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 is the article entitled "Stories about Fractal Art." It shows a host of fractal images intended to be the main attraction and offers the text as optional references. The text occasionally touches upon certain areas of advanced mathematics, but it is written in a way that is accessible to people who have not experienced beyond introductory calculus.

The website has additional galleries of fractal art: Gallery 2D mainly shows two-dimensional (2D) images given by the basic programming algorithms described in "Stories about Fractal Art," while Gallery 3D comprises images generated by a variety of techniques based on college mathematics. The latter includes such 3D objects as fractal mountains and 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 © 1997-2023 Junpei Sekino 
The website was last updated on October 1, 2023 

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   Preface Prep Math: Orbits, Dynamical Systems and Topological Properties
Previews of the Central Sections:
The Divergence Scheme The Mandelbrot Set The Convergence Scheme
Julia Sets Generalizations The Logistic Equation Newton Fractals

Additions  Fractal Coloring Algorithms Sekino's Home Page
Galleries  Gallery 2D Gallery 3D

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 0.1-0.8 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. The Mandelbrot Set Figure 0.2.  A Julia Set Figure 0.3. A Newton Fractal 
"Twin Dragon Gatekeepers" for the Website


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 almost 100 years before Mandelbrot published his book entitled "The Fractal Geometry of Nature." During the early 20th century, Pierre Fatou and Gaston Julia laid the foundations for fractals generated by "dynamical systems."


Figure 0.4. Classical Fractals by Geometric Constructions


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


Pascal's Triangles with Modular Arithmetic



Figure 0.5. "Atomic Fusion" in Fractal Geometry
Cf. Atoms and Pictorial Interpretation of the Fatou-Julia Theorem (1918-1919)


It was in 1980 when Mandelbrot showed the famous fractal now called the Mandelbrot set generated by a simple dynamical system and a computer. Almost immediately after that, the novelty and complexity of the Mandelbrot set reinvigorated the interests in fractals and stimulated mathematicians to develop further theories in fractal geometry.

On the other hand, chaos often associated with fractals 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. 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 received a great sensation especially because there appeared very little difference between chaotic and random outcomes even though the former resulted from deterministic processes. It soon became known that fractals and chaos are closely related and together they provide applications in science as well as in art.


Figure 0.6.  Fractals Near the Area where Chaos was Discovered




Technical Descriptions: Mandelbrot Fractals of z0 = 0.1 on p-Canvases
Top Left: Centered at p = (3.567150, 0.002194); Top Right: p = (3.567174, 0.002195)
Bottom: p = (3.570063528376, 0.00000001192)
Generated by the Logistic Equation


Through Google, we find numerous websites that display stunningly beautiful computer-generated fractal art images. It indicates that a large population not only appreciates the digital art form but also participates in the eye-opening creative activities of fractal plotting. A fairly large part of what follows is dedicated 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 or coding but instead tells the general principles for fractal plotting in everyday language assuming the readers' basic programming experience.

Particularly exciting is the moment the fractal image generated by our personal program emerges in our computer screen because of its possibly "amazing" artistry and built-in chaos-related unpredictability. The reader who may be merely intrigued by the general idea behind fractal plotting is encouraged to try it. Many of the images will stir our imaginations in the part of mathematics that is in fact quite deep and still filled with unknowns. Best of all, though, it is plain fun.


Figure 0.7.  "Esmeralda Lion"

Technical Description: The Julia Set of p = (0.935638029, 0.355810484)
On a z-Canvas Centered at z0 = (0.5, 0)
Generated by the Logistic Equation
Cf. "Esmeralda Lion" in Gallery 2D



Figure 0.8.  "Cloisonné Elephant"

Technical Description: Subset of the Julia Set of p = (0.971218547, 0.27795678)
On a z-Canvas Centered at z0 = (0.5, 0)
Generated by the Logistic Equation
Cf. "Cloisonné Lion II" in Gallery 2D





Preparatory Mathematics

People who wish to understand fractals in some depth 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 some basic computer programming experience.

(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 the distance of z from the origin O 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, it follows that |w| ≤ |w + v| + |v|; hence, another way of writing the triangle inequality is

    |w + v| ≥ |w| - |v|.

(b) includes the basic ideas about the derivative and a critical point of a function where the derivative vanishes.


Figure 1.0(A). A Local Image of the Mandelbrot Set

Technical Description: See Figure 2.3 in § 2



Figure 1.0(B).  "Dancing Seahorses"

Technical Description: The Julia Set of p = (-0.77146, -0.10119)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation
See Fractal Coloring for the Coloring Algorithm


We now introduce a few 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. We refer to the index n as time or instant, so, e.g., at any instant n ≥ 2, we have zn = 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. It is important to note that the orbit of p (with a fixed initial value z0) 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 in a seemingly unpredictable way as time n progresses.



Figure 1.0(C). "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(D).  "Mosaic Lion"

Technical Description: The Julia Set of p = (0.25000316315, -0.00000000895)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation


In classical analysis, an orbit generally meant the "range" of a sequence of numbers, but in fractal geometry, the orbit is defined to be the sequence itself and may "move around" like the aforementioed orbit of p = -1.9. In that sense, our orbit is more like the position of a comet orbiting the Sun which changes with time and less like the elliptical "orbit" of the comet which is a fixed path.

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).  "Lavender Dragon"
Technical Description: The Julia Set of p = (0.0830158, 0.5347594)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Dynamical System zn+1 = zn3 + zn + p



Figure 1.1(B).  "Pearly Dragon"
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 x jmax = 50 x 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.

p-Canvases and z-Canvases: Although the next four paragraphs apply to any dynamical systems, consider the Mandelbrot equation (1.1) for convenience. As we have seen, it comprises infinitely many orbits zn of complex numbers, one for each choice of values of z0 and p, both varying through the complex plane. Recall that a canvas mentioned above is a rectangle R 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 R a p-canvas or view these complex numbers as values of z0 and call the canvas R a z-canvas.


Figure 1.2(B). "Jellyfish Queue"

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


Plotting a fractal on a p-Canvas: Roughly speaking, we use the dynamical system (1.1) and plot a fractal on a p-canvas as follows: Choose a value of z0, say z0 = 0. For each pixel (i, j) on the p-canvas R, use its representative parameter p and (1.1) to generate the orbit of p with the initial value z0 = 0. We then pick on certain behavior of the orbit and use it to color the pixel (i, j). As we have seen, the orbits from adjacent pixels on the p-canvas may have drastically different behaviors, possibly causing dramatic color changes in the image painted on the p-canvas. The details will be shown in § 2 and § 4. The image shown above is an example.

Plotting a fractal on a z-Canvas: We now make a convenient name change and call the orbit of p with an initial value z0 the orbit of z0 with a parameter value p. The rest is symmetric to what we have just seen: Choose a value of p, say p = (-1.1128, 0.23076). For each pixel (i, j) on the z-canvas R, use its representative initial value z0 and (1.1) to generate the orbit of z0 with the fixed parameter value p. Then use its behavior to color the pixel (i, j). The details are shown in § 5.

