figure 6.9(B)
Painted on a z-Canvas with different colors




Figure 1.2(G). "Running Corolla"

Technical Description: The Julia Set of p = (-1.1128, 0.23076)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation



Geometric Similarity:  We say that two objects in a plane are geometrically similar if one can be obtained from the other by uniform scaling (enlarging or reducing), translation, rotation and/or reflection; see Wikipedia for detail. Geometrically similar objects are said to be congruent if one can be obtained from the other without uniform scaling. We learn the concept of geometric congruence in high school geometry mostly using triangles and conditions like "side-angle-side." In this article, we don't distinguish geometrically similar fractals and treat the images such as the ones shown below to be identical.



Technical Description: See Figure 6.0 in § 6


In geometry, we generally imagine that objects such as triangles are made of a rigid material like a metal plate. If they are made of something totally elastic like pizza dough that can be deformed by kneading then we are in the realm of topology instead of geometry. Here are some of the basic topological ideas that will appear in the upcoming sections.

Topological Ideas:  We say that nonempty sets A and B of points in the complex plane are topologically equivalent if there is a continuous function h mapping A onto B in a one-to-one fashion such that its inverse h-1 is also continuous. Such a function h is called a homeomorphism, which is a formal notion of "kneading a pizza dough to change its shape from A to B." For example, a triangle and a square are topologically equivalent or homeomorphic while they are not geometrically similar.




Technical Description: See Example 5 in § 5


A topological property means a property of a set that is preserved under a homeomorphism. For example, being connected as "one piece" is a topological property because if A and B are homeomorphic and A is connected then B must be connected as well. Similarly, having no holes is a topological property. "Tearing" or "poking a hole" on a pizza dough is not part of "kneading." In § 3, we will define connectedness more carefully as it plays an important role in fractal geometry.

To see a few more topological ideas which we will encounter later on, consider a circle in the complex plane. The disk that comprises all of the points inside the circle but none of the points on the circle is called an open disk. Let A be a set of points in the complex plane and call the set of points not in A the complement of A. Then a point b is called a boundary point of A if every open disk about b contains a point belonging to A and a point belonging to the complement of A. The set of all boundary points of A is called the boundary of A and the largest subset of A without any of its boundary points is called the interior of A.

We say that A is closed if it contains all of its boundary points, and A is open if it contains none of its boundary points. Thus, closed and open sets are generalizations of closed and open intervals on the real number line. We also say that A is bounded if there is a circle in the complex plane that encloses A, and A is compact if A is closed and bounded. Compactness, closedness and openness are all topological properties but boundedness is not.


Figure 1.4(B).  The Mandelbrot Set
With its Complement (Left Green), Boundary (Left Amber) and Interior (Right)

Plotted on a p-Canvas
by the Divergence Scheme of § 2 		Plotted on a p-Canvas
by the Convergence Scheme of § 4


As we'll see, the Mandelbrot set generated by the Mandelbrot equation is bounded so its global figure fits in a circle or a rectangle like in Figure 1.4(B), and from the shape of its black silhouette, it is often nicknamed "Warty Snowman" lying sideways. A part of the global image is called a local image, but a local image in fractal art is usually given by zooming in on a small rectangular neighborhood of a point that is near or on the border of the snowman's silhouette.

Because the Mandelbrot set is also known to be closed, its boundary, which coincides with the snowman's silhouette's border, is a part of the Mandelbrot set. It is also known that the boundary has the "topological dimension" of 1, meaning intuitively that it comprises razor thin "filaments" without thickness, just like the boundary of a circular disk. So, the boundary is mostly invisible in local images unless we "light up" their filaments like in a tungsten light bulb; see Figure 1.4(C) shown below.

As Figure 1.4(B) shows, the filaments are like branched hairs growing outwards from the warty snowman and known to carry infinitely many miniature copies of the snowman, namely mini-Mandelbrot sets. One of them is visible in Figure 1.4(C) and, as we have seen, the boundary of the Mandelbrot set is incredibly intricate.


Figure 1.4(C).  A Local Image of the Mandelbrot Set
With its Complex Boundary Highlighted by the Goldish Color

Painted on p-Canvases Centered at p = (0.281229249, 0.011344208)
See "Daytime View" and "Nighttime View" of a Fractal in § 3


Go to   Top of the Page Introduction § 1. Prep Math
§ 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
§ 5. Julia Sets and the Fundamental Dichotomy
§ 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
Fractal Coloring Algorithms Gallery 2D Gallery 3D



§ 2.  The Divergence Scheme

We say that a sequence zn of complex numbers diverges to ∞ if the real sequence |zn| diverges to ∞, i.e., if zn gets further away from the origin of the complex plane without bound as n gets larger. The object of § 2 is to introduce a fractal plotting technique, called the "Divergence Scheme," associated with the notion of divergence of orbits of complex parameters p generated by the Mandelbrot equation (1.1).


Figure 2.0(A). A Sample Fractal Generated by the Divergence Scheme

A Local Image of Figure 2.3.


We first view the complex plane as the set of all (complex) parameters p, and for each p in the complex plane, define a function fp of a complex variable z by setting fp(z) = z 2 + p. Since p is a constant in each fp, its derivative is fp'(z) = 2 z, hence, its critical point is z = 0. Yes, we can apply the familiar rules of differentiation from elementary calculus on fp. We then write the dynamical system (1.1) as

(2.1)   zn+1 = fp(zn) = zn2 + p

and set

(2.2)   z0 = 0,

which is the critical point of fp. Thus, for each p, (2.1) together with (2.2) constitutes the orbit zn of p with the fixed initial value z0 = 0. Because its initial value is the critical point, we call the orbit the critical orbit of p. Throughout § 2 - § 4, we assume, unless otherwise stated, that every orbit zn is a critical orbit of p satisfying (2.1) and (2.2).

We now have the following surprisingly simple definition of what NOVA dubbed "the most famous object in modern mathematics." In § 5, we'll explain its relation with the fundamental dichotomy we mentioned in Introduction.

The Mandelbrot Set, which we will denote by ℳ, means the set of all parameters p in the complex plane whose (critical) orbits do not diverge to ∞.


Figure 2.0(B). Another Sample Fractal Generated by the Divergence Scheme

Technical Description: A Local Image of The Mandelbrot Set on a p-Canvas
Centered at p = (0.28212348434375, 0.0110096504375)


If a and b are real numbers, let max{a, b} denote the largest of a and b. To develop our fractal plotting method, we need:

Proposition A: Any orbit zn of p (critical or noncritical) diverges to ∞ if and only if |zm| > max{2, |p|} for some m.

Proof: Suppose |zm| > max{2, |p|} for some m. Then by (2.1) and the triangle inequality, we have
   |zm+1| = |zm2 + p| ≥ |zm|2 - |p| ≥ |zm|2 - |zm| = |zm|(|zm| - 1) = α|zm|,
where α = |zm| - 1 > 1. Since |zm+1| ≥ α|zm| > |zm| > max{2, |p|}, we may repeat the above argument to get
   |zm+2| ≥ |zm+1|(|zm+1| - 1) ≥ α|zm|(|zm| - 1) ≥ α2|zm|.

By induction, it follows that |zm+k| ≥ αk|zm| for any k ≥ 1. Since α > 1, we conclude that if |zm| > max{2, |p|} for some m then the orbit zn diverges to ∞. The converse of the statement is trivial.

Proposition B: If |p| > 2, then the critical orbit zn of p diverges to ∞, i.e., p does not belong to the Mandelbrot set ℳ.

Proof: If |p| > 2, (2.1) and (2.2) imply |z1| = |p| > 2; hence, by the triangle inequality, we have
    |z2| = |z12 + p| ≥ |z1|2 - |p| = |p|2 - |p| = |p|(|p| - 1) > |p| = max{2, |p|}.

Hence, by Proposition A, the orbit zn diverges to ∞.

Figure 2.1(A). The Mandelbrot Set


With zm > θ, θ = 2


With zm > θ, θ = 10

Propositions A with |p| ≤ 2 and Proposition B together imply:

The Divergence Criterion:  |zm| > 2 for some m if and only if the (critical) orbit zn of p diverges to ∞.

Here, we note that if |p| > 2 the divergence criterion is trivial because of Proposition B, and if |p| ≤ 2 then max{2, |p|} = 2 in Proposition A. We also note that |p| > 2 means p is outside of the circle with radius 2 and |zm| > 2 for some m means: The orbit zn of p gets out or "escapes" from the circle at some instant or time n = m.

We now use the divergence criterion and a computer to plot the Mandelbrot set ℳ. Let R be a square canvas comprising 2,000 × 2,000 = 4,000,000 pixels centered at the origin (0, 0) of the complex plane with radius 2, i.e., R is bounded by xmin = -2, xmax = 2, ymin = -2 and ymax = 2. Defining a canvas is always the first step of fractal plotting.

We then regard R as a p-canvas so as to identify each pixel (i, j) in the canvas with a unique parameter p belonging to the pixel.

The Divergence Scheme: Plotting ℳ on the p-canvas is now easy. Paint the entire canvas R, say, white initially, and let M = 1000 and θ = 2. For every pixel (i. j), er, parameter p, in the p-canvas R, iterate (2.1) with (2.2) at most M times and paint the canvas R as follows:

  •   If |z1| > θ then color the pixel p black ,
  • else if |z2| > θ then color the pixel p red ,
  • else if |z3| > θ then color the pixel p black ,
  • · · ·
  • else if |zM| > θ then color the pixel p red .

    • Thus, the above scheme assigns the color, red or black, to each pixel p in the p-canvas R according to how quickly the orbit zn of the parameter p escapes from the circle of radius θ = 2 before taking a long journey toward ∞; see the     divergence criterion. For example, if p = (2, 0) then |z1| = θ and |z2| > θ, so the "escape time" is m = 2 and the pixel p is colored red.

    We call the plotting process given by the if-statement the divergence scheme, so as to contrast it with the convergence scheme, which we will introduce in § 4.

    Of course, an actual computer program based on the divergence scheme can be streamlined in many ways. Probably the most important is to use |zm|2 > θ2 instead of |zm| > θ to avoid using the hidden square root in |zm| and shorten the computing time as it is used millions, if not billions, of times while running the program.

    Figure 0.1(A) shown at the outset of this article is the output image of the computer program in which the circle of radius θ = 2 is visible. The portion that retains the white canvas color and resembles a "snowman" figure is precisely an approximation of ℳ plotted on the canvas with finitely many pixels and by replacing ∞ in the definition of ℳ by "up until M = 1000."

    The first of the two images in Figure 2.1(A) shows a closeup of the approximated Mandelbrot set ℳ.

    Now, simple logic shows that the     divergence criterion remains true if we replace θ = 2 by any real number θ ≥ 2, and it implies that the divergence scheme is valid for any θ ≥ 2. The second image in Figure 2.1(A) is given by increasing the "threshold" from θ = 2 to θ = 10. Looking at the red-black stripes of both images, it appears that ℳ is better (albeit marginally) approximated if the threshold θ gets greater.

    In fact, the accuracy of a computer plot by the divergence scheme depends on the size (or image resolution) of the canvas, the maximum number of iterations M and the threshold θ. The computer plot gets more accurate if any of the three gets greater but with "diminished returns" and with the cost of increasing the computing time. We generally keep the threshold low between 2 and 10 and increase M and the canvas size for a better image. The best way of finding good numbers is to engage in frequent trial-and-error computer experiments. It gets easier quickly as it is similar to figuring out the amount of time needed to cook something in a microwave oven. We generally omit mentioning "approximation," understanding that all computer-generated fractal images are approximations of "real" things.
    Figure 2.1(B). The Mandelbrot Set


    Coloring Fractals by 24-Bit Colors:  Modern computers show graphics in the "24-bit colors" comprising 224 ≈ 16 million colors and we can modify the divergence scheme to take advantage of the capacity to plot colorful fractals such as Figures     2.0(A) and 2.0(B); see Fractal Coloring Algorithms. For example, the image on the left is painted by a basic technique described in the website.

    Important Fact:  Figures 2.1(A) and 2.1(B) show that the divergence scheme paints the     complement of ℳ while leaving ℳ in a single canvas color like white and black.

    Basic     Topological Properties of the Mandelbrot Set ℳ: Proposition B implies that ℳ is enclosed in the circle of radius 2 so it is bounded. It can be also shown that ℳ is closed, i.e., it contains its boundary as its subset. Therefore, ℳ is compact. Note that the boundary of ℳ is the boundary of the complement of ℳ as well.

    Zooming In On Local Images: Even though the images in     Figure 2.1(A) are painted in a primitive way that uses only three colors, it gives us valuable information about ℳ. For example, the red-black stripes in the images get more and more complex when they get nearer the boundary of ℳ. What kind of world do we have in the area that is extremely close to the boundary of ℳ ?

    The question leads to our common practice of zooming in on a small rectangular neighborhood of a point extremely near or on the boundary of ℳ. Here, the sides of the small rectangle are parallel to the coordinate axes of the complex plane so we can use the rectangle as a canvas with a large number of pixels to magnify the "local" image by the divergence scheme.

    In fractal art, the boundary of ℳ plays a role of utmost importance for its connection with chaos as described in Introduction.

    Example 1:  The image shown below on the left is a local image of ℳ given by zooming in on the microscopic square neighborhood of the complex parameter p = (-0.688497, 0.279885) with radius 0.000073. p is very near or on the     boundary of ℳ but not in the interior of ℳ. It is generated by the divergence scheme with the coloring technique similar to the one used to paint the "global" Mandelbrot set shown in Figure 2.1(B). The global image hides infinitely many intricate local images and we try to get them like treasure hunters.


       Figure 2.2. A Local Image of ℳ Generated by the Divergence Scheme





    People familiar with multivariable calculus can find a fun project of painting the fractal on a nonplanar surface like a sphere.

    Example 2:  Figure 2.3 is a cropped and resized image from a computer plot on the large square p-canvas with 6,400 × 6,400 pixels centered at the complex number (0.28206125, 0.011014375) with radius 0.0000011. M = 100,000 is used as the maximum number of iterations for the divergence scheme. The image contains several (deformed) replicas of the "snowman" painted black, namely, mini-Mandelbrot sets. They look like small isolated islands but as we'll find out in the next section, they are actually connected to ℳ by razor-thin "filaments" belonging to the     boundary of ℳ.


    Figure 2.3.  Another Local Image of the Mandelbrot Set



    The zooming process can be repeated on the local image to capture additional local images. It is time consuming to compute a fractal on such a large canvas, but it gives us an option of finding additional local images as well as an option of making a high resolution printout of the image. For example,     Figure 0.4(B) is given by zooming in on a microscopic rectangle in Figure 2.3 and applying the divergence scheme, and so is Figure 2.0(A)

    This and That:  (1) Here, we show that the zooming process is fairly easy. Use graphic software such as Photoshop and place the mouse cursor on the point on the image like Figure 2.3 we want to zoom in on and get its pixel coordinates (i, j). Then use the     conversion formulas (1.2) and (1.3) to convert the pixel coordinates (i, j) to the Cartesian coordinates (x, y) like (0.28122928, 0.01134422). This can be done by a simple computer program.

    (2) It is not too hard to generalize the divergence criterion and find a threshold like θ = 2 for a general polynomial dynamical system. We have also seen that any number θ ≥ 2 can be used as a threshold for the divergence scheme with possibly an improved output image. This provides us with a nice tool for computer experiments, when we use a polynomial dynamical system other than the Mandelbrot equation and want to avoid calculating a threshold. But be careful: Blindly increasing the threshold also increases the computing time without notably improving the output image. There is a good reason why Mandelbrot used the smallest threshold θ = 2 in plotting the Mandelbrot set when the computers were much slower.

    Multibrot Set:  A Multibrot set is a straightforward extension of the Mandelbrot set given by the Mandelbrot equation (2.1) with 2 replaced by an integer k ≥ 2. For example, shown below is a local image of the Multibrot set with k = 7 given by the seventh degree Mandelbrot equation

    (2.3)   zn+1 = fp(zn) = zn7 + p

    with z0 = 0 painted on a plane and a torus.


    Figure 2.4.  Seventh Degree mini-Mandelbrot Set



    Go to   Top of the Page Introduction § 1. Prep Math
    § 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
    § 5. Julia Sets and the Fundamental Dichotomy
    § 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
    Fractal Coloring Algorithms Gallery 2D Gallery 3D



    § 3.  The Mandelbrot Set

    In 2008, PBS broadcast a NOVA program proclaiming that the Mandelbrot set had become "the most famous object in modern mathematics."  Since its birth in 1980, the Mandelbrot set, denoted ℳ, popularized fractal plotting by computers and has been the gold standard for all types of fractals. In this section, therefore, we'll discuss some of its notable properties.

    In § 2, we noted that the divergence scheme paints the complement of ℳ in full color and ℳ itself by a single color like black or white with a cautionary remark that a computer plot is an approximation which is not 100% accurate. We have also seen that ℳ is     closed so it contains its boundary as its subset. As observed throughout in Introduction, the boundary is an extremely important object in fractal art as its complexity and chaotic nature have profound and direct impacts on the structure and aesthetics of the imagery.

    Because the topological dimension of the boundary is known to be 1 like the boundary of a circular disk, the boundary is, intuitively speaking, made of "razor-thin filaments" without thickness and normally invisible in a image like Figure 2.3. Using the fact that the boundary of ℳ coincides with the boundary of the complement of ℳ, we can actually use the divergence scheme to "light up" the filaments while darkening the complement to visualize the boundary. The following example shows that the boundary is indeed all over the image and dictates its composition in the manner illustrated by Figures 0.6(A) and 0.6(B):


    Figure 3.1.  The Boundary of the Mandelbrot Set in Figure 2.3



    Figure 3.1 shows that the boundary of ℳ in the rectangular area is vividly self-similar, making it a fractal as per our     informal definition, and the infinitely recursive self-similarity makes many areas of the boundary look fuzzy and two-dimensional. It is no wonder why nobody knows what the area of the boundary is. Here's a celebrated theorem regarding the boundary of ℳ and the Hausdorff dimension we discussed in Introduction:

    Shishikura's Theorem (1998): The Hausdorff dimension of the boundary of the Mandelbrot set is 2 (which is the topological dimension of the plane).

    As pointed out in Introduction, the theorem effectively proves that no figures on the plane are more complex than the boundary of the Mandelbrot set. Shishikura's theorem also makes the boundary a fractal according to

    Mandelbrot's Definition (1975): A fractal means a set for which the Hausdorff dimension strictly exceeds the topological dimension.

    For example, the boundary (perimeter) of the Koch Snowflake in Figure 0.3 is a fractal because its topological dimension and Hausdorff dimension are 1 and 1.26, respectively. Note that amaong all line art images, the difference between the Hausdorff dimension and the topological dimension of the boundary of ℳ is maximum, again hinting that it has a "maximum fractal complexity."


    Another Local Image of ℳ and its Boundary (Right)

    This is another local image of Figure 2.3.


    One of the most important     topological properties in fractal geometry is "connectedness" of a set and Figures 2.1(B) and even 3.1 appear to show that the Mandelbrot set ℳ with its complex boundary is "connected" as "one piece." To give precision to the intuitive concept involving "one piece," R. C. Buck adopts the following formal definition in his classical textbook for Advanced Calculus: Suppose S is a nonempty set of points in the xy-plane. S is said to be connected if it is impossible to split S into two disjoint sets, neither one empty, without having one of the sets contain a boundary point of the other.

    For example, it is known that the "neck" of the "snowman" in Figure 2.1 is the point (-3/4, 0), and if we cut the head off the body of the snowman with the vertical line x = -3/4, then either the head or the body contains the boundary point of the other, namely (-3/4, 0). Thus, the particular attempt fails to show that ℳ is disconnected (and the validity of the argument is assured by the theorem given below). Because of the complexity of its boundary, proving whether or not ℳ is connected is by no means a simple task, as evidenced by the fact that Mandelbrot initially conjectured ℳ to be disconnected and reversed it later without substantiation.

    The Douady-Hubbard Theorem (1982): The Mandelbrot set is connected.

    Adrien Douady and John H. Hubbard also proved that ℳ is "simply connected," which means ℳ has no holes. Topologically speaking therefore, ℳ is well-behaving as a     compact set in one piece without a hole. As described by Wikipedia, Douady and Hubbard established many of the fundamental properties of ℳ at an early stage and created the name "Mandelbrot set" in honor of Mandelbrot. They were the pioneers of the mathematical study of ℳ.

    "Who Discovered the Mandelbrot Set?" is the title of an interesting read that appeared in Scientific American in 2009. It writes: Douady now says, however, that he and other mathematicians began to think that Mandelbrot took too much credit for work done by others on the set and in related areas of chaos. "He loves to quote himself," Douady says, "and he is very reluctant to quote others who aren't dead."


    Figure 3.2.  "Daytime" and "Nighttime Views" of a Fractal

    Local Images of ℳ in a Neighborhood of p = (0.28212348434375, 0.0110096504375)



    "Daytime" and "Nighttime Views" of a Fractal:  Shown above on the left is another fractal generated by the Mandelbrot equation (2.1) and the divergence scheme, where the razor-thin filaments of the boundary of the Mandelbrot set ℳ are invisible. When its colors are darkened and the thin filaments are lit up, we get the "nighttime view" of the fractal on the right, vividly showing the presence of the complex boundary of ℳ hidden in the "daytime view" on the left.

    Note that the two views may appear totally different, mainly because the daytime view shows the     complement of ℳ. Figure 2.3 we have seen is a daytime view while Figure 3.1 is a nighttime view. The nighttime views are not as colorful but they show the underlying complexity of ℳ and make it easier for us to visualize the important theorems established by Shishikura, Douady and Hubbard. In terms of fractal art, it is again important to stress that the boundary of ℳ dictates the composition of the daytime view in the manner illustrated by Figures 0.6(A) and 0.6(B).

    Technical Remark:  Plotting the complex boundary of ℳ with reasonable accuracy may demand days and weeks of computing time even with a fast modern computer. Figure 3.3 shown below is a resized and cropped image from a fractal on the p-canvas with 4,000 × 4,000 pixels centered at the point

    p = (0.25000316374967, -0.00000000895972)

    with a microscopically small radius ≈ 0.0000000000003 = 3 × 10-13.  We note that p is very near the cusp (0.25, 0) of the cardioid in     Figure 2.1.


    Figure 3.3.  A mini Mandelbrot Set under the Microscope


    M = 1,500,000 M = 500,000


    For the above image on the left, we used whopping 1,500,000 iterations of the Mandelbrot equation for each black pixel. If we use M = 500,000 (still a large number) instead, the outline of the mini-Mandelbrot set becomes blurry as shown in the above picture on the right. Fortunately, computers (especially used ones) are inexpensive nowadays and we can easily afford a second or third computer to do tedious jobs. Programming carefully so as to minimize computing time is not as important as it used to be.

    Shown below is a nighttime view of the fractal on the left that reveals the boundary of the mini Mandelbrot set. In Figure 3.1, the boundary of ℳ appears to be made of razor-thin filaments and ℳ certainly looks simply connected (with no holes), but it is not the case in Figure 3.4. That is because the maximum number of iterations, M, is not large enough and still made the boundary near the mini Mandelbrot set a little blurry.


    Figure 3.4.  The Boundary of the mini-Mandelbrot Set



    Connected Components and the Interior of ℳ: In addition to all the wonders of the boundary of ℳ we have witnessed, we'll see in the next section that it is exactly where chaos occurs in ℳ. As mentioned in     Introduction, chaos adds unpredictability and irregularity to the imagery and makes it more exciting and creative. So, if we remove the boundary from ℳ, we may think the remainder, namely the interior of the Mandelbrot set, is rather boring. As we will soon find out, that's not the case at all.

    We stated earlier a precise definition of a set in the complex plane being "connected" as "one piece" and now wish to dig into the notion of "pieces." The earlier example shows that the "snowman" of Figure 2.1 cannot be split into "two pieces," the head and body, without having either one of them contain a boundary point of the other.

    If we restrict our attention to the interior of ℳ which does not contain any of the boundary points, the situation changes completely. Not only can we split the head from the body without worrying about the boundary points, we can actually decompose the snowman into numerous disjoint connected body parts including all those (circular) disks attached to the cardioid body. Note that each of the disks is an open set without a boundary point and it is maximal in the sense that it is not a proper subset of a larger connected subset of the interior of ℳ.

    In general, if S is any nonempty set of points in the complex plane, a nonempty maximal connected subset of S is called a connected component of S. It is an easy task for people familiar with elementary set theory to prove that S can be partitioned into the disjoint union of its connected components. Thus, S is connected if and only if it consists of exactly one connected component (or "piece"). By virtue of the Douady-Hubbard theorem, ℳ has exactly one connected component, but its interior is disconnected and has infinitely many connected components including the aforementioned open disks.