Note that a z-canvas is associated with a fixed parameter value p, hence the orbits used to paint a fractal on the z-canvas have variable initial values z0 while the orbits used to paint a fractal on a p-canvas have variable parameter p and fixed initial value z0. Here's an example:


Figure 1.2(C). "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


As we'll see in § 5, a fractal called a "Julia set" is a set of points in the complex plane plotted on a z-canvas with a fixed parameter value p.

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 concepts of geometric similarity and 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 Examples 4 and 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


Figure 1.4(C).  Local Images 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





§ 1.  Previews of Upcoming Sections: § 2 ― § 8

In 2008, PBS broadcast a NOVA program proclaiming that the Mandelbrot set, which is a fractal generated by the Mandelbrot equation, had become "the most famous object in modern mathematics." Naturally then, a large part of the article, § 2 - § 5 and § 7, is devoted to the Mandelbrot set and a myriad of its attributes that fascinate mathematicians and artists alike. We'll define the Mandelbrot set carefully in § 2 and denote it by ℳ.

In § 2 and § 4, we explain how to plot ℳ globally and locally on p-canvases using algorithms called the divergence and convergence schemes. Figure 1.4(B) shows global images of ℳ contrasting the difference between the two algorithms. Because fractal art is wide open to new ideas and discoveries, people learning fractal plotting should be free-spirited and carry out frequent computer experiments. The next two images are examples that popped up from rather aberrant experiments by a computer and the algorithms. The cause for the latter has been documented as in the eyeball effect but the former is still under investigation.


Figure 1.5(A). "Crooked Mandelbrot Set"




Figure 1.5(B). "Partying Cuttlefish"

Technical Description: Julia Fractal of p = (0.25000316374967, -0.00000000895902)
On a z-Canvas Centered at z0 = (-0.073, 0.01)
Generated by the Mandelbrot Equation


The Mandelbrot set ℳ turned out to be the most complex object ever plotted on a plane and we show why from an intuitive viewpoint in § 3. We also show the meticulous numerical structure of the interior of ℳ interweaving its atoms with periods in § 4; see the periodicity diagram. Despite all these astounding intricacies, ℳ remains, topologically, a relatively simple object, which is compact and connected and without any holes.

In § 5, we define the Julia set of a parameter p generated by the Mandelbrot equation and show how to plot it on a z-canvas using the divergence and/or convergence schemes. Frequently, we observe striking similarities in appearance between local images of ℳ and Julia sets and learn that it is no accident. To further consolidate the close-knit relations between ℳ and Julia sets, we also learn that ℳ is completely characterized as the set comprising parameters whose Julia sets are connected.

If a parameter p belongs to an atom of ℳ, then the period of the atom affects the shape of the Julia set of p in an amazing and sometimes amusing way, although its cause is shrouded in mystery. It makes the aforementioned periodicity diagram all the more important in plotting Julia sets. For example, the image shown below is the Julia set of a parameter chosen from an atom of period 9, while Figure 1.4(A) shows two Julia sets born from the same atom of period 85 = 17 × 5.


Figure 1.6(A). "Hydra of Lerna" Born from an Atom of Period 9

Technical Description: The Julia Set of p = (-0.6663, 0.3289)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation



Figure 1.6(B). "Dancing Beans" Born from an Atom of What Period?

Technical Description: The Julia Set of p = (0.262, 0.5701)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Mandelbrot Equation


In § 6, we extend the ideas of the Mandelbrot and Julia sets to Mandelbrot fractals and Julia fractals generated by general dynamical systems. We take quite a bit of time on a comical fractal we call the "Speared Mandelbrot Set" generated by a cubic dynamical system with two critical points. By comparison, the Mandelbrot equation has a single critical point that provides us with a simple and elegant classification of its Julia sets called the Dichotomy Theorem.

As we'll see, the two critical points make the matter more challenging as well as interesting and help us forecast highly convoluted circumstances if the dynamical system has three or more critical points. The next image shows a disconnected non-Cantor Julia set defying the Dichotomy Theorem.


Figure 1.7(A). "Green Monster"

Technical Description: The Julia Set of p = (0.00033, -2.0006785)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Dynamical System zn+1 = zn3 + zn + p



Figure 1.7(B). "Burst Mandelbrot Set"

A Local Image of the "Speared Mandelbrot Set"
Quiz: Find it in Figure 6.1; hint: Colors have been altered



Figure 1.7(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



Figure 1.7(D). "Dancing Metabo Seahorses"

Technical Description: The Julia Set of p = (-1.022, 0.14846)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Dynamical System zn+1 = zn(zn2 + p)



Figure 1.7(E). "Cloisonné Turtle"

Technical Description: Subset of the Julia Set of p = (-1.0158, 0.1449)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Dynamical System zn+1 = zn(zn2 + p)



Figure 1.7(F).  "Gold Dragon"

Technical Description: The Julia Set of p = (0.0618974, 0.5400784)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Dynamical System zn+1 = zn3 + zn + p


Figure 1.7(G).  "Pearly Dragon"
Technical Description: The Julia Set of p = (0.0352236, 0.5448064)
On a z-Canvas Centered at z0 = (0, 0)
Generated by the Dynamical System zn+1 = zn3 + zn + p


In § 7, we turn our attention to the logistic equation, which became famous with the advent of chaos in the 1970s, and show (using high school algebra) that the Mandelbrot equation and the logistic equation are "conjugate" to each other, which implies that they share the same Julia sets. The process effectively indicates that all quadratic dynamical systems are in fact conjugate to the Mandelbrot equation and explains part of the reasons why the Mandelbrot equation in such a simple form generates such a complex and fascinating object as the Mandelbrot set.


Figure 1.8(A). "Cloisonné Elephants"

Technical Description:
A Julia Fractal of p = (2.99999597, 0.0079408)
On a z-Canvas Centered at z0 = (0.5, 0)
Generated by the Logistic Equation


Figure 1.8(B). "Pearly Elephants"

Technical Description: A Mandelbrot Fractal of z0 = 0.1 on a p-Canvas
Generated by the Logistic Equation
See "Elephant Bay"


There is a special subset of the Julia fractals consisting of fractals generated by so-called "Newton's rootfinding method." We call them Newton fractals and discuss them in § 8. Here are sample fractals:


Figure 1.9(A). "Butterflies"

Newton Fractal on a z-Canvas



Figure 1.9(B). "Spiders"

Newton Fractal on a z-Canvas



Figure 1.9(C). "Newton Garden"

Newton Fractal on a z-Canvas



Figure 1.9(D). "Firefly Forest"

Newton Fractal on a z-Canvas



Figure 1.9(E). "Crab Shell"

Newton Fractals on a z-Canvas with Different Colorings



Figure 1.9(F). "Barn Owl"


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


People who are familiar with multivariable calculus can venture into plotting fractals in a 3D space. One of the possibilities is to map a fractal from the plane to various surfaces such as a sphere and a torus. We will throw in 3D examples here and there in the upcoming sections.





§ 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. 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)


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). With that we have the following surprisingly simple definition of probably the best known object in modern mathematics:

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 ∞.