    Figure 3.5.  The Mandelbrot Set
    With its Complement (Left Green), Boundary (Left Amber) and Interior (Right)

    Plotted on a p-Canvas
    by the Divergence Scheme of § 2         		Plotted on a p-Canvas
    by the Convergence Scheme of § 4


    In § 4, we will develop our second and last fractal plotting algorithm called the convergence scheme and use it to paint some of the connected components of the interior of ℳ in various colors as shown in the above image on the right. There we will find that the components are subject to fascinating numerical patterns that make the full color painting possible.

    The set S is said to be totally disconnected if it is disconnected and every connected component of S comprises just one point. As we have seen, the topological dimension of a curve is 1, but the topological dimension of a totally disconnected set is 0. In § 5 and onward, we'll see fractals composed of the latter as well as those composed of curves. Just like in art, therefore, mathematics has its own stippling art as well as line art.

    Compactness, connectedness, the number of connected components, being simply connected without a hole and being totally disconnected are all topological properties. Topologists generally identify homeomorphic objects and use topological properties to distinguish objects. In the 3D space, for example, a donut and a coffee cup with a handle are the same to topologists but the "broken taiko drum" shown below and a ping pong ball are different.




    "Broken Taiko Drum"

    Here, we have the mini-Mandelbrot set of Figure 3.3 flipped vertically and painted in different colors and its application in multivariable calculus.


    Go to   Top of the Page Introduction § 1. Prep Math
    § 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
    § 5. Julia Sets and the Fundamental Dichotomy
    § 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
    Fractal Coloring Algorithms Gallery 2D Gallery 3D



    § 4.  The Convergence Scheme

    The Mandelbrot set has become so illustrious, everybody interested in fractals knows its "warty snowman" shape by heart. To its main cardioid-shaped body, a bunch of (circular) disks are tangentially attached, and to each of these disks another bunch of disks are tangentially attached. The pattern repeats as if the cardioid has children, grandchildren, great grandchildren and so on and so forth. Here, a "cardioid" means, instead of the familiar curve, the curve together with all the points inside the curve.

    As     Figure 3.1 shows, ℳ also contains infinitely many mini Mandelbrot sets, each of which again comprises a cardioid (which may be distorted) with infinite generations of disks (which may be distorted) and even smaller mini Mandelbrot sets. If we remove the boundary of ℳ from ℳ, we are left with the interior of ℳ comprising the interiors of these disks and cardioids, etc., which are the connected components of the interior of ℳ.

    Atoms and Molecules: Let's use Mandelbrot's idea shown in his article as a cue and call each connected component of the interior of ℳ an atom of ℳ and a (disjoint) union of one or more atoms a molecule. Thus, atoms include the interiors of all those disks and cardioids with various degrees of distortion and possibly other shapes nobody have encountered yet. An atom and the interior of ℳ are examples of molecules.

    As we saw in § 2, the divergence scheme cannot distinguish these atoms and paints them in a single color like black or white; see Figure 3.5. Our current goal is to develop another simple algorithm called the convergence scheme which will be used to color ℳ like in Figure 4.1 and many other fractals in upcoming sections. Along the way, we will see that the atoms are associated with "periods" like in chemistry (but in a totally different way). It will in turn lead us into a complex world portrayed by one of the most important open questions in fractal geometry called the "density of hyperbolicity" conjecture.


    Figure 4.1.  The Mandelbrot Set with Colorful Atoms



    Cycles and Periods: A sequence cn of complex numbers is called a cycle if there is a positive integer k satisfying cn = cn+k for any index n. The smallest such integer k is called the period of the cycle, and a cycle with period k is called a k-cycle for short. For example, the sequence

        1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, · · ·

    is a 3-cycle but not a 6-cycle or a 9-cycle. The sequence  0, 0, 0, 0, · · ·  is a 1-cycle, which we identify with the constant 0.

    A sequence zn is said to converge to a k-cycle cn provided that zn gets arbitrarily close to cn as n gets bigger, or more precisely, for any (tiny) real number ε > 0, there is an integer N > 0 such that n ≥ N implies |zn - cn| < ε.  For example, the sequences  1/2, 1/3, 1/4, 1/5, 1/6, · · ·  and  1/2, 2/3, 3/4, 4/5, 5/6, · · ·  converge to the constants 0 and 1, or equivalently, to the 1-cycles  0, 0, 0, 0, · · ·  and  1, 1, 1, 1, · · ·,  respectively. Therefore, the sequence

        1/2, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 4/5, 1/6, 5/6, · · ·, 1/1000, 999/1000, 1/1001, 1000/1001, · · ·

    converges to the 2-cycle  0, 1, 0, 1, 0, 1, 0, 1, 0, 1, · · ·, 0, 1, 0, 1, · · ·.
    Figure 4.2. The Convergence Scheme


    With k = 1



    With k = 1, 2, 3, 4

    With k = 1, 2, 3, ..., 90

    Again, suppose ε is a (small) positive real number. Using the aforementioned definition of a k-cycle and the     triangle inequality, it is easy to prove the following.

    Proposition:  Assume that a sequence zn converges to a k-cycle. Then there is a positive integer N such that for all n ≥ N, |zn+k - zn| < ε.

    The Convergence Scheme: Our new algorithm called the convergence scheme with period index k is based on this proposition and given by replacing the inequality |zm| > θ  of the     red-black divergence scheme of § 2 by the inequality |zm+k - zm| < ε. Thus, for each pixel (i, j), er parameter p, on the p-canvas R and its (critical) orbit zn, it is given by the if-statement:

  •   If |z1+k - z1| < ε then color the pixel p col1,
  • else if |z2+k - z2| < ε then color the pixel p col2,
  • else if |z3+k - z3| < ε then color the pixel p col3,
  • · · ·
  • else if |zM+k - zM| < ε then color the pixel p colM.

    • Here, col1, col2, ... , colM are prescribed colors and ε a small positive real number like 10-6, Δx or Δy; see     (1.2). ε works like the threshold θ of the divergence scheme, except that the output image is more accurate when ε is smaller.

    For simplicity, let's call a parameter whose critical orbit converges to a k-cycle a parameter of period k and call an     atom an atom of period k if it comprises parameters of period k. With that we have:

    Example 1 (The Mandelbrot Set): Start with the p-canvas R, which is the rectangle in the complex plane with center (-0.52, 0) and horizontal radius 1.65 and comprises 3,000 × 2,500 pixels.

    We first apply the divergence scheme with M = 20000 and θ = 2 on R and extract the Mandelbrot set ℳ comprising the pixels p whose critical orbits do not diverge to ∞. Then apply various convergence schemes with ε = 10-8 on ℳ. Figure 4.2 shows the (resized) output images of three molecules.

    The first image is generated by the convergence scheme with period index k = 1 and shows that the interior of the cardioid is an atom of period 1. Painting in subtle shades of red is done by a basic technique included in the Fractal Coloring site.

    The second image is given by the convergence scheme with period indices k = 1, 2, 3, 4, which is basically defined as the natural sequence of the four convergence schemes, the one with period index k = 1 followed by the one with period index k = 2, etc. It shows that the interior of the largest disk is an atom of period 2 and painted in subtle shades of orange. Similarly, the green and purple atoms are of periods 3 and 4, respectively.

    The third image is given by a straightforward extension of the scheme described in the preceding paragraph. Because there aren't enough colors that are easily distinguishable, the correspondence between the periods and colors of the atoms is not one-to-one. For example, the atoms of periods 2 and 5 are painted orange in the third image.

    Example 1 shows that the convergence scheme may mean the one with a single period index or multiple period indices. Note that the convergence scheme with, say, period index 6 cannot distinguish parameters of periods 1, 2, 3 and 6 that are divisors of 6. Therefore, we need to be a little careful when we program a computer to carry out the convergence scheme, especially the one with multiple period indices.

    Important Conjecture: We have just seen that each of the atoms visible in Figure 4.2 appears to have a period according to our computer experiments but we are, of course, uncertain if it actually holds, let alone for all of the atoms of ℳ. It is an extremely complex problem but we are not completely in the dark.

    An atom (or a connected component) of the interior of ℳ is said to be hyperbolic, using the technical term in the field of complex dynamics, if the atom has a single period. One of the most important conjectures in fractal geometry, called the density of hyperbolicity, states that every atom is a hyperbolic component and that every parameter on the boundary of ℳ is arbitrarily close to a parameter in a hyperbolic component.

    We may look at the second part of the conjecture in terms of what we stated in Introduction: Every segment of the boundary of ℳ tangles with infinitely many mini Mandelbrot sets. To see how amazing it is, we should look at the boundary on ℳ depicted in     Figure 3.1.

    In this article, we assume that the conjecture is true. For example, it implies that the critical orbit of every parameter in the interior of ℳ converges to a cycle and it resolves the fundamental weakness of the convergence scheme based on the aforementioned proposition. We now "know" that the hypothesis of the proposition is true in terms of the critical orbits of the parameters in the interior of ℳ, cementing the validity of the convergence scheme.

    Mini-Mandelbrot Sets: Recall that the last image of Figure 4.2 is painted on a large canvas, and it includes the parts given below in Figure 4.3. The closeup images show not only meticulously aligned circular atoms but also several molecules that look like little flying insects. Looking at his computer printout of ℳ created by a dot matrix printer of the 1970s, Mandelbrot initially thought they were "dirt."


    Figure 4.3.  Closeups of the Interior of the Mandelbrot Set



    The dirt turned out to be well structured molecules, namely mini Mandelbrot sets, which often play     important roles in fractal art. Here is a little and nondescript conjecture we have made based on our computer experiments: If ℳ ' is a mini Mandelbrot set, there is a one-to-one correspondence between the atoms of ℳ ' and the atoms of ℳ and an integer λ ≥ 2 such that for every atom of ℳ ', its period is given by λ k, where k is the period of the corresponding atom of ℳ. We call λ the base period of ℳ '.

    For example, the most visible mini-Mandelbrot set of Figure 4.3 happens to have the base period λ = 4 and is shown in Figure 4.4 and the inset of the periodicity diagram. There, it is painted by the convergence scheme with period indices λ k = 4, 8, 12, ..., 100 and with the colors of the Mandelbrot set of Figure 4.2 so as to emphasize the one-to-one correspondence between the atoms of ℳ and the atoms of the mini-Mandelbrot set.

    Figure 4.4 shows that the boundary of the mini-Mandelbrot set of the base period λ = 4 is more intricate than the boundary of the Mandelbrot set seen in     Figure 4.1. Unfortunately, most of the mini Mandelbrot sets we use in fractal art have base periods λ greater than 100 and the convergence scheme is too slow to paint them in full color using currently available computers. That's why the mini Mandelbrot set of say, Figure 3.3, is left in a single color black.


    Figure 4.4.  The the mini-Mandelbrot Set of Base Period λ = 4 and Its Boundary
    Compare with the Mandelbrot set of Figure 4.1


    Chaos in the Mandelbrot Set: In general, a critical orbit either diverges to ∞ or converges to a cycle, or else it is called a chaotic orbit. As we know, the complement of ℳ comprises the parameters whose critical orbits diverge to ∞. Because of the     density of hyperbolicity conjecture, each parameter with chaotic orbit falls into the boundary of ℳ and the orbit of each parameter p near the boundary of ℳ reacts "sensitively" to a minuscule change of the value of p and totally alters its dynamic behavior. In other words, the boundary is exactly where chaos occurs in the Mandelbrot set (provided that the conjecture is true).

    In fractal art, the "sensitive dependence on a parameter" near the boundary of ℳ causes the drastic changes of patterns in a nighttime view of the fractal in Figure 0.4(C) and the drastic changes of colors in a daytime view of the fractal in Figure 0.4(B).

    The "Eyeball Effect": We     have seen that the convergence scheme is valid for painting the interior of ℳ. What happens if we apply it on the complement of ℳ ?

    The picture shown below on the left is essentially the same as Figure 3.3 (but in different colors) and is given by the divergence scheme alone, while the one on the right is painted by the divergence scheme followed by the convergence scheme with period index k = 1 (using different colors) on the complement of ℳ. The "eyeballs" painted by the convergence scheme are caused by its "mistake" of confusing some of the slowly divergent orbits as convergent. The images show which parameters are affected. Figures 1.8(D) and 5.8 illustrate the "eyeballs" more vividly. Note: The "eyeball effect" shows that the converse of the proposition is false.


    Figure 4.5.  The Eyeball Effect (Right) Given by the Convergence Scheme




    Artist's Renderings: As we saw in § 2, any computer generated image of the Mandelbrot set ℳ is an approximation of ℳ, but it is probably more appropriate now to call the image an artist's rendering, as it gets increasingly more colorful and artistic. Here are a couple of artist's renderings of ℳ given by the convergence schemes with different coloring.


    Figure 4.6.  "Mandelbrot Platters"




    Periodicity Diagram: If we label the atoms of the Mandelbrot set in     Figures 4.2 and 4.4 by their periods instead of colors, we get the following periodicity diagram. The periods in the diagram show meticulously aligned numerical patterns that are easy to recognize and will play an important role in plotting many of the "Julia sets" in the next section. The numerical patterns are yet another amazing property of the Mandelbrot set ℳ.


    Figure 4.7.  Periodicity Diagram of ℳ

    Note: λ is the period of a mini-Mandelbrot set


    Go to   Top of the Page Introduction § 1. Prep Math
    § 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
    § 5. Julia Sets and the Fundamental Dichotomy
    § 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
    Fractal Coloring Algorithms Gallery 2D Gallery 3D


    § 5.  Julia Sets and the Fundamental Dichotomy

    As mentioned in     Introduction, fractals called "Julia sets" preceded the Mandelbrot set by about sixty years and appeared as a part of the work by Pierre Fatou and Gaston Julia. To show the main ingredients of their work, we begin with a polynomial function of a complex variable z,

    (5.1)  f(z) = cm zm + cm-1 zm-1 +  · · · + c2 z2 + c1 z + c0,

    where m ≥ 2, and cm, cm-1, · · ·, c1, c0 are complex constants with cm ≠ 0. Then consider the dynamical system

    (5.2)  zn+1 = f(zn) = cm znm + cm-1 znm-1 +  · · · + c2 zn2 + c1 zn + c0,   n = 0, 1, 2, · · · ,

    consisting of all orbits of z0 in the complex plane. It is important to note that here we regard the complex plane as the set of all possible initial values z0 (instead of parameters p as in the previous sections). We may also view any of the coefficients cj as a constant parameter p in (5.2) when such a need arises.

    Define the filled-in Julia set of f denoted by 𝒦(f) to be the set of all initial values z0 in the complex plane whose orbits do not diverge to ∞ and define the Julia set of f denoted by 𝒥(f) to be the boundary of 𝒦(f). Then it can be shown that 𝒦(f) and 𝒥(f) are both compact, so 𝒦(f) contains 𝒥(f) as its subset. Just like the boundary of the Mandelbrot set, the Julia set is a fractal most of the time. It can be connected or disconnected, and if it is totally disconnected, we call it a Cantor set or Cantor dust.

    Julia and Fatou independently established the following powerful theorem in 1918-1919:

    The Fatou-Julia Theorem:  (1) If all of the critical points of f belong to 𝒦(f), then 𝒥(f) is connected as one piece, and (2) if none of the critical points of f belong to 𝒦(f), then 𝒥(f) is totally disconnected and forms a Cantor set.


    Figure 5.0(A).  "Cloisonné Lion"

    The Julia Set of p = (0.26898, 0.004) in color; See Example 1.


    Henceforth in this section, we deal exclusively with the Mandelbrot equation with a constant parameter p, which is a special case of (5.2):

    (5.3)   zn+1 = fp(zn) = zn2 + p,   n = 0, 1, 2, · · · .

    For simplicity, we now write 𝒦(p) = 𝒦(fp) and 𝒥(p) = 𝒥(fp) and call them the filled-in Julia set of p and the Julia set of p, respectively. Because the definition of 𝒦(p) is almost identical to the     definition of the Mandelbrot set ℳ in § 2, we expect that the divergence and convergence schemes work for 𝒦(p) as well; see Fractal Coloring.

    Example 1:      Choose p = (0.26898, 0.004) from outside of the Mandelbrot set ℳ near its cusp p = (0.25, 0). Then Figure 5.0(A) is painted on a z-canvas (instead of a p-canvas as in the previous sections) centered at the critical point z0 = (0, 0) of the function fp by the divergence scheme with the threshold

          θ = max{2, |p|} = 2;  cf. proposition A.

    As in the case of ℳ, therefore, the image other than "the Cloisonné metal wires" shows the complement of the filled-in Julia set 𝒦(p), and hence, the critical point of fp at its center does not belong to 𝒦(p). Because it is the only critical point of fp, it follows from the Fatou-Julia theorem that the Julia set 𝒥(p) is a Cantor set.

    Figure 5.0(B) is given by highlighting the Julia set (the Cloisonné metal wires) in Figure 5.0(A). Because it is a Cantor set, it is supposed to be a stippling art, but the computer image looks more like a line art. Also, when the Julia set is a Cantor set, the filled-in Julia set has an empty interior and coincides with the Julia set.


    Figure 5.0(B).  A Nighttime View of "Cloisonné Lion"


    Just like the boundary of the Mandelbrot set, a Julia set was found to be chaotic. Its "sensitive dependence on the initial values z0" causes the drastic and unpredictable changes of colors and patterns and makes an image like Figure 5.0(A) exciting and creative.

    Example 2:      Choose p = (0.32835, 0.05695) from a circular atom of period 10 near the cusp of the main cardioid atom of the Mandelbrot set ℳ. The "lion," shown below in Figure 5.0(C), is painted on a z-canvas centered at the critical point z0 = 0 of the function fp by the convergence scheme with period index k = 10 and the background by the divergence scheme with the threshold θ = max{2, |p|} = 2.

    Recall that the center of the z-canvas used in the example is the critical point z0 = 0 of the function fp, whose orbit coincides with the critical orbit of p. Because the period of p is 10, the critical orbit converges to a cycle of period 10 at the center of the canvas. Therefore, the convergence scheme with period index k = 10 is a natural choice in decorating the filled-in Julia set 𝒦p. The image clearly shows that the critical point belongs to 𝒦(p); hence, the Julia set 𝒥(p) is connected by the Fatou-Julia theorem.

    It is another fascinating fact about the Mandelbrot set that the period of the parameter p is always reflected in the shape of the filled-in Julia set of p, as in the number of "snake-shaped lion's manes" although why it is so is not completely understood. The image in Figure 5.0(D) is similar except that the parameter is chosen from a nearby atom of period 21.


    Figure 5.0(C) "Medusa Lion" Figure 5.0(D) "Medusa Lion"


    Born from an Atom of Period 10 Born from an Atom of Period 21


    The most consequential theorem in fractal geometry is an immediate corollary to the     Fatou-Julia theorem and comes from the fact that the base function fp of the Mandelbrot equation (5.3) has exactly one critical point z = 0. Here is the theorem:

    The Fundamental Dichotomy: For any parameter p in the complex plane, the Julia set of p is either connected or totally disconnected.

    For example, the Julia set of     Figure 1.1(A) is neither connected nor totally disconnected and the dichotomy assures us that a Julia set like this never arises from the dynamical system (5.3). Mandelbrot, who once studied under Gaston Julia and later became an "IBM fellow," used a computer to visualize the fundamental dichotomy that divides up the complex plane into two parts. He initially defined the Mandelbrot set to be

    (†)     the set ℳ comprising all parameters p in the complex plane whose Julia sets are connected.

    He knew how to compute ℳ as the Fatou-Julia theorem clearly implies that the Julia set of p is connected if and only if the critical orbit of p does not diverge to ∞; see the computer-friendly definition of § 2. Thus, the famed Mandelbrot set ℳ was born from the dichotomy of the Julia sets. It also explains why the critical point of (5.3) is indispensable in computing ℳ.

    Henceforth, we call (†) the alternative definition of ℳ. It shows that the Julia sets 𝒥(p) of Figures 5.0(C)(D) (and hence the "Medusa Lions" as well) are connected, while "Cloisonné Lion" of Figure 5.0(A) is a Cantor set. Thus, from its birth, the Mandelbrot set is a "field guide map" on the complex plane to show where we can find connected Julia sets (in the line art form) as well as totally disconnected Cantor sets (in the stippling art form).


    Figure 5.0(E) "Twin Lions" Figure 5.0(F) 


    Filled-In Julia Sets Born from the Same Atom of Period 17 × 5.


    Today the Mandelbrot set ℳ as a "field guide map" is considerably more detailed because of the     periodicity diagram, and all experienced fractal artists know how to use the map and find Julia sets with many of the specific and fascinating shapes. As we have seen, "lions," "elephants" and "seahorses" are among the Julia set shapes.

    Example 3:  Start with an atom of period 17 in ℳ (as a map) between the two atoms used in Example 2. We know what to expect from a Julia set born from the atom of period 17. Then consider a "    second generation" atom of Period 17 × 5 attached to the atom of period 17. The filled-in Julia sets called "Twin Lions" and shown above are given by parameters belonging to the atom of period 85 = 17 × 5. Note that both factors 17 and 5 are clearly visible in the "Twin Lions."

    The two "lions" are painted by the convergence scheme with period index 85 and the background by the divergence scheme with the threshold θ = max{2, |p|} = 2. The curling directions of the mane of the "Twin Lions" are opposite to each other and depend on the locations of the parameters in the atom. All these details can be found easily through computer experimeents.

    "Esmeralda Lion" is an enlarged version of the filled-in Julia set shown above on the left and "Ruby Lion" shown below is an enlarged version of the filled-in Julia set shown above on the right. The topological structure of a Julia set gets more complex with the period of its generating parameter.


    Figure 5.0(G).  "Ruby Lion"

    The Filled-in Julia Set of p = (0.282311250, 0.012143125)


    We continue with yet another fascinating attribute of the Mandelbrot set, which is its role as the "field guide map" for finding all types of Julia sets. We have found that the area around the atoms near the cusp of the main cardioid is a "lion sanctuary" where the Julia sets roam disguising "lions" of various shapes. As     Figure 5.0(A) shows, if a parameter is outside of ℳ but near one of the atoms, it still generates a Julia set of a "lion shape" albeit a weaker resemblance. As the parameter moves further away from these atoms, its Julia set gradually loses the atoms' "gravity" and the "lion shape."

    An area around the atom of period 4 painted purple and located directly above the cusp of the main cardioid of ℳ in the periodicity diagram is a "bean field." The image shown below is a Julia set born from that atom. It is interesting to see, through computer experiments, how the "lion shape" gradually changes to the "bean shape" as a parameter moves from the lion territory to the bean territory using the atoms attached to the cardioid as stepping stones.


    Figure 5.1(A). "Dancing Beans" Born from an Atom of Period 4

    The Filled-In Julia Set of p = (0.262, 0.5701) on a z-Canvas Centered at z0 = (0, 0)


    The next three images are from an area around the shoulder of ℳ (as the snowman), which we call "Lerna" of the Mandelbrot set.


    Figure 5.2(A). "Hydra of Lerna" Born from an Atom of What Period?

    The Filled-in Julia Set of p = (-0.661, 0.3434) on a z-Canvas Centered at z0 = (0, 0)



    Figure 5.2(B). "Hydra of Lerna" Born from an Atom of Period 11

    The Filled-In Julia Set of p = (0.262, 0.5701) on a z-Canvas Centered at z0 = (0, 0)



    Figure 5.2(C). "Hydra's Ash"

    The Julia set of p = (-0.6891, 0.27896)
    which is near an atom of period 11 but outside of ℳ


    Again, if p is near but not in an atom, the Julia set of p still resembles the shape of a Julia set born from the atom as shown in Figure 5.2(C). As p gets further away from the atom, the "gravity" of the atom on p naturally wanes and the Julia set of p gradually loses the resemblance. Thus, the "field guide map" based on the Mandelbrot set is not confined to ℳ and instead stretches all over the complex plane.

    Example 4 (Jordan Curves): It can be shown that the Julia set of p = -2, which is the leftmost tip of the Mandelbrot set, is the closed interval [-2, 2] on the real axis in the complex plane. The next simplest is the Julia set of p = 0 belonging to the cardioid atom of the Mandelbrot set, which is the unit circle centered at z0 = 0.

    All other Julia sets of p belonging to the cardioid atom turn out to be, just like the unit circle, non-self-intersecting continuous loops in the complex plane called Jordan curves, but they are, unlike the unit circle, fractals without smooth segments and seen only on computer plots. For example, the filled-in Julia set and the Julia set of the parameter p = (-0.32, 0.25) belonging to the cardioid atom are shown below.


    Figure 5.3(A).  The Filled-in Julia Set and Julia Set of a Parameter of Period 1


    Jordan Curve


    The Jordan curve theorem states that a Jordan curve divides the plane into two parts, a bounded region called "inside" and an unbounded region called "outside." The theorem seems utterly obvious from a typical image like the one shown above, but the Julia set as a Jordan curve can get extremely convoluted geometrically if the parameter gets arbitrarily close to the boundary of the cardioid. In fact, the proof of the Jordan curve theorem is far from obvious involving algebra, analysis and topology and provides one of the fascinating topics in mathematics.