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.
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 of A: Suppose |zm| > max{2, |p|} for some m. Then by (2.1) and the triangle inequality (see § 1(a)), 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.

Proof of B: If |p| > 2, (2.1) and (2.2) imply |z1| = |p| > 2; hence, by the triangle inequality (see § 1(a)), 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. 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 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 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 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     Figure 2.0; see Fractal Coloring Algorithms. For example, the image on the left is painted by a basic technique described in the website.

    Figures 2.1 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 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.

    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, which we call 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 1.0(A) is given by zooming in on microscopic rectangles in Figure 2.3 and applying the divergence scheme.

    Example 3:  Here's another local image of the Mandelbrot set painted on a p-canvas centered at p = (0.281229249, 0.011344208).


    Figure 2.4.  Another Local Image of the Mandelbrot Set



    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) We have 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 dynamical system other than the Mandelbrot equation and don't know what threshold to use. 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 2.5.  Seventh Degree mini-Mandelbrot Set




    .

    § 3.  The Mandelbrot Set

    Benoit Mandelbrot rocked the mathematics world in 1980, when he introduced his computer-generated images of the fractal now called the Mandelbrot set. Its novelty and intricacy and the fact that it can be generated by such a simple     process invigorated a great many mathematicians and scientists to engage in their research and numerous articles on its extraordinary properties appeared in mathematics books and journals, popular magazines and major newspapers. In 2008, PBS broadcast a NOVA program proclaiming that the Mandelbrot set had become "the most famous object in modern mathematics." In addition to the fact that it has limitless varieties of astounding local images as indicated in § 1 and § 2 (rather modestly), we will see many more reasons for the fame in § 3, § 4, § 5 and § 7.

    Recall that the Mandelbrot set is closed so it contains its boundary as its subset. It is known that the topological dimension of the boundary is 1 like the boundary of a circular disk, so we intuitively picture it as an object made of "razor-thin filaments" without thickness. Does it mean that the area of the boundary is zero? Nobody can find the answer, and we suddenly realize that it is considerably more complex than it appears in a global image like Figure 2.1.

    Although it may not sound obvious unless we know something about     fractal dimensions, the following celebrated theorem implies that no figures on the plane are more complex than the boundary of the Mandelbrot set, boosting the Mandelbrot set to be the most complex object ever plotted on a plane.

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

    Let's pause for a moment, recall that the Mandelbrot set is denoted by ℳ and look at its local image in, say, Figure 2.3, in which a part of ℳ is visible. The intricate image surely looks impressive, but exactly where is the boundary of ℳ and what does it have to do with the colorful patterns? It turned out that the boundary of ℳ is all over the image as we can see in Figure 3.1 given by darkening the entire Figure 2.3 and lighting up its razor-thin filaments:


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



    The image shows that the boundary of ℳ in the rectangular area is vividly self-similar, making it a fractal as per our informal definition. Shishikura's theorem also makes it a fractal according to Mandelbrot's definition: A fractal means a set for which the Hausdorff-Besicovitch dimension (aka the fractal dimension) strictly exceeds the topological dimension.

    Through the self-similarity, we observe that the luminous filaments of the boundary of ℳ get so dense they work like space-filling curves in infinitely many areas of the plane. It provides us with an intuitive idea as to why the "fractal dimension" of the boundary of ℳ is the same as the dimension of the plane and why it is equated with the complexity of ℳ.

    The self-similar boundary image also shows that it works like the basic monochromatic line art of the multicolored Japanese manga, anime and ukiyo-e woodblock prints, which are completed by the additional and secondary step of coloring between the lines. The finer the line art, the more elaborate the woodblock print. Quite similarly, the boundary image controls the shape and quality of a multicolored fractal image like     Figure 2.3. Here's another dramatic image:


    Figure 3.1(B).  The Boundary of the Mandelbrot Set in Figure 2.4



    One of the most important     topological properties in fractal geometry is "connectedness" of a set and Figures 3.1 and 3.1(B) 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. 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―before Adrien Douady and John H. Hubbard settled it:

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

    They 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 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. Figires 2.3 and 2.4 we have seen are daytime views while Figires 3.1 and 3.1(B) are nighttime views. The nighttime views are not as colorful but they make it easier for us to visualize the important theorems established by Shishikura, Douady and Hubbard.

    Plotting the complex boundary of ℳ with reasonable accuracy may demand 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.


    Figure 3.4.  The Boundary of the mini-Mandelbrot Set



    Topological Properties     (continued): We stated earlier the precise definition of a set being "connected" as "one piece" and now wish to dig into the notion of "pieces" as a preparation for the upcoming sections. We showed, while discussing the definition by Buck, 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 easy for people familiar with elementary set theory to use the idea of an "equivalence relation" and 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.

    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'll see, many fractals are totally disconnected, but the interior of ℳ is not one of them.

    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.





    § 4.  The Convergence Scheme

    We are not done yet with the complex nature of the Mandelbrot set ℳ and still stay with it. In § 2 and § 3, we discussed the     complement and the boundary of ℳ; see "Daytime and Nighttime Views" of a Fractal. We now turn our attention to its interior, namely, ℳ minus its boundary.

    The Mandelbrot set has become so illustrious, everybody interested in fractals knows its "snowman" shape by heart. To its main body, which is a heart-shaped "cardioid," a bunch of (circular) disks are tangentially attached, and to each of these disks another bunch of disks are tangentially attached; see "Mandelbrot Set" by Wikipedia for detail. The fractal 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 Figures     3.1 and 3.1(B) show, ℳ also contains infinitely many mini-Mandelbrot sets, each of which is a smaller copy of ℳ, again comprising 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, as we have discussed, 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 we have not recognized. Atoms and the interiors of mini-Mandelbrot sets and 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. 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).


    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 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:  If 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 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.

    Example 2 (mini-Mandelbrot Sets): Recall that the last image of Figure 4.2 is painted on a large canvas, and it includes the images given below in Figure 4.3. The closeup images show not only elaborately aligned circular atoms but also several mini-Mandelbrot sets that look like little flying insects. Looking at his black and white computer printout of the Mandelbrot set, Mandelbrot initially thought they were "dirt" created by the printing process of the 1970s.

    The dirt turned out to be well structured molecules, of course, and each of the molecules, namely the mini-Mandelbrot sets, happens to be made up of atoms of periods

    (4.1)  k = λ, 2 λ, 3 λ, 4 λ, · · · ,

    for some positive integer λ ≥ 2. For convenience, we call λ the period of the mini-Mandelbrot set.

    For example, the most visible mini-Mandelbrot set in the closeup on the right is near the largest green atom and it has period λ = 4 = 3 + 1, where 3 happens to be the period of the green atom. Similarly, the mini-Mandelbrot set seen near the largest purple atom has period λ = 5 = 4 + 1 while the purple atom has period 4.


    Figure 4.3.  Closeups of the Interior of the Mandelbrot Set




    Figure 4.4 shows the mini-Mandelbrot set of period λ = 4 plotted by the convergence scheme with period indices k = 4, 8, 12, ..., 100. It is painted by the colors used for the Mandelbrot set of     Figure 4.2 so as to emphasize that it is indeed a (slightly distorted) copy of the Mandelbrot set. For example, the cardioid atom of period 4 is painted red.

    Here's an additional technical detail: For the convergence scheme with k = 4, 8, 12, ..., 100, we used the variable maximum number of iterations

    (4.2)  M = 20000 - 150k,

    rather than a constant like M = 20000 to speed up the computation without notably sacrificing the appearance of the output image.

    Weakness of the Convergence Scheme: The convergence scheme with period index k is slow especially when k is large. That's mainly because it uses |zn+k - zn| < ε to test if zn reached the     threshold (instead of |zn| > θ in the divergence scheme), which requires up to k × M extra iterations of (2.1) for each pixel on the canvas belonging to ℳ.


    Figure 4.4.  The Interior and Boundary of the mini-Mandelbrot Set of Period λ = 4




    Aside from its aforementioned slowness, the convergence scheme may have a potential issue as it relies on the proposition whose converse is false (unlike the divergence criterion on which the divergence scheme is based). For the convergence scheme to be fully applicable on the interior of ℳ, the critical orbit of every parameter in the interior of ℳ has to be convergent to some cycle, thereby satisfying the hypothesis of the proposition. The next two paragraphs address this issue.

    Ghost Atoms: Recall that an     atom of the Mandelbrot set ℳ is by definition a connected component of the interior of ℳ and that every atom we have dealt with seems to have a period k, i.e., it comprises parameters whose critical orbits converge to a k-cycle for some k; see the periodicity diagam. Any other type of atom is called a ghost atom (or "ghost component" per Wikipedia) as its existence or nonexistence has not been proven.

    Famous Conjecture and the Validity of the Convergence Scheme: It has been conjectured that ghost atoms do not exist; see     density of hyperbolicity. It implies that every parameter in the interior of ℳ has the critical orbit that converges to a cycle, satisfying the hypothesis of the proposition.

    In general, an orbit either diverges to ∞ or converges to a cycle, or else it is called a chaotic orbit. Because of the conjecture and the fact that the complement of ℳ comprises the parameters whose critical orbits diverge to ∞, the parameters with chaotic critical orbits fall into the boundary of ℳ.

    We now know that the convergence scheme is valid for painting the interior of ℳ. What happens if we apply the convergence scheme on the complement of ℳ ?


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




    The "Eyeball Effect": The following is a testament that even a programming bug can create a beautiful output image sometimes and help us discover interesting ideas. The picture shown above 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. Figure 1.5(B) and Figure 5.8 illustrate the "eyeballs" more vividly.

    Artist's Rendering: We saw earlier an approximation of the Mandelbrot set in an image like     Figure 2.1 and a more colorful version in Figure 4.1. The latter should probably be called an artist's rendering of the Mandelbrot set rather than its approximation, because its definition says nothing about atoms and periods. In fact, almost all images in this website, especially those involving the convergence scheme, are meant to be artist's renderings of certain mathematical objects. Here are additional artist's renderings of the Mandelbrot set.


    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 have interesting 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





    § 5.  Julia Sets

    We have so far viewed the complex plane as the set of parameters p and the Mandelbrot equation     (2.1) as the collection of all orbits of p varying through the complex plane while the initial value z0 is fixed at the critical point z0 = 0 of the function fp. As shown in § 2, we use these critical orbits to define the Mandelbrot set ℳ, which possesses the dazzling features, mathematical and artistic, we have witnessed in § 2 through § 4. In this section, we will show, as yet another fascinating attribute of the Mandelbrot set ℳ, that almost every parameter p on or near ℳ gives rise to an intricate fractal called the "Julia set" of p.

    For that purpose, we now view the complex plane as the set of all possible initial values z0 for the Mandelbrot equation

    (5.1)  zn+1 = fp(zn) = zn2 + p,

    while keeping the value of p fixed at a constant. Recall the     name change and call the orbit of p with an initial value z0 the orbit of z0 with a parameter value p.

    We now define the filled-in Julia set of p to be the set of all initial values z0 in the complex plane whose orbits zn (with the fixed value of p) do not diverge to ∞. Because the definition of the filled-in Julia set is almost identical to the definition of the Mandelbrot set, we expect that the divergence and convergence schemes are again effective in plotting the filled-in Julia sets, this time on a z-canvas instead of a p-canvas. The use of these plotting schemes is explained in detail in Fractal Coloring Algorithms.

    If p is a parameter in the interior of ℳ then p belongs to a unique atom of ℳ by what we have seen in the preceding section. So, we say (rather fancifully) that the filled-in Julia set of such a parameter p is "born from the atom of ℳ." Here are examples:


    Figure 5.0.  "Medusa Lions"



    Figure 5.1(A).  "Hydra of Lerna with 11 Heads"

    A Filled-in Julia Set Born from an Atom of Period 11 of ℳ
    For Figures 5.0 and 5.1(A), see the Technical Detail


    Example 1: The first image of Figure 5.1(B) shown below is the filled-in Julia set of the parameter p = (-0.68938, 0.27896) chosen from a circular     atom of period 11. The atom belongs to the first image of Figure 4.3 and is attached to the red cardioid of the Mandelbrot set. The white "Hydra" with green background is painted on a z-canvas centered at the critical point z0 = 0 of the function fp by the divergence scheme with the threshold

          θ = max{2, |p|} = 2,

    where |p| ≤ 2; see proposition A. Like the Mandelbrot set of Figure 2.1, the white "Hydra" is an approximation of the filled-in Julia set of p.

    The second image of Figure 5.1(B) is given by applying the convergence scheme with period index k = 11 on the filled-in Julia set. It is an artist's rendering of the filled-in Julia set. 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 "Hydra's heads," although why it is so is not completely understood. From now on, we will normally omit mentioning "approximation" and "artist's rendering."


    Figure 5.1(B).  Another "Lernaean Hydra with 11 Heads"

    An Approximation of
    the Filled-in Julia Set of p         		An Artist's Rendering of
    the Filled-in Julia Set of p


    Remarks on Plotting the Filled-in Julia Set: We quickly learn by computer experiments that a filled-in Julia set gets less interesting when its generating parameter p gets further away from the boundary of the Mandelbrot set, just like the     local images of the Mandelbrot set shown in § 2. Since the Mandelbrot set is bounded by the circle of radius 2, we always choose parameters p satisfying |p| ≤ 2 in this section. As shown in Example 1, therefore, the threshold of the divergence scheme in plotting filled-in Julia sets is θ = 2, which coincides with the smallest threshold in plotting the Mandelbrot set.

    We also 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 11, the orbit converges to a cycle of period 11 at the center of the canvas. Therefore, the convergence scheme with period index k = 11 is a natural choice in decorating the filled-in Julia set of Example 1.
    "Hydra's Ash"


    The Cantor Set of p = (-0.6891, 0.27896)

    The filled-in Julia set of     Figure 5.1(A) is painted by the same divergence and convergence schemes with p = (-0.692712, 0.273012) belonging to the same atom of period 11. People with sharp eyes may have noted, however, that the "Hydra" coils differently from the "Hydra" of Figure 5.1(B). This is because the coiling direction depends on the position of p in the atom. "Medusa Lions" of Figure 5.0 are generated by parameters chosen from the atoms of periods 10 and 21 near the cusp of the cardioid of the Mandelbrot set.