    Around the "ear of the snowman" containing the "    second generation" atoms of periods 2 × 3, 2 × 4, 2 × 5 and the "third generation" atoms of periods 2 × 3 × 9, etc. is a "runner park." "Running Corolla" of Figure 1.2(G) is a Julia set born from the atom of period 2 × 3 × 9 and the image shown below from the atom of period 2 × 4. Both factors 2 and 4 are visible in the latter.


    Figure 5.3(B). "Run for the Sun"

    A Filled-in Julia Set Born from an Atom of Period 2 × 4


    Many other shapes of Julia sets that can be found on the Mandelbrot set are well documented and shown on the Internet. The locations for the "lion sanctuary," etc. relative to the Mandelbrot set can be translated to the locations relative to any mini Mandelbrot set and provide us with Julia sets often with dazzling backgrounds. The image shown below is a Julia set born in the "hydra territory" of the mini Mandelbrot set in Figure 0.5(A) (which is essentially the same as Figure 3.3); see Introduction and Gallery 2D for more examples.


    Figure 5.4(A). "Hidden Hydra"

    The Julia Set of p = (0.250003163749637436, -0.000000008959716416)
    on a z-Canvas Centered at z0 = (0, 0)


    Local Images of (Filled-in) Julia Sets: One of the many wonders of the Mandelbrot set is that it has infinitely many dazzling local images. Not all Julia sets have such a magical power, but we can still dig into many of them to find nice local images. For example, consider the Julia set shown below. Because the Julia set is compact, the global image fits into a rectangle just like in Figure 5.5(A).


    Figure 5.5(A). Nighttime View of "Cuttlefish Lion"


    The Julia Set of p = (0.25000316374967, -0.00000000895972) on a z-Canvas Centered at z0 = (0, 0)


    Shown below is a local image of the Julia set shown above but it is painted by using various colors and the eyeball effect. The eyeball effect makes it easier to identify the numerous "cuttlefish" swimming in the global image. Note that one of the cuttlefish is at the center of the global image.


    Figure 5.5(B). "Partying Cuttlefish"



    If we zoom in on the center of the global image between the eyes of the central cuttlefish using different colors but without the eyeball effect, we get another local image of Figure 5.5(A) revealing the cuttlefish's mouth painted black. The black mouth is a part of the interior of the filled-in Julia set, showing that the Julia set is indeed not a Cantor set.


    Figure 5.5(C). Center of "Cuttlefish Lion"



    Local Similarities between Julia sets and the Mandelbrot Set: People with interests in fractal plotting inevitably observe striking resemblance between a local image of the Mandelbrot set and a local image of a Julia set from time to time. For example, if we zoom out from     Figure 3.3 slightly and paint the local image of the Mandelbrot set by the coloring scheme used for Figure 5.5(C), we get the image shown in Figure 5.5(D). Is there an explanation for the resemblance?
    Figure 5.5(D).
    mini-Mandelbrot Set of Figure 3.3


    Recall that the Mandelbrot set and a (filled-in) Julia set belong to two different complex planes, one comprising parameters p and the other initial values z0 of the Mandelbrot equation (1.1). The Mandelbrot set is by definition the set of all parameters p whose critical orbits do not diverge to ∞ and a filled-in Julia set is similarly defined in the other complex plane.

    A parameter p is called a Misiurewicz point if the critical orbit of p is not a cycle but becomes a cycle after finitely many iterations. For example, while discussing (1.1), we saw that the critical orbit of p = -2 is

      z0 = 0,  z1 = -2 ,  z2 = 2 ,  z3 = 2 ,  z4 = 2 , · · · .

    Because it is not a cycle but becomes a 1-cycle after two iterations, the parameter p = -2 is a Misiurewicz point.

    Some of the known facts are: (1) Misiurewicz points belong to the boundary of the Mandelbrot set. (2) If p is a Misiurewicz point, then the filled-in Julia set of p has no interior points, hence, coincides with the Julia set of p. (3) Misiurewicz points are "dense" in the boundary of the Mandelbrot set, i.e., every open disk about a point on the boundary of the Mandelbrot set contains a Misiurewicz point.

    Tan Lei's Theorem (1990): If p is a Misiurewicz point, the Julia set of p centered at z0 = 0 and a local image of the Mandelbrot set centered at p are asymptotically similar through uniform scaling (enlarging and reducing) and rotation; see Wikipedia and geometric similarity.

    At first glance, the scope of Tan Lei's theorem seems to be rather limited because of the aforementioned properties (1) and (2), but (3) boosts the theorem to be enormously powerful: Let p be a parameter on or near the boundary of the Mandelbrot set. Then it is either a Misiurewicz point or near a Misiurewicz point, and consequently, in a local image of the Mandelbrot set centered at p, we are likely to see a shape resembling the Julia set of p near its center z0 = 0. For this reason, the Mandelbrot set is sometimes called an "index" to all Julia sets.

    This probably explains why the local images like Figures 5.5(C) and 5.5(D) are strikingly similar even though the parameter p belonging to the interior of the mini-Mandelbrot set is not a Misiurewicz point. The sidenote to Figure 3.3 shows that the distance between p and a nearby Misiurewicz point is much less than 10-13. Figure 5.5(E) shows we can zoom out from Figures 5.5(C) and 5.5(D) while retaining some degree of similarity.


    Figure 5.5(E). "Cuttlefish"
    Swimming in the Mandelbrot Set (Left) and in the Julia Set (Right)




    Go to   Top of the Page Introduction § 1. Prep Math
    § 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
    § 5. Julia Sets and the Fundamental Dichotomy
    § 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
    Fractal Coloring Algorithms Gallery 2D Gallery 3D



    § 6.  Generalizations

    By the 1990s, hardware for personal computers greatly improved for fractal plotting, both in terms of computing speed and generating images with higher resolutions and a wider range of colors. Fractal generating software such as Fractint also became more accessible. As a result, plotting the Mandelbrot set became so popular that a great many computer hobbyists, digital artists, mathematicians and scientists have explored around it and shown their images on posters, T-shirts, coffee mugs and websites. In the 1990s, almost all calculus textbooks appeared to have fractals on the covers.

    Although the hidden beauty of the Mandelbrot set is inexhaustible, it has become quite a challenge to unearth attractive local images of the Mandelbrot set or Julia sets that look markedly different from what have been published by using available computers and software. An easy way to find a new pattern such as the one shown below is to use a dynamical system other than the Mandelbrot equation and there are infinitely many of them.


    Figure 6.0(A).  "Gold Dragon"
    The Connected Julia Set of p = (0.0618974, 0.5400784)
    Generated by (6.2) on a z-Canvas Centered at z0 = (0, 0)


    By a holomorphic function, we mean a complex-valued function of a single complex variable which is differentiable on some domain of the complex plane. Suppose fp is a holomorphic function of a complex variable z containing a complex parameter p. Then fp gives rise to a dynamical system

    (6.1)   zn+1 = fp(zn),   n = 0, 1, 2, · · · ,

    which is viewed as the collection of infinitely many sequences of complex numbers, one sequence zn for each choice of the parameter value p and the initial value z0. As     before, we call each sequence the orbit of p (with the fixed initial value z0) or the orbit of z0 (with the fixed parameter p).

    If z0 is any initial value, we define the Mandelbrot set of z0, denoted by ℳ(z0), to be the set of all parameters p in the complex plane whose orbits with the fixed initial value z0 do not diverge to ∞. Note that ℳ(z0) is a straightforward extension of the Mandelbrot set we saw in § 2, where z0 is a unique critical point of fp. Similarly, if p is any parameter, we define the Julia set of p, denoted by 𝒥(p) as shown in     § 5.

    As we have seen, the Mandelbrot set and the Jula set live in fundamentally different complex planes, the former in the complex plane as the set of all complex parameters p and the latter in the complex plane as the set of all initial values z0. As we have also seen, using a critical point z0 of fp in the Mandelbrot set has a historical and mathematical significance, but nothing prevents fractal artists from computing it using a noncritical point z0. In the next section, we'll show several Mandelbrot sets of noncritical points with interesting effects.

    Figure 6.0(B).  "Turquoise Dragon"
    The Disconnected Julia Set of p = (0.0371542, 0.5501254)
    Generated by (6.2) on a z-Canvas Centered at z0 = (0, 0)


    Holomorphic functions comprise a wide variety of functions including familiar polynomials, rational functions, trigonometric and exponential functions on which we can apply the familiar rules of differentiation. It means that we have infinite possibilities for explorations and quests for new varieties of fractals in fractal art and geometry.

    On the other hand, a transcendental function like an exponential or trigonometric function involves infinite series to evaluate its function values, and computing its fractal with thousands or millions of iterations per pixel is way more time consuming than computing a fractal of a simpler polynomial or rational function. It also raises more challenges that have to be dealt with in an ad hoc manner, like determining a threshold for the divergence scheme.

    In this website, we will show the first step of generalization by using a simple cubic function with two critical points. As we'll see, it will greatly increase the variations of fractals from what we have seen earlier with the quadratic function. It will also provide us with a unique "Mandelbrot set" or a "field guide map" for finding all types of Julia sets; see     Figures 0.2(A) in Introduction. Here we take advantage of the Fatou-Julia Theorem. The first step of generalization will clearly indicate what the second step will be; see Figures 0.2(B).

    Figure 6.1(A)
    The "Speared Giant"
    Figure 6.1(B)
    The "Toddler"



    Consider the cubic dynamical system

    (6.2)   zn+1 = fp(zn) = zn3 + zn + p ,   n = 0, 1, 2, · · · ,

    where the function fp has a conjugate pair of critical points ± i / √3.

    The comical image shown in Figure 6.1(A) is the Mandelbrot set ℳ (i / √3) of the critical point z0 = i / √3 given by (6.2), which we call the "Speared Giant."

    ℳ (i / √3) is painted by the convergence scheme with period indices k = 1, 2, 3, ..., 50 and is the complement of the dark green background painted by the divergence scheme with a sufficiently large     threshold.

    As in § 4, we again define an atom to be a connected component of the interior of ℳ (i / √3) and a molecule to be a (disjoint) union of atoms. Thus, the atoms are in a wide variety of shapes and include the interior of a purple disk as well as the interior of the red "Spearhead" (from the stone age with jagged edges).

    Notable molecules include the interior of the "Giant," who was speared, and the interior of the "Toddler," who launched the big spear at the Giant, seen near the bottom of Figure 6.1(A) like an isolated island. We name some of the molecules and areas partly for fun but mainly for necessity and convenience just as we name people and places.

    A closeup of the Toddler is shown in Figure 6.1(B). Compared to the     Mandelbrot set, the Toddler has a proportionately larger head (like a toddler) and its boundary is disconnected from the boundary of the Speared Giant. The interior of the Toddler is painted by the convergence scheme with period indices k = 2, 4, 6, ..., 50 and the colors matched with the colors of the Mandelbrot set emphasizing the one-to-one correspondence between their atoms.

    Not surprisingly, local images we find around the boundary of the Toddler are similar to those found near the Mandelbrot set. Here's one of them, which can be used as a night sky of 3D landscapes such as "    Mandelbrot Moon" in Gallery 3D.


    Figure 6.2(A). A Local Image Near the Toddler



    The origin (0. 0) of the complex plane is at the tip of the Spearhead, which coincides with the upper left corner of the closeup image shown below. We call the area "Spearhead Bay." Like the Mandelbrot set, the Giant contains infinitely many circular atoms that satisfy the numerical pattern of the periodicity diagram. These circular atoms include the largest and the second largest blue atoms shown in Spearhead Bay, whose periods happened to be 7 and 8, respectively. Note that the "Seaweed" growing out of the blue atom of period 7 contains seven-way junctions and likewise for the "Seaweed" around the atom of period 8.

    We also note that in Spearhead Bay, the Seaweed grows only on the side of the Giant Mandelbrot Set and tangles with infinitely many extra atoms that look like tropical fish. Interestingly, the fish-like atoms begin to disintegrate near the circular atom of period 6, which is painted purple at the mouth of Spearhead Bay, and they become extinct near the circular atom of period 5, which is located just outside of the bay. Apparently, the phenomena are related to the "Fusion Diagram" of Figures 6.6(A).


    Figure 6.2(B). "Spearhead Bay"


    The boundary of the Giant near the circular atom of period 5 is depicted in the image shown below. It shows no signs of fish but, like in the Mandelbrot set, it contains five-way junctions and tangles with numerous mini-Mandelbrot sets. Unlike the Mandelbrot set however, the boundary now appears to be disconnected.