    By a Cantor set or Cantor dust is meant a totally disconnected set with infinitely many components and a fractal structure. It was named after Georg Cantor, the pioneer of set theory, who discovered the early form of the fractal in 1883.

    By the Julia set of p, we mean the boundary of the filled-in Julia set. The filled-in Julia set is known to be compact, hence, it contains the Julia set as its subset. Like the boundary of the Mandelbrot set, the Julia set dictates the complexity of the filled-in Julia set, which has a strong association with chaos theory. It is named after Gaston Julia, who was one of the main pioneers of fractals generated by dynamical systems along with Pierre Fatou.

    The elegant theorems stated below are immediate consequences of the     Fatou-Julia theorem we'll discuss later in § 6.

    Theorem: The Julia set (and the filled-in Julia set) of p is     connected if and only if p belongs to the Mandelbrot set.

    Most mathematicians define the Julia sets first and then define the Mandelbrot set as the set of all parameters p whose Julia sets are connected. To us, this is the alternative definition of the Mandelbrot set.

    The Dichotomy Theorem: The Julia set of p is either a connected set or a totally disconnected Cantor set.

    Remark: Despite the clearly stated dichotomy theorem involving two opposite topological structures and despite the superior modern computer displays, it is generally still difficult to determine if a given Julia set is connected by looking at its computer plot; see     Gallery 2D. In other words, we rely on the theorem for this practical purpose. In a dynamical system other than the Mandelbrot equation, the situation may be more complicated as we may have a Julia set that is neither connected nor a Canter set; e.g., see Figure 6.9(B).

    Example 2:  The parameter p = (-0.6891, 0.27896) used for "Hydra's Ash" shown above is near the atom of the period 11 but lies outside of the Mandelbrot set; hence, the Julia set of p is a Cantor set.

    Remark: A Julia set which is the boundary of a filled-in Julia set borders between the filled-in Julia set and its complement and is known to have an empty interior. Intuitively speaking therefore, we imagine that a Julia set is composed of razor-thin filaments or totally disconnected points. If we light up the filaments or the points and darken its complement, we get a nighttime view of the filled-in Julia set. Each of the filled-in Julia sets shown in Figures 5.0 and 5.1 is a daytime view, where the Julia set is invisible and its complement is painted by various colors.

    Example 3:  The next two images are given by p = (0.2514122, 0.000094) chosen from outside of the Mandelbrot set near its cusp p = (0.25, 0). Hence, the Julia set is a Cantor set (and so is the filled-in Julia set). In the first image, the Julia set is painted in a goldish color and is somewhat visible. In the second image, its complement is darkened to make it more visible.


    Figure 5.2.  "Cloisonné Lion"



    Figure 5.3.  Nighttime View of "Cloisonné Lion"


    Example 4:  The filled-in Julia sets called "Twin Lions" and shown below are given by parameters belonging to an atom of period 85 = 17 × 5. The atom is attached to an atom of period 17 which is attached to the main cardioid of the Mandelbrot set near its cusp (0.25, 0); see the     periodicity diagram. Note that both factors 17 and 5 are clearly visible in the "Twin Lions."

    The two images are painted by the convergence scheme with period index 85 and its background by the divergence scheme with the threshold θ = 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.


    Figure 5.4.  "Twin Lions" born from the same atom of period 17 × 5



    Example 5: "    Esmeralda Lion" with a technical description in Gallery 2D is an enlarged version of the filled-in Julia set shown above on the left. "Ruby Lion" shown below is an enlarged version of the filled-in Julia set shown above on the right.



    Figure 5.5.  "Ruby Lion"

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


    The Julia set which is     compact and connected is not simply connected and has infinitely many holes, namely the interiors of "rubies," cut out from the filled-in Julia set. Topologically, it is a very complex object.

    Example 6 (Jordan Curves): We now move from topologically complex Julia sets to topologically simpler Julia sets. 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. It is not a fractal. 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 5.6.  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.

    Example 7: The image shown below is the filled-in Julia set of the parameter p = (-1.0073, 0.2552) chosen from a circular     atom of period 2 × 4. The atom is the leftmost blue disk shown in the first image of Figure 4.3 and is attached to the orange atom of period 2. Both factors 2 and 4 are visible in the Julia set.


    Figure 5.7. "Run for the Sun"

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



    Local Images of (Filled-in) Julia Sets: As we did in § 2 with the Mandelbrot set, we can find interesting local images from the global image of a (filled-in) Julia set. Here's an example:


    Figure 5.8. "Partying Cuttlefish"



    It is a local image of the global filled-in Julia set shown below in Figure 5.9 but is painted by using different colors and the     eyeball effect. The eyeball effect makes it easier to identify the numerous "cuttlefish" swimming in the global image, in which their eyes are closed. Note that one of the cuttlefish is at the center of the image.


    Figure 5.9. "Cuttlefish Lion"


    The global image is the filled-in Julia set of the parameter p = (0.25000316374967, -0.00000000895972), which is the center of     Figure 3.3 belonging to the mini-Mandelbrot set, hence the Julia set is connected as well as compact. It is painted using the divergence scheme alone, on a z-canvas centered at the critical point z0 = 0 of the function fp in (5.1). If we zoom in on the center of Figure 5.9 between the eyes of the central cuttlefish, we get another local image, this time without the eyeball effect:


    Figure 5.10. Center of "Cuttlefish Lion"



    Since Figures 5.10 is painted by the divergence scheme alone, it is a daytime view of the filled-in Julia set, where its complement is painted in various colors other than black. Thus, the portion painted black at the center of Figure 5.10 is a part of the filled-in Julia set and its interior is cut out from the Julia set as a "black hole."

    Local Similarities of a Julia set and the Mandelbrot Set: People who regularly plot fractals 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 5.10, we get the image shown below in Figure 5. 11. Is there an explanation for the resemblance?
    Figure 5.11.
    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 5.10 and 5.11 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 5.12 shows we can zoom out from Figures 5.10 and 5.11 while retaining some degree of similarity. The cuttlefish on the left has the mini-Mandelbrot set of Figure 5.11 at the midpoint between its "eyes" and the cuttlefish on the right the "black hole" of the Julia set of Figure 5.10 instead. Each figure contains, as we can see, infinitely many cuttlefish pointing exactly where we can find other mini-Mandelbrot sets and "black holes" through the self-similarity. The "black holes" that are not part of the Julia set indicate an enormous topological complexity of "Cuttlefish Lion" of Figure 5.9.