    Figure 6.2(C). "Seaweed" Rooted on an Atom of Period 5



    The next image, "Cheetah," is a local image of the Speared Giant painted on a p-canvas centered at the complex number (0.04886516, -1.20677998) by the divergence scheme. Since it is disconnected, it cannot be found from the Mandelbrot equation.


    Figure 6.3(A).  "Cheetah"



    The image shown below is again a local image of the Speared Giant and is painted on a p-canvas centered at (0.04978, 1.094143) mainly by the convergence scheme with period index k = 21. The interior of each rose petal is an atom of period 21. Because it is highly unlikely that the Mandelbrot set has an atom like that, the image cannot be found from the Mandelbrot equation.


    Figure 6.3(B).  "Mini Mandelbrot Set with Roses"



    The next image is a close-up of the green molecule located between the Spearhead and the Toddler in     Figure 6.1(A) that looks like a pair of balloons. We call it "Broken Balloons" because of its "bursted lips" with jagged edges and small fragments. It is generated by the convergence scheme with period indices k = 3, 6, 9, ..., 60. Like the cardioid body of the Mandelbrot set, we again painted the atoms of the smallest period 3 red.


    Figure 6.4(A).  "Broken Balloons"



    The next image is a close-up of the green molecule seen near the top of     Figure 6.1(A). It is again generated by the convergence scheme with period indices k = 3, 6, 9, ..., 60 with k = 3 corresponding to the red atoms, just like in the "Broken Balloons."


    Figure 6.4(B).  "Broken Boxing Gloves"



    "Atomic Fusion" and the Mandelbrot Set of the Dynamical System: Recall that our dynamical system     (6.2) differs from the Mandelbrot equation (2.1) in the important fact that it comes with two critical points z0 = ± i / √3 = ± (0, 1 / √3) and hence without the dichotomy. Let ℳ1 = ℳ (i / √3), which is the Mandelbrot set of the critical point i / √3 depicted in Figure 6.1(A), which is in turn copied as the first image of Figure 6.5(A) shown below with the Toddler omitted for simplicity.

    We have noted that ℳ1 contains "irregularities" not seen in the Mandelbrot set from the earlier sections such as the jagged edges of the spearhead and flying fragments in the broken balloons and the broken boxing gloves. Let ℳ2 = ℳ (-i / √3), which is the Mandelbrot set of the critical point -i / √3. ℳ2 turns out to be symmetric to ℳ1 through the real axis of the complex plane and is shown in the second image of Figure 6.5(A) shown below.


    Figure 6.5(A).
    "Atomic Fusion" of the Mandelbrot Sets ℳ1 and ℳ2 of the Two Critical Points.

    ℳ1
    ℳ2
    ℳ1 ∪ ℳ2


    Interestingly, all of the fragments and jagged edges in ℳ1 and ℳ2 disappear in the third image of Figure 6.5(A), which we call "Atomic Fusion" given by superimposing ℳ1 and ℳ2. The figure shows that ℳ1 and ℳ2 fit so perfectly under the superimposition they are made for each other to be "fused together." Colors used in the third image are different from those used in the first and second images.

    Figure 6.5(B) shows the fusion of the Giant in ℳ1 with the Spearhead in ℳ2 and the fusion of the Broken Balloons in ℳ2 with the Broken Boxing Gloves of ℳ1. The fusion repaired the Broken Balloons beautifully.


    Figure 6.5(B).  Portions of the "Atomic Fusion"


    Figure 6.6(A)
    Simplified Fusion Diagram


    So, what would be the Mandelbrot set of the dynamical system (6.2) with two critical points?

    We first recall that the famed Mandelbrot set ℳ was born as a visualization of the     Fatou-Julia Theorem with a single critical point z0, namely, the Fundamental Dichotomy. For example, suppose p is any constant parameter and 𝒦(p) and 𝒥(p) stand for the filled-in Julia set and the Julia set of p, respectively. Recall that the orbit of z0 with the parameter p is the same as the orbit of p with the initial value z0.

    Thus, if z0 belongs to 𝒦(p), then the orbit of z0 with the parameter p does not diverge to ∞ and hence, p belongs to ℳ, and similarly, if z0 does not belong to 𝒦(p), then p does not belong to ℳ. Thus, the Fatou-Julia Theorem with a single critical point is equivalent to saying the following in terms of ℳ:

    (1) if p belongs to ℳ then 𝒥(p) is connected;
    (2) if p belongs to ℳc, then 𝒥(p) is a Cantor set,

    where ℳc stands for the complement of ℳ. (1) and (2) are exactly what Mandelbrot intended the Mandelbrot set to satisfy.

    Figure 6.6(A) shows a simplified version of the "Atomic Fusion" in Figure 6.5(A). Here, the union ℳ1∪ℳ2 is painted by yellow or red and the intersection ℳ1∩ℳ2 by red. Thus, the dark background is the complement of ℳ1∪ℳ2 denoted by [ℳ1∪ℳ2]c, and the yellow zone is the difference between the union and the intersction denoted by

       ℳ1∇ℳ2 = ℳ1∪ℳ2 - ℳ1∩ℳ2.

    Note that ℳ1∇ℳ2 happens to be the "symmetric difference" of ℳ1 and ℳ2 but it's just a coincidence for the current case with two critical points.

    We now consider the Fatou-Julia Theorem with two critical points, and suppose z0 is any one of them. Repeating the above argument allows us to conclude that z0 belongs to 𝒦(p) if and only if p belongs to ℳ(z0), namely, the Mandelbrot set of the critical point z0.

    It is now straightforward to prove that the Fatou-Julia Theorem applied on (6.2) can be written in terms of ℳ1 and ℳ2 as follows:

    (1) If p belongs to ℳ1∩ℳ2 then the Julia set of p is connected;
    (2) if p belongs to [ℳ1∪ℳ2]c then the Julia set of p is a Cantor set.

    (1) and (2) imply:

    (3) If the Julia set of p is disconnected but not a Cantor set, then p belongs to ℳ1∇ℳ2.

    All of our relevant computer output seems to show that the converse of (3) is true, but the Fatou-Julia Theorem does not confirm it.

    Note that if ℳ1 = ℳ2 then (1), (2) and (3) coincide with the earlier (1) and (2). Note also that we have just shown that the "Atomic Fusion" comprising ℳ1∪ℳ2 and ℳ1∩ℳ2 illustrates the Fatou-Julia theorem applied on (6.2) with exactly two critical points. It is natural, therefore, that we call ℳ1∪ℳ2 together with ℳ1∩ℳ2 the Mandelbrot set of the dynamical system (6.2).

    In terms of pictures, the fusion diagrams of both Figure 6.6(A) and Figure 6.5(A) represent the Mandelbrot set of (6.2). It is akin to the situation where we have two versions of the original Mandelbrot sets Figure 2.1(B) and Figure 4.1, the first as a visual interpretation of the Fatou-Julia theorem and the second as a detailed map for finding Julia sets.

    Example 1: The Julia set of Figure 0.1(B) called "Twin Dragons" and shown at the outset of this website is given by the parameter p = (0.185, 0.00007666) belonging to ℳ1∩ℳ2; hence, it is connected. Figure 6.7(A) shown below contains three topologically distinct "Twin Dragons."


    Figure 6.7(A).  "Twin Dragons"

    p = (0.2011575, 0.00002) in ℳ1∩ℳ2


    p = (0.21828, -0.00230) in [ℳ1∪ℳ2]c p = (0.2176, 0.0128) in ℳ1∇ℳ2


    Figure 6.8(A) "Connected Roses"


    It is hard to tell from the picture if the first "Twin Dragons" is connected but the connectedness is assured by the Fatou-Julia Theorem. Similarly, the second image is a Cantor set. The third image shows a kind that does not appear in the dichotomy theorem, namely a disconnected Julia set which is not a Cantor set.

    Example 2: Recall that the     Broken Balloons is a molecule comprising atoms of periods

        k = 3 × 1, 3 × 2, 3 × 3,  · · · .

    It can be seen near the bottom of Figure 6.6(A) that it intersects with both ℳ1∩ℳ2 (the red zone) and ℳ1∇ℳ2 (the yellow zone).

    The connected "Roses" of Figure 6.8(A) is the Julia set of the parameter p = (0.02912, -1.093853) belonging to an atom of period 3 × 7 in the Broken Balloons. The parameter p belongs to ℳ1∩ℳ2, so the numerous "roses" seen in the image are connected by the "stems." We can clearly see the number 7 in the picture but where do we see the number 3 ?

    The disconnected "Roses" of Figure 6.8(B) shows the Julia set of the parameter p = (0.07761, -1.12427) belonging to an atom of period 3 × 4 in the Broken Balloons.  The parameter p is chosen from ℳ1∇ℳ2, so the Julia set is disconnected, which we can see in the broken "stems." Note that the Julia set is not a Cantor set.



    Figure 6.8(B).  "Disconnected Roses"



    "Elephants" also pop up along with many other shapes in and around the Broken Balloons.  The next two images show examples of the Julia sets of parameters belonging to [ℳ1∪ℳ2]c near the Broken Balloons. They are both Cantor sets.


    Figure 6.8(C).  "Cantor Elephants"


    p = (0.087, -1.1848) p = (0.092, -1.1728)


    Example 3: The Toddler seen near the bottom edge of     Figure 6.1(A) belongs to ℳ1∇ℳ2 but it is omitted from the Fusion Diagram for simplicity. Recall that it comprises atoms of periods k = 2 × 1, 2 × 2, 2 × 3, · · · . It produces a great many attractive fractals but they are naturally similar to the fractals coming out from the Mandelbrot set for the quadratic system—except that they are all disconnected. For example, the image which is shown below and resembles the "Hydra" of Figure 5.2(A) in its basic form is the Julia set of p = (0.00399109,-1.98545775) belonging to an atom of period 2 × 13. It contains numerous dots in its background each of which is a baby hydra.


    Figure 6.9(A).  "Lernaean Hydra with Thirteen Heads and Offsprings"



    The image shown below is a Julia set born from the Toddler's cardioid atom of period 2 × 1 near its cusp. It has a basic shape familiar from the famed Mandelbrot set, but like the image shown above, it is disconnected.


    Figure 6.9(B). "Flying Lion"

    The Julia Set of p = (0.00033, -2.0006785)
    On a z-Canvas Centered at z0 = (0, 0) Generated by (6.2)



    Example 4: While the     Toddler generate Julia sets that resemble Julia sets of the Mandelbrot set seen in § 5, the Giant produces Julia sets that do not resemble anything from the Mandelbrot set, apparently affected by the Spearhead. "Twin Dragons" of Figure 6.8 are such examples. Here is another, this time from near the neck of the Giant.

    Figure 6.10(A).  "Pearly Dragon"

    The Julia Set of p = (0.0352236, 0.5448064)
    On a z-Canvas Centered at z0 = (0, 0) Generated by (6.2)
    The Julia set is totally disconnected.


    Consider the quartic dynamical system

    (6.3)   zn+1 = fp(zn) = zn4 + zn + p ,   n = 0, 1, 2, · · · ,

    where the function fp now has three critical points  α  and  α(1 ± i √3)/2  with  α = -(1/4)1/3.  Figure 6.11(A) shows the Mandelbrot set, ℳ1, of the first critical point. The poor "Giant" is shot by two spears this time launched by two "Toddlers," where the tips of the spearheads is again at the origin of the complex plane. The Mandelbrot sets of the other two roots turned out to be the same as the clockwise rotations of ℳ1 120o and 240o about the origin. Call them ℳ2 and ℳ3, respectively.

    The three Mandelbrot sets are again beautifully     fused together to generate the "fusion diagram" shown in Figure 0.2(B). A simplified version of the fusion diagram is shown below in Figure 6.11(B), where ℳ1∩ℳ2∩ℳ3 is painted green and ℳ1∪ℳ2∪ℳ3 is painted yellow or green.


    Figure 6.11(A)
    The Mandelbrot Set of α         		Figure 6.11(B)
    The Mandelbrot Set of (6.3)




    It is again easy to show that the     Fatou-Julia Theorem applied on (6.3) can be written in terms of ℳ1, ℳ2 and ℳ3 as follows:

    (1) If p belongs to ℳ1∩ℳ2∩ℳ3 then the Julia set of p is connected;
    (2) if p belongs to [ℳ1∪ℳ2∪ℳ3]c then the Julia set of p is a Cantor set.

    For this reason, we call ℳ1∪ℳ2∪ℳ3 together with ℳ1∩ℳ2∩ℳ3 the Mandelbrot set of the dynamical system (6.3). In terms of pictures, it is again represented by both Figure 6.11(B) and Figure 0.2(B). The former illustrates the historical significance of the Mandelbrot set tied with the Fatou-Julia theorem and the latter the modern role of the Mandelbrot set as a "field guide map" on the complex plane to show where we can find a variety of the Julia sets given by (6.3).