    Figure 5.12. "Cuttlefish"
    Swimming in the Mandelbrot Set (Left) and in the Julia Set (Right)








    § 6.  Generalizations

    Soon after Mandelbrot published its computer plot in 1980, the Mandelbrot set became so popular that a great many computer hobbyists, digital artists, mathematicians and scientists have explored around it and shown their fractal images on a variety of objects including posters, book covers, T-shirts, coffee mugs and webpages. Although the hidden beauty of the Mandelbrot set is inexhaustible, it has become quite a challenge to unearth local images of the Mandelbrot set or Julia sets that look drastically diffferent 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.  "Turquoise Dragon"
    The Julia Set of p = (0.0371542, 0.5501254)
    Generated by the Dynamical System (6.2)
    Cf. Figure 1.1(A)    , Figure 1.1(B), Figure 1.7(F), Figure 1.7(G)


    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. Holomorphic functions comprise a wide variety of functions including familiar polynomials, rational functions, exponential and trigonometric functions on which we can apply the familiar rules of differentiation.

    Figure 6.1
    The Speared Mandelbrot Set

    Suppose fp is a holomorphic function of a complex variable z involving a complex parameter p. Then fp gives rise to a dynamical system

    (6.1)   zn+1 = fp(zn),

    which is viewed as the collection of infinitely many sequences of complex numbers, one sequence zn for each choice of the values of p and the initial values 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); see the "    name change."

    If R is a p-canvas associated with a constant initial value z0, then for each pixel (i, j), er, parameter p, on R, (6.1) defines the orbit of p, which allows us to use the divergence and/or convergence schemes to plot a fractal on R called a Mandelbrot fractal of z0. For example, all of our images involving the Mandelbrot set are Mandelbrot fractals of the critical point z0 = 0 given by the dynamical system (2.1). Thus, z0 is often a critical point of fp but it is not a requirement. In § 7, we will see quite a few Mandelbrot fractals of noncritical points.

    Similarly, if R is a z-canvas associated with a constant parameter p, then for each pixel (i, j), er, initial value z0, on R, (6.1) defines the orbit of z0, which allows us to plot a fractal on R. The fractal is called a Julia fractal of p plotted on the z-canvas. For example, all images of the preceding section involving the (filled-in) Julia sets are Julia fractals generated by the dynamical system (5.1).

    The Speared Mandelbrot Set:  The comical image shown in Figure 6.1 is a Mandelbrot fractal of z0 = i / √3 given by the cubic dynamical system

    (6.2)   zn+1 = fp(zn) = zn3 + zn + p ,

    where the initial value z0 is one of the two conjugate critical points ± i / √3 of fp.

    Like in     Figure 4.1, the central object in Figure 6.1 is the portion comprising the parameters whose orbits do not diverge to ∞ called the "Speared Mandelbrot Set." It is the complement of the green background comprising the parameters whose orbits diverge to ∞.

    As in § 4, we again define an atom to be a connected component of the interior of the Speared Mandelbrot set 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 blue disc and the interior of the red "spearhead" (from the stone age with jagged edges).

    Notable molecules include the interior of the "Giant Mandelbrot Set," who was speared, and the interior of the "Toddler Mandelbrot Set," who launched the big spear at the giant, seen near the bottom of Figure 6.1 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.

    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 Mandelbrot Set 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.


    "Spearhead Bay"


    The boundary of the Giant Mandelbrot Set 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 encloses numerous mini-Mandelbrot sets. Unlike the Mandelbrot set however, the boundary now appears to be disconnected.


    "Seaweed with Five-Way Junctions"



    A closeup of the Toddler Mandelbrot Set is shown below. Compared to the Mandelbrot set, the Toddler Mandelbrot Set has a proportionately larger head (like a toddler) and its boundary is disconnected from the boundary of the Giant Mandelbrot Set. The interior of the Toddler Mandelbrot Set 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.


    Figure 6.2.  The "Toddler Mandelbrot Set"



    Not surprisingly, the Mandelbrot and Julia fractals we find around the boundary of the Toddler Mandelbrot Set 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.






    Another area in     Figure 6.1 that provides a rich fishing ground for attractive fractals is in and around the blue molecule located between the spearhead and the Toddler Mandelbrot Set" that looks like a pair of balloons. We call it "Broken Balloons" because of its "bursted lips" with jagged edges and small fragments; see the image shown below. 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 6.3.  "Broken Balloons"



    The three fractals shown below are given by zooming in on microscopic rectangles near "Broken Balloons" and are generated by the divergence scheme alone. Like "Broken Balloons," "Cheetah" is a Mandelbrot fractal of z0 = i/√3 ≈ 0.57735i, which is painted on a p-canvas centered at the complex number (0.04886516, -1.20677998). "Elephant" is a Mandelbrot fractal of the noncritical point z0 =0.53i, which seems to have an effect of simplifying the output fractal. The p-canvas is centered at (0.092504, -1.1722) and "Rhino" is a local image of "Elephant."


    "Cheetah"


    "Rhinoceros" "Elephant"




    Figure 6.1 also contains two "Squished Mandelbrot Sets," each of which has a "bursted" cardioid. The molecule can be seen near the top of Figure 6.1, but its magnified image shown below uses different colors. It is 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.  "Squished Mandelbrot Sets"



    "Squished Mandelbrot Sets" break down near the "bursted" cardioids and the flying fragments provide interesting fractals. For example, the image shown below is given by zooming in on microscopic rose-shaped fragments in a p-canvas centered at p = (0.04978, 1.094143) and by the convergence scheme with period index k = 21 = 7 × 3. The interior of each rose petal is an atom of period 21.


    Figure 6.5.  "mini-Mandelbrot Set and Roses"



    Recall that     Figure 6.1 is a Mandelbrot fractal of z0 = i / √3, which is a critical point of fp in (6.2), and it turned out that the Mandelbrot fractal of the conjugate critical point z0 = - i / √3 given by the same fractal plotting process is the mirror image of Figure 6.1 through the real axis. If we superimpose the two mirror images, we get a surprising results as shown in Figure 6.6: The big Spearhead in one image fits perfectly in the cardioid body of the "the Giant Mandelbrot Set" in the mirror image and the lips of the "Broken Balloons" in Figure 6.3 are beautifully repaired by the "Squished Mandelbrot Sets" of Figure 6.4.


    Figure 6.6.  The "Giant Mandelbrot Set" with the "Mandelbrot Balloons"

    Figure 6.7
    "Venn Diagram"


    Figure 6.7 shows the situations in Figure 6.6 more globally and is explained a little later when we classify the Julia sets of     (6.2).

    The concept of Julia set naturally extends from the Mandelbrot equation to a more general dynamical system (6.1). Thus, the filled-in Julia set of a parameter p in (6.1) is the set of all possible initial values z0 of (6.1) in the complex plane whose orbits with the fixed value of p do not diverge to ∞ and the Julia set of p is the boundary of the filled-in Julia set. A lot of things about the general Julia sets are still in mystery, however, and belong to experimental mathematics by the use of computers.

    Here is a fascinating and useful fact, however. Gaston Julia and Pierre Fatou independently proved it in 1918-1919, way before the computer era. It is a version of what is now known as the Fatou-Julia Theorem.

    Theorem:  Consider a dynamical system of the form

    (6.3)  zn+1 = fp(zn) = cm znm + cm-1 znm-1 +  · · · + c2 zn2 + c1 zn + p,

    where m ≥ 2 and cm, cm-1, · · ·, c2, c1 and p are complex constants.  Then the Julia set of p is connected if and only if every critical orbit of p stays within a finite bound. Moreover, if every critical orbit of p diverges to ∞, then the Julia set of p is a Cantor set.