    Note that (1) and (2) imply (3): If the Julia set of p is disconnected but not a Cantor set, then p belongs to the difference

        ℳ1∇ℳ2∇ℳ3 = ℳ1∪ℳ2∪ℳ3 - ℳ1∩ℳ2∩ℳ3.

    All of our relevant computer output seems to show that the converse of (3) is also true.


    Figure 6.12(A).  "Unicorns"

    The Julia Set of p = (-0.39674, 0.01012) belonging to [ℳ1∪ℳ2∪ℳ3]c
    See also Figure 0.2(C) born from ℳ1∩ℳ2∩ℳ3 and Figure 1.2(E) born from ℳ1∇ℳ2∇ℳ3


    There are infinitely many polynomials generating fractals that cannot be found from the Mandelbrot set. Here are a few of them:


    Figure 6.13(A).  "Phoenix"



    Figure 6.13(B). "Dancing Metabo Seahorses"



    Figure 6.13(C). "Cloisonné Turtle"



    Go to   Top of the Page Introduction § 1. Prep Math
    § 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
    § 5. Julia Sets and the Fundamental Dichotomy
    § 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
    Fractal Coloring Algorithms Gallery 2D Gallery 3D



    § 7.  The Logistic Equation

    In § 2 - § 5, we saw many wonders of the Mandelbrot set ℳ. Despite the fact that it is generated by a simple quadratic equation     (1.1), it exhibits stunningly intricate and beautiful patterns when visualized. For math students of all ages, ℳ is visually engaging, captures their attention and makes various abstract concepts in mathematics such as infinity and topological properties more tangible.

    The boundary of ℳ is self-similar as well as chaotic and has unsurpassed fractal complexity, while the interior of ℳ comprises atoms with astounding geometric and numeric attributes that affect the shapes of the Julia sets in a mysterious way. "Infinite generations" of circular atoms are lined up around the cardioid ofℳ in a miraculous precision and at the same time subject to well-defined numeric patterns given by their periods. Because of its boundary, ℳ is a champion of complexity in mathematics and because of its interior atoms,ℳ is an ideal "field guide map" showing where to find the Julia sets of varying shapes.

    The vibrant and complex visuals of the Mandelbrot and Julia sets have inspired artists to seek both abstract and natural art forms and mathematicians to discover hidden theories in fractal geometry and complex dynamics.


    Figure 7.1(A).  "Spring Reflection"

    A Local Image of the Mandelbrot Set, ℳ(0.1), of the Noncritical Point z = 0.1
    Generated by the Logistic Equation



    Figure 7.1(B).  "Bowl and Apple"

    Possible Projects for Multivariable Calculus Students



    In this section, we discuss the eye-catching simplicity of (1.1), which may be the most charming feature of the Mandelbrot set. When we paint a fractal, we use a canvas comprising millions of pixels and thousands or even millions of iterations of the dynamical system per pixel. So, adding an extra term in (1.1) can make a significant difference in the computer's runtime. It is especially true when mathematicians or artists want to explore deeper into the changing world of the Mandelbrot set by increasing the number of iterations per pixel to tens or hundreds of millions in the future.

    Fortunately, it turned out that (1.1) with its simple form is not as constraining in fractal geometry as it first appears. The reason is that if     (5.2) is any quadratic dynamical system, then we can use high school algebra to show that it is "conjugate" to (1.1), guaranteeing that any Julia set generated by (5.2) is geometrically similar to a Julia set generated by (1.1) and vice versa. Thus, all Julia sets of (5.2) are essentially the same as Julia sets of (1.1). The Tan Lei-like theorems further consolidate the similarity of the Mandelbrot set of (1.1) with that of (5.2).

    Rather than showing the "conjugacy" in full generality, we will verify it using a special quadratic dynamical system called the logistic equation. As mentioned in Introduction, the logistic equation became famous with the advent of chaos and is interesting in its own right.



    Figure 7.1(C). "Circus Elephants"

    A Local Image of the Mandelbrot Set, ℳ(0.1), of the Noncritical Point z = 0.1
    Generated by the Logistic Equation



    What is the logistic equation? In 1838 Pierre Verhulst introduced a     differential equation called the "logistic equation," which became widely used to describe the population dynamics with self-limiting growth. If we replace the derivative in the differential equation by its approximating difference quotient and do some algebra, we get the following "difference equation," which is more suitable for computer applications and again called the logistic equation:

    (7.1)   zn+1 = fp(zn) = p(1 - zn) zn .

    Suppose (7.1) is a dynamical system comprising all functions fp of a complex variable z where p varies through the complex plane. Since p is a constant for each fp, the critical point of fp is z = 0.5. Note that when p = 0, all complex numbers are critical points of fp including z = 0.5. Such a critical point is sometimes called "degenerate," but it is nonetheless a critical point.

    So, as the standard procedure, let ℳ ' = ℳ(0.5), namely, the "Mandelbrot set" of z = 0.5 defined by (7.1). Figure 7.2(A) shown below is the global image of ℳ ' painted by the divergence and convergence schemes on a p-canvas centered at the point (1, 0). For convenience, we call the entire molecule ℳ ' comprising the atoms together with its hairy boundary the logistic set.


    Figure 7.2(A). The Logistic Set



    The origin (0, 0) of the complex plane coincides with the center of the red circular atom on the left and the point (1, 0) is the intersection point of the figure 8. The real axis of the complex plane is the horizontal line through the two straight antennas of the logistic set and the intersection of the right-hand antenna and the real axis is the closed interval [α, 4] on the real axis with α ≈ 3.569945672. α is at the end of the bifurcation given by the atoms of periods 1, 2, 4, 8, ... located on its left. The logistic set is symmetric with respect to the 180o rotation about the point (1, 0) and the vertical flipping about the real axis.

    As partly mentioned in     Introduction, biologist Robert May discovered chaotic orbits of p belonging to the interval [α, 4] in 1974. It caused the mathematical term chaos to appear for the first time in 1975, the year in which Mandelbrot coined the term fractal purely coincidentally. In 1993, a chaotician showed up in Steven Spielberg's hit movie, "Jurassic Park," tacitly suggesting possible chaos in the controlled dinosaur populations. So, it is natural that we plot various local images of ℳ ' by zooming in on the interval [α, 4]. Figure 0.7(A) is one of them and Figure 7.1(A) is another given by a noncritical point z = 0.1 instead.

    The intersection point of the largest and second largest circular atoms near the right-hand antenna is (3, 0), and we call the area directly above the point "Seahorse Bay" as it gives birth to Julia sets that look like seahorses or elephants. The Julia sets of Figures     7.4(A), 7.4(C), 7.5(A) and 7.5(B) are such examples.


    Figure 7.3(A). "Birth of Seahorses"

    A Local Image of the Mandelbrot Set, ℳ(0.2), of the Noncritical Point z = 0.2
    Generated by the Logistic Equation


    Because of the aforementioned conjugacy and the     Tan Lei-like theorems, local images of the logistic set tend to resemble local images of the Mandelbrot set, often revealing mini Mandelbrot sets. In this section, therefore, we deviate from the tradition and plot several local images of the Mandelbrot sets of noncritical points z = 0.1 and z = 0.2 generated by the logistic equation. Images similar to those in Figures 7.1(C), 7.3(A) and 7.3(B) can be easily found in or near the Seahorse Bay but can never be found using the critical point z = 0.5.


    Figure 7.3(B).  "Pearls"









    Local Images of the Mandelbrot Set, ℳ(0.1), of the Noncritical Point z = 0.1
    Generated by the Logistic Equation


    Julia Sets by the Logistic Equation:  We now show several     (filled-in) Julia sets from the Seahorse Bay generated by the logistic equation. The readers should not dismiss these Julia sets citing the aforementioned "conjugacy" and thinking we can find them using the simpler Mandelbrot equation (1.1).

    The reason is that finding a certain fractal is, to put it in a nutshell, a chance encounter and nobody can find the parameter p of, say, Figure 7.4(A), from scratch. It means that it is impossible for someone to come up with the Julia set of Figure 7.4(A) using the logistic equation, let alone using the Mandelbrot equation. It is always a good idea to retain our curiosity and try all kinds of venues as we never know when an artistic masterpiece suddenly turns up.

    Figure 7.4(A) shows the filled-in Julia set of the parameter p = (2.994915, 0.1) belonging to the red atom of period 1 in the logistic set. Hence, it is decorated by the convergence scheme with period index 1 and its complement by the divergence scheme. Figure 7.4(B) illustrates the boundary of the filled-in Julia set, namely, the Julia set of p. It is a Jordan curve which is homeomorphic to a circle.


    Figure 7.4(A). "Circus Elephants"

    The Filled-In Julia Set of p = (2.994915, 0.1) by the Logistic Equation (7.1)



    Figure 7.4(B). "Circus Elephants"

    The Julia Set of p = (2.994915, 0.1) by the Logistic Equation (7.1)


    Figure 7.4(C) shows the Julia set of the parameter p = (3.0014564, 0.08) belonging to the     Seahorse Bay and the complement of the logistic set. It is a Cantor set.


    Figure 7.4(C). "Pearly Elephants"

    The Julia Set of p = (3.0014564, 0.08) Generated by the Logistic Equation (7.1)


    The parameter p = (3.0237615, 0.1) that generates "Dancing Seahorses" shown below belongs to the orange atom of period 2 in the     Seahorse Bay. So, the filled-in Julia set is painted by the convergence scheme with period index 2. The Seahorse Bay is sandwiched by the red atom of period 1 and the orange atom of period 2, and interestingly, "seahorses" are born from the orange shore while "elephants" are born from the red shore.


    Figure 7.5(A). "Dancing Seahorses"

    The Filled-In Julia Set of p = (3.0237615, 0.1) Generated by the Logistic Equation
    On a z-Canvas Centered at z0 = (0.5, 0)



    Figure 7.5(B). "Cloisonné Elephants"


    The Julia Set of p = (2.99997742, 0.0100200333) Generated by the Logistic Equation
    On a z-Canvas Centered at z0 = (0.5, 0)



    Conjugacy of the Logistic and Mandelbrot Equations:  As indicated at the outset of this section, the logistic equation and the Mandelbrot equation (2.1) are "conjugate" to each other, and hence, by knowing the Julia sets of the Mandelbrot equation, we effectively know all     Julia sets of the logistic equation and vice versa. To discuss these ideas in detail, let's rewrite the logistic equation (7.1) as

    (7.2)  ζn+1 = q(1 - ζn) ζn ,

    and reserve zn and p for the Mandelbrot equation

    (7.3)   zn+1 = zn2 + p ,

    where p and q ≠ 0 are constant parameters, while the initial values ζ0 and z0 vary through the entire complex plane. It is important to remember that the filled-in Julia set of p by (7.3) is by definition the set of all z0 in the complex plane whose orbits zn do not diverge to ∞ and likewise for the filled-in Julia set of q by (7.2). Also, the Julia set of q means the boundary of the filled-in Julia set of q.

    Figure 7.6(A).
    "Dancing Seahorses" II

    Filled-In Julia Set of q = (3.02382, 0.1)
    by the Logistic Equation (7.2)



    Filled-In Julia Set of p ≈ (-0.77146, -0.10119)
    by the Mandelbrot Equation (7.3)


    We say that (7.3) is conjugate to (7.2) if there are complex constants a ≠ 0 and b such that the change of variables,

    (7.4)   zn = a ζn + b,

    transforms (7.3) to (7.2) for all n ≥ 0.

    Suppose for a moment that (7.3) is conjugate to (7.2) under (7.4) to see what it leads to. Then firstly, (7.4) has its inverse ζn = (zn - b)/a that transforms (7.2) back to (7.3); hence, (7.2) is conjugate to (7.3) as well, i.e., (7.2) and (7.3) are conjugate to each other.

    Secondly, applying the     triangle inequality on the transformation (7.4) and its inverse, it is easy to show that ζn diverges to ∞ if and only if zn diverges to ∞; hence, the transformation (7.4) with n = 0 maps the (filled-in) Julia set of q onto the (filled-in) Julia set of p in a one-to-one fashion. 

    It is not particularly difficult to show that the transformation (7.4) with n = 0 is not only a homeomorphism but also a "similarity transformation" from the complex plane as the set of ζ0 to the complex plane as the set of z0 so that the aforementioned Julia sets are geometrically similar.