    In case of the Mandelbrot equation     (5.1),  fp has just one critical point. Hence, each parameter p corresponds to a unique critical orbit, which either stays within a finite bound or diverges to ∞. It stays within a finite bound if and only if p belongs to the Mandelbrot set, and consequently, the Fatou-Julia Theorem implies the theorem and the dichotomy theorem as corollaries.

    Pictorial Interpretation of the Fatou-Julia Theorem:  Consider, for example, the cubic dynamical system     (5.2), namely,

       zn+1 = fp(zn) = zn3 + zn + p

    with two critical points i/√3 and -i/√3 of fp. Let A be the set of parameters p whose critical orbits with the initial value i/√3 stay within a finite bound and B the same with the initial value -i/√3. A is precisely the "Speared Mandelbrot Set" depicted by Figure 6.1 and B the mirror image of A through the real axis. As we saw in Figure 6.6, A and B have interesting relations when they are superimposed.

    Figure 6.7 shows a pseudo Venn Diagram of A∪B and A∩B which we call the "Venn Diagram." Here, the union A∪B is painted by colors other than black and the intersection A∩B by colors other than black and green. Thus, the black zone is the complement of A∪B denoted by [A∪B]c and the green zone is the symmetric difference

       AΔB = A∪B - A∩B.

    Note that the "Venn diagram" quickly gets a lot more complex if fp has three or more critical points.

    In terms of the "Venn Diagram," the Fatou-Julia theorem states:

    (1)  If p belongs to A∩B then the Julia set of p is connected;
    (2)  if p belongs to AΔB then the Julia set of p is disconnected;
    (3)  if p belongs to [A∪B]c then the Julia set of p is a Cantor set.

    Thus, the symmetric difference in (2) plays the role not seen in the dichotomy theorem. Although the "Venn Diagram" works well for our purpose of forming examples that illustrate the Fatou-Julia Theorem, we need to be a little careful as the diagram does not include the hairy boundaries of A and B seen in Figure 6.1. Also omitted in the diagram are the Toddler Mandelbrot Set in A and its mirror image in B. Both of them belong to the green zone AΔB.

    Example 1: The Julia set of Figure 0.2 called "Twin Dragons" and shown at the outset of this website is given by the parameter p = (0.185, 0.00007666) belonging to A∩B; hence, it is connected. It is actually given by rotating the output image 90o to better fit on the webpage; see geometric similarity. If we move the parameter to p = (0.185, 0) that lies on the real axis, the output image becomes symmetric about the center horizontal line providing us with "Identical Twin Dragons." Figure 6.8 shown below contains three topologically distinct "Twin Dragons."


    Figure 6.8.  "Twin Dragons"

    p = (0.2011575, 0.00002) in A∩B


    p = (0.21828, -0.00230) in [A∪B]c p = (0.2176, 0.0128) in AΔB


    Figure 6.9(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 "    Broken Balloons" is a molecule comprising atoms of periods

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

    It can be seen near the bottom of the "Venn diagram" and it intersects both A∩B and the symmetric difference AΔB which is the green zone.

    The connected "Roses" of Figure 6.9(A) is a Julia fractal of the parameter p = (0.02912, -1.093853) belonging to an atom of period 3 × 7 in "Broken Balloons." The parameter p also belongs to A∩B, 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 6.9(B) is a Julia fractal of the parameter p = (0.07761, -1.12427) belonging to an atom of period 3 × 4 in "Broken Balloons."  The parameter p also belongs to the symmetric difference AΔB, so the Julia set is disconnected, which we can see in the broken "stems." Note that the Julia set is not a Cantor set. Where in the picture do we see the number 3 ?



    Figure 6.9(B).  "Disconnected Roses"



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


    Figure 6.10.  "Cantor Elephants"


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



    Example 3: The "    Toddler Mandelbrot Set" seen near the bottom edge of Figure 5.1 belongs to AΔB but it is omitted from the "Venn diagram."  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—except that they are all disconnected as "the Toddler Mandelbrot Set" belongs to AΔB. For example, the image which is shown below and resembles the "Hydra" of Figure 5.1 is a Julia fractal 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 6.11.  "Lernaean Hydra with Thirteen Heads and Offsprings"



    Example 4: While the "Toddler Mandelbrot Set" generate Julia sets that resemble Julia sets of the Mandelbrot set seen in § 5, the "Giant Mandelbrot Set" 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 from near the real axis through the Giant Mandelbrot set. Here is another, this time from near the neck of the giant.


    Figure 6.12.  "Pearly Dragon"
    The Julia Set of p = (0.00618, 0.54433) in [A∪B]c which is a Cantor Set
    Cf. Figure 6.0    , Figure 1.1(A), Figure 1.1(B), Figure 1.7(F), Figure 1.7(G)






    § 7.  The Logistic Equation

    There is one remaining and possibly the biggest selling point of the Mandelbrot set we would like to discuss and it is the striking simplicity of the     Mandelbrot equation (1.1) from which the Mandelbrot set is defined and all the wonders we have witnessed are generated. It turned out that the simplicity is a disguise and any quadratic dynamical system is "conjugate" to (1.1) for some p in the sense that any Julia set of the former is geometrically similar to the Julia set of the latter (and vice versa). We recall that the Mandelbrot set is completely characterized as the set comprising parameters p whose Julia sets are connected per its alternative definition.

    Mathematically speaking, therefore, the Mandelbrot set is loaded with information on all quadratic dynamical systems. Rather than showing the "conjugacy" in full generality which merely involves completing the square used in high school algebra, we will verify it using an example of quadratic dynamical systems called the logistic equation. The logistic equation became famous in the 1970s with the advent of chaos and it is interesting in its own right.

    Artistically speaking, the "conjugacy" may be a letdown but we should still retain our interests in plotting fractals using the logistic equation. Figure 7.11 shows that the "conjugacy" that preserves the geometric shape of a Julia set need not preserve the colors and textures of the fractal. Besides, getting just the right shapes and colors in fractal plotting is very often a chance encounter, and it is unlikely that what we get from the logistic equation someday emerges from the Mandelbrot equation.

    Figure 7.1 depicting a mini-Mandelbrot set given by the logistic equation may well be a fractal that cannot be found by the Mandelbrot equation. It also shows that a coloring change alone may drastically alter the appearance of a fractal, indicating that art is more sensitive to a variety of factors than mathematics.


    Figure 7.1. Daytime and Nighttime Views of a mini-Mandelbrot Set by the Logistic Equation



    The image shown below is a 3D rendering of the 2D fractal shown above and suggests a possible project to people familiar with multivariable calculus.


    "Mandelbrot Volcano"



    What is the logistic equation? In 1838 Pierre Verhulst introduced a differential equation called the "logistic equation" 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 .