    Now, without assuming conjugacy, we wish to show that (7.2) can be written in the form

    (7.5)   a ζn+1 + b = (a ζn + b)2 + p,

    which is the result of applying (7.4) on (7.3). The process involved is precisely the same as finding the vertex of the parabola given by a quadratic function in high school algebra. Rewrite (7.2) as

       -q ζn+1 = q2 ζn2 - q2 ζn ,

    i.e.,   a ζn+1 = (a ζn)2 + 2b(a ζn) ,

    where a = -q and b = q/2. Completing the square with respect to a ζn , we get

       a ζn+1 = (a ζn + b)2 - b2,

    which is equivalent to (7.5) with

    (7.6)   p = q(2 - q)/4.

    Since q ≠ 0, it follows that (7.4) is defined by a = -q and b = q/2 and transforms (7.3) to (7.2), as was to be shown, provided (7.6) is true. Summarizing, we have:

    Theorem: If p = q(2 - q)/4 then the logistic equation (7.2) and the Mandelbrot equation (7.3) are conjugate to each other and the Julia set of q by (7.2) and the Julia set of p by (7.3) are     geometrically similar.

    Example 1: The first image of Figure 7.6(A) shows the filled-in Julia set of the parameter q = (3.02382, 0.1) generated by the logistic equation (7.2) and the second image the filled-in Julia set of p = q(2 - q)/4 ≈ (-0.77146, -0.10119) generated by the Mandelbrot equation (7.3). The Julia sets are indeed geometrically similar, but the appearances of the filled-in Julia sets are a little different, even though they are painted by exactly the same coloring routine. Thus, the conjugacy relation preserves the geometric shape of the filled-in Julia set but not necessarily its coloring.

    As we have seen, the logistic set and the Mandelbrot set generated by (7.2) and (7.3) lie in two different complex planes, one comprising parameters q and the other comprising parameters p, respectively. Note that the quadratic function (7.6) maps the first complex plane onto the second complex plane continuously in a two-to-one fashion except for the center (1, 0) of the logistic set. Figure 7.7(A) shows some of the corresponding parameters. Particularly interesting is the fact that the center of the logistic set works like doubling the noteworthy cusp of the Mandelbrot set.


    Figure 7.7(A). Relationship
    Between the Logistic Set and the Mandelbrot Set
    The corresponding parameters generate geometrically similar Julia sets.


    If we solve (7,6) for q by the quadratic formula, we get two q-values

    (7.7)  q = 1 ± √ (1 - 4 p)

    that correspond to a single p-value and are symmetric about the center (1, 0) of the logistic set. It means that the Julia set of each p actually corresponds to two Julia sets of q that are geometrically similar, unless p is the cusp of the Mandelbrot set.

    Because of the symmetric shape of the logistic set and the two-to-one correspondence shown in Figure 7.7(A), the "    Lion Sanctuary" of the Mandelbrot set located near its cusp corresponds to two "Lion Sanctuaries" of the logistic set, one located above and the other below its center. Similarly, the logistic set actually has four areas that can be called "Seahorse Bays" that correspond to two such areas of the Mandelbrot set near its "neck."

    Example 2: Figure 7.8(A) shows the Julia set of q = (0.9975, 0.0752) belonging to the Lion Sanctuary of the logistic set above its center. By the aforementioned theorem, it is geometrically similar to the Julia set of p = q(2 - q)/4 ≈ (0.2514121975, 0.000094) belonging to the Lion Sanctuary of the Mandelbrot set near its cusp. By virtue of (7.7), the value of p in turn corresponds to the second value of q, namely, q = (1.0025, -0.0752) belonging to the Lion Sanctuary of the logistic set below its center and providing the second Julia set of q that is again geometrically similar to the Julia set of p.


    Figure 7.8(A).  "Cloisonné Lion"

    The Julia Set of q = (0.9975, 0.0752) Generated by the Logistic Equation (7.2)



    Go to   Top of the Page Introduction § 1. Prep Math
    § 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
    § 5. Julia Sets and the Fundamental Dichotomy
    § 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
    Fractal Coloring Algorithms Gallery 2D Gallery 3D


    § 8.  Newton Fractals

    The idea of the     Julia set of a polynomial we saw in § 5 naturally extends to a rational function with minor modifications. So, suppose g is a polynomial of degree greater than one and consider a dynamical system of the form

    (8.1)   zn+1 = zn - g(zn)/g'(zn),

    which generates, like in § 5, a (filled-in) Julia set of (8.1). The reader may have noted already that (8.1) is nothing but the Newton-Raphson Root-Finding Algorithm, aka Newton's Method for finding roots of g. For this reason, we call a Julia Fractal featuring the Julia set of (8.1) a Newton fractal of g.

    As a benefit of using Newton's method, each orbit of (8.1) converges to a root of g quickly more often than not, and it allows us to plot most of the Newton fractals by the convergence scheme (with period index k = 1) alone with a relatively small maximum number of iterations like M ≤ 500. Also, because g is a polynomial, we may take advantage of the time-saving scheme called Horner's Method to efficiently evaluate both g and g' that appear in (8.1). Horner's method is nothing but "synthetic division" taught in high school algebra, and it should be interesting for the reader to see how (differently) it is applied in computer programming.

    Furthermore, if we know all the roots of g prior to the fractal plotting, we can modify the convergence scheme fairly easily so as to add more colors to Newton fractals of g; see Example 1 below. Because a Newton fractal is a Julia fractal, "orbit" and "canvas" always mean an orbit of z0 and z-canvas, respectively, in this section. It is important to remember that z0 is an initial value for computing a root by Newton's Method (8.1).

    Example 1 (Roots of Unity): Among all attractive Newton fractals, probably the simplest to plot are generated by a polynomial of the form

        g(z)  =  z n -  1 ,

    as its roots r0, r1, r2, ... , rn-1, called the nth roots of unity, are given in a trigonometric expression by

       rk = cos(2kπ/n) + i sin(2kπ/n)  with   r0 = 1.

    The fact that each rk is indeed a root of the polynomial g(z) follows immediately from De Moivre's formula.


    Figure 8.1.  Newton Fractals of g(z)  =  z 5 -  1


    The image on the left is a Newton fractal for g(z)  =  z 5 -  1 painted on a square canvas centered at the origin with radius 1.1. It uses five essentially different colors, sky blue, purple, red, amber, and blue, associated with the five roots of g. For example, the sky blue region represents the region comprising the initial values z0 in the complex plane whose orbits converge to the root r0 = (1, 0) called the basin of attraction of Newton's method for the root.

    Thus, there are five basins of attraction in the fractal and they are separated by the basin boundary. The basin boundary is precisely the Julia set of the Newton fractal and the union of the five basins plays the role of the interior of the filled-in Julia set we saw in § 4. Unlike the Julia set of § 4, the basin boundary is not     compact but it is closed. The basin boundary is known to be a Cantor set and it is where Newton's rootfinding algorithm behaves in a "chaotic" fashion.

    The second image of Figure 8.1 is a variation of the first. The image shown below, called "Crab Queue," is given by zooming in on one of the "bands" in the second image. It is accompanied by a fractal showing the Julia set in "Crab Queue."








    Example 2 (Cyclotomic Polynomials): Another interesting example with known roots is a Cyclotomic Polynomial. The picture on the left in Figure 8.2 is a Newton fractal of the "30th cyclotomic polynomial"

        g(z)  =  z 8 +  z 7 -  z 5 -  z 4 -  z 3 +  z  +  1

    with the unit disk highlighted. Since g happens to be a factor of  z 30 -  1,  its roots are among the 30th roots of unity that lie on the unit circle. In the picture, the thirty dots on the unit circle show where the roots of unity are located and eight of them colored yellow show the whereabouts of the roots of g. The picture on the right is a Newton fractal of the "20th cyclotomic polynomial"

        g(z)  =  z 8 -  z 6 +  z 4 -  z 2 +  1.

    Figure 8.2.  Cyclotomic Polynomials with Eight Roots (and Application in 3D Plotting)





    Finding the Roots: Examples 1 and 2 show that what makes Newton fractals stand out with an abundance of colors are the roots of the polynomial g(z) and the Julia set that divides the basins of attraction to the roots. The more intricate the Julia set, the more attractive the Newton fractal, but ironically, the Julia set is the biggest culprit that complicates the rootfinding process by Newton's method.

    Newton's method for finding all roots of g(z) requires us to choose (almost blindly) an intial value z0 to (8.1) hoping it belongs to the basin of attraction to one of the roots, say r1. Once r1 is found, we use the aforementioned Horner's method (aka synthetic division) to divide g(z) by z - r1 and repeat the process on the "deflated" polynimial, starting again with a new initial value z0 to find a second root r2. The process is repeated until we find all roots, but it is likely to fail if any of the initial values hits the Julia set.

    So, what should we do if we don't know the roots of the input polynomial? One way is to use Muller's Method instead of Newton's method; e.g., see     Wikipedia. Although Muller's Method lacks the simplicity of Newton's method and still requires the "deflating" steps, it works fairly quickly without the burden of finding initial values. All computer programs for the Newton fractals shown in this website use Muller's Method—even for Example 1 whose roots are well known (so we don't have to write down the roots in our computer program).

    Once our computer program starts running smoothly, plotting Newton fractals provides us with great entertainment. It is easy to pick an input polynomial from infinitely many choices with anticipation from not knowing what to expect in the output. Furthermore, a high-res output image generally emerges within minutes rather than hours and days of runtime. Figures 8.3 through 8.8 shown below are among numerous Newton fractals for which we randomly chose the input polynomials.

    Here's an example given by a fifth degree polynomial. Just for fun, we painted it on a sphere and a torus as well as on a plane.


    Figure 8.3.  "Fireflies"






    The next example, which is given by a seventh degree polynomial, is painted on a plane and an apple.


    Figure 8.4.  "Fruit Flies"


    Similarly, we have:

    Figure 8.5.  "Newton's Apple"



    Figure 8.6 shows a Newton fractal of a 12th degree polynomial painted on a plane and a sphere. The second image which is painted on a sphere is intended to give a 3D appearance. All of the twelve roots are again found by Muller's Method quickly.


    Figure 8.6.   "Dragonfly"




    Part of the Julia Set of "Dragonfly"



    Figure 8.7 illustrates two images given by highlighting different parts of essentially the same Newton fractal. We can see from the images that the Newton fractal is generated by a fifth degree polynomial that has two pairs of conjugate complex roots and a simple real root. Each image accompanies a fractal that emphasizes the intricate Julia set.


    Figure 8.7.   "Ghost Fish" and "Ghost"







    Shown below is a fractal similar to the "Ghost" but given by a slightly different fifth degree polynomial. The Julia set appears to be bounded but it is in fact a part of an unbounded Julia set highlighted like the "Ghost."


    Figure 8.8.  "Spiderman"



    Shown below are additional Newton fractals.


    Figure 8.9(A). "Butterflies"

    Newton Fractal on a z-Canvas



    Figure 8.9(B). "Spiders"

    Newton Fractal on a z-Canvas



    Figure 8.9(C). "Dragonflies"

    Newton Fractal on a z-Canvas



    Figure 8.10(A). "Newton Garden"

    Newton Fractal on a z-Canvas



    Figure 8.10(B). "Firefly Forest"

    Newton Fractal on a z-Canvas



    Figure 8.10(C). "Crab Shell"

    Newton Fractals on a z-Canvas with Different Colorings



    Figure 8.10(D). "Barn Owl"


    Newton Fractal on a Plane (z-Canvas) and a Sphere


    Newton Fractals of Rational Functions: Through the quotient rule of differentiation and the aforementioned Horner's method, it is straightforward to extend the fast plotting methods we have seen from the polynomials to the rational functions. Figure 7.7 shows a simple example, called "Five Crabs in a Circle," where the "huddling crabs" are the basins of attraction for the fifth roots of unity and the Julia set is, unlike in the earlier Newton fractals, bounded.


    Figure 8.11.  Newton Fractals of g(z)  =  (z 5 -  1) / (z 5 +  1)



    Go to   Top of the Page Introduction § 1. Prep Math
    § 2. The Divergence Scheme § 3. The Mandelbrot Set § 4. The Convergence Scheme
    § 5. Julia Sets and the Fundamental Dichotomy
    § 6. Generalizations § 7. The Logistic Equation § 8. Newton Fractals
    Fractal Coloring Algorithms Gallery 2D Gallery 3D


    Links

    Pacific Northwest Section, Mathematical Association of America

    Wikipedia Horner's Method Newton's Method Muller's Method
    Mandelbrot Set Julia Sets

    Google Fractal Art Gallery Fractal Plotting Fractal Coloring

    Go to Willamette University Mathematics Department Sekino's Home Page