    If we expand its variables and parameters to complex numbers and apply the divergence and convergence schemes on the     critical orbits of p with the fixed initial values z0 = 0.5, we get the Mandelbrot fractal shown below. For convenience, we call the entire molecule with its hairy boundary the logistic set. Figures 7.1 and 7.2 show that the interior of the logistic set is again the disjoint union of all atoms with a variety of shapes, just like the interior of the Mandelbrot set.


    Figure 7.2. 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 right-hand antenna coincides with the closed interval [α, 4] on the real axis with α ≈ 3.569945672. The logistic set is symmetric with respect to the 180o rotation about the point (1, 0) and the horizontal flipping about the real axis.

    In 1974, while conducting a computer simulation of certain population changes with the logistic equation, biologist Robert May discovered     chaotic orbits of p belonging to the closed interval [α, 4]. Research by stimulated mathematicians followed and the mathematical term chaos appeared 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 Mandelbrot fractals of the dynamical system (7.1) by zooming in on the interval [α, 4]. Figure 7.3 is one of them and uses a noncritical point z0 = 0.1 (10% of the sustainable population) of fp as the initial value for the orbits of various species p. Note the bifurcation pattern on the leaves.


    Figure 7.3.  "Spring Reflection"



    A nearby vicinity contains numerous attractive fractals including the three images in     Figure 0.6. Mapping the fractal shown above on a bowl and an apple, we get:






    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 "Elephant Bay" as it contains many fractals that look like elephants. Although attractive fractals can be found everywhere near or on the boundary of the logistic set, we will limit our attention to Elephant Bay for the rest of the section. While     Figure 7.1 is a Mandelbrot fractal from Elephant Bay given by the critical point z0 = 0.5 of fp, we find a large number of interesting fractals using noncritical initial points instead.


    Figure 7.4. "Birth of Elephants"



    Figure 7.4 is a Mandelbrot Fractal of z0 = 0.2 generated by the Logistic equation. The use of the noncritical point as the initial value makes the circular atoms of the logistic set crack like eggs and give birth to elephants. The next three images are Mandelbrot fractals of the noncritical point z0 = 0.1 of fp in (7.1) from Elephant Bay. For example, the third image comes from a microscopic rectangular neighborhood of the parameter p = (2.999997892, 0.0079284853).


    Figure 7.5. "Circus Elephants"



    Figure 7.6. "Pearly Elephants"



    Figure 7.7. "Cloisonné Elephants"




    Julia Sets by the Logistic Equation:  We now show a few     (filled-in) Julia sets generated by the logistic equation. Figure 7.8 shows (an artist's rendering of) 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 at Elephant Bay. Hence, it is decorated by the convergence scheme with period index 1 and its complement by the divergence scheme. The second image shown below is the boundary of the filled-in Julia set, namely, the Julia set of p = (2.994915, 0.1). It is a Jordan curve which is homeomorphic to the second image of Figure 5.6.


    Figure 7.8. "Circus Elephants"







    Figure 7.9 shows the Julia set of the parameter p = (3.0014564, 0.08) belonging to Elephant Bay and the complement of the     logistic set. By virtue of the "conjugacy" which we will show a little later, the dichotomy theorem also holds for the logistic equation, hence it is a Cantor set.


    Figure 7.9. "Pearly Elephants"

    Julia Fractal of q = (3.0014564, 0.08) by the Logistic Equation (7.2)



    The parameter p = (3.0237615, 0.1) that generates "Dancing Seahorses" shown below belongs to the orange atom of period 2 in the logistic set at Elephant Bay, hence the filled-in Julia set is painted by the convergence scheme with period index 2. Elephant bay is sandwiched by a red atom of period 1 and an orange atom of period 2, and interestingly, a parameter from the orange shore generates "seahorses" instead of "elephants."


    Figure 7.10. "Dancing Seahorses I"




    Conjugacy of the Logistic and Mandelbrot Equations:  As indicated at the outset of this section, the logistic equation and the Mandelbrot equation are "conjugate" to each other, and as a result, the two dynamical systems share     geometrically similar Julia sets. 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 7.11.
    "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.

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

    (7.7)  q = 1 ± √ (1 - 4 p).

    These two q values are symmetric about the intersection point (1, 0) of the figure 8 in the     logistic set and generate geometrically similar (filled-in) Julia sets.

    Example: The first image of Figure 7.11. 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). By the aforementioned theorem, they are geometrically similar. Although the two images are painted by exactly the same coloring routine, the artist's renderings of the filled-in Julia sets turned out to be a little different. It shows that the conjugacy relation preserves the geometric shape of the filled-in Julia set but not necessarily its coloring.

    Finally, the dynamical system

    (7.8)   zn+1 = fp(zn) = p(1 - zn2) zn

    is an extension of the logistic equation (7.1), which we call the third degree logistic equation for convenience.

    Figure 7.12 is a global     Mandelbrot fractal of the critical point z0 = 1/√3 of the function fp. Figure 7.13 shows two local Mandelbrot fractals of noncritical points z0 = 0.1 and z0 = 0.5. The circular atoms of the global image are cracked and deformed by the use of the noncritical points and give birth to interesting figures like the ones shown in Figure 7.13. These figures often have strong resemblance to Julia fractal born from the atoms. Figure 7.14 shows a closeup of a crack painted on a plane and on an egg.


    Figure 7.12.  The Third Degree Logistic Set



    Figure 7.13.  "Moray Eels"



    Figure 7.14.  Crack in an Atom and "Shellfish"



    The dynamical system (7.8) also generates Julia sets. The seahorse dancers in Figure 7.15 line up a little differently from the ones shown in     Figure 7.11. A part of the first image of Figure 7.15 shows a striking resemblance with the first image of Figure 7.13.


    Figure 7.15.  "Dancing Seahorses" by the Third Degree Logistic Equation
    p = (1.18, 0.376) p = (1.1565, 0.3688)


    Here is another dancer from the fifth degree logistic equation zn+1 = fp(zn) = p(1 - zn4) zn. The Julia set is emphasized in the nighttime fractal on the right.


    Figure 7.19.  "Dancing Bouquet" by the Fifth Degree Logistic Equation





    § 8.  Newton Fractals

    A     Julia Fractal is called a Newton fractal if it is given by a dynamical system of the form

    (8.1)   zn+1 = zn - g(zn)/g'(zn)

    where the parameter p = 0 is invisible and g is a holomorphic function with its derivative g'. In this section, g(z) is a polynomial in complex variable z which allows us to take advantage of the time-saving scheme called Horner's Method to efficiently evaluate both g and g' that appear in the dynamical system. 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.

    The reader may have noted already that the dynamical system (8.1) is nothing but the Newton-Raphson Root-Finding Algorithm, aka Newton's Method. Hence, each orbit of (8.1) converges to a root of g quickly more often than doing something else, 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.

    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 = con(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 divided by the basin boundary. The basin boundary is precisely the Julia set of the Newton fractal, and that is where Newton's rootfinding algorithm behaves in a "chaotic" fashion. It is known that the Julia set is a Cantor set.

    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 fails when any of the initial values chosen hits the Julia set, causes (8.1) to diverge or is a critical point of g(z).

    So, is there a hassle-free way 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"



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





    Links

    Pacific Northwest Section, Mathematical Association of America

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

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