This website is a gallery of computer-generated fractal art as well as a text that explains what it is and how it is created. Central to the website are a myriad of fractal images shown in "Stories about Fractal Art" augmented by the text as optional references. In addition, there are two galleries of fractal art: Gallery 2D is a collection of two-dimensional (2D) images made recently, while Gallery 3D comprises such 3D objects as fractal mountains (some are realistic and others imaginative), fractal forests and fractals painted on various nonplanar surfaces. Here are examples:
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.
Like "fractal," the word "chaos" was used as a mathematical term for the first time in 1975 when the American Mathematical Monthly published "Period Three Implies Chaos" by T.Y. Li and James Yorke. The paper inspired by May's discovery received a great sensation especially because there appeared very little difference between chaotic and random outcomes even though the former resulted from deterministic processes.
The idea of chaos quickly evolved into comprehensive chaos theory in science and at the same time its affinity with fractal art became evident. It may sound paradoxical, but the more elaborate and dazzling a fractal image, the more likely its entanglements with chaos. For example, the Julia sets became known to be chaotic and shape a great many fractal art pieces; e.g., see Figures 0.6(B), 0.6(C), 0.6(D) and 0.6(E).
Googling we can find a host of websites displaying computer-generated fractal art images, some of which are stunningly beautiful. It indicates that a large population not only appreciates the digital art form but also participates in the eye-opening creative activities. A fairly large part of this article is devoted to show how to program a computer and plot popular types of fractals generated by simple dynamical systems. It is not a text on computer programming but instead tells the general principles for fractal plotting in everyday language so as to entice more people into trying it.
Particularly exciting is the moment the fractal image generated by our personal program emerges in our computer screen, because of its potentially "astounding" artistry and built-in chaos-related unpredictability. Many of the fractal art images will stir our imaginations in the part of mathematics that is in fact quite deep and still filled with unknowns. It is plain fun.
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
p-Canvases and z-Canvases: Consider any dynamical system like the Mandelbrot equation (1.1) comprising 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 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.
Plotting a fractal on a p-canvas is, roughly speaking, as follows: Choose a value of z0, say z0 = 0, and an appropriate p-canvas following the direction in § 2 or § 4. For each pixel (i, j) on the p-canvasR, use its representative parameter p and the dynamical system to generate the orbit of p with the initial value z0 = 0. We then use a certain property of the orbit 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.
Plotting a fractal on a z-Canvas is similar except that here we use orbits of z0 (with a fixed parameter value p) corresponding to all z0 on the z-canvas instead of orbits of varying p with a fixed initial value z0. We call a fractal plotted on a z-canvas a Julia fractal, as it is typically a colorful fractal art image depicting a Julia set; see Figure 0.6(B). Because Julia sets are colorless subsets of the complex plane belonging to fractal geometry, there is a clear difference between a Julia set and a Julia fractal, but after a while, we'll follow the common practice of identifying them and calling the colorful version a Julia set rather than a Julia fractal.
For the similar reason, a fractal plotted on a p-canvas is called a Mandelbrot fractal. It is normally an artistic view of the Mandelbrot set, but again, we will call the colorful version the Mandelbrot set rather than a Mandelbrot fractal. There are some exceptional cases, however, like when we discuss a "Mandelbrot-like" set generated by a dynamical system other than the Mandelbrot equation. In such a case, the image on the p-canvas will be called a Mandelbrot fractal instead of the Mandelbrot set.
As we'll see, the Mandelbrot set generated by the Mandelbrot equation is bounded so its global figure fits in a circle or a rectangle like in Figure 1.4(B), and from the shape of its black silhouette, it is often nicknamed "Warty Snowman" lying sideways. A part of the global image is called a local image, but a local image in fractal art is usually given by zooming in on a microcroscopic rectangular neighborhood of a point that is very near or on the border of the snowman's silhouette.
Because the Mandelbrot set is also known to be closed, its boundary, which coincides with the snowman's silhouette's border, is a part of the Mandelbrot set. It is also known that the boundary has the "topological dimension" of 1, meaning intuitively that it comprises razor thin "filaments" without thickness, just like the boundary of a circular disk. So, the boundary is mostly invisible in local images unless we "light up" their filaments like in a tungsten light bulb; see Figure 1.4(C) shown below.
As Figure 1.4(B) shows, the filaments are like branched hairs growing outwards from the warty snowman and known to carry infinitely many miniature copies of the snowman, which we call "mini-Mandelbrot sets." One of them is visible in Figure 1.4(C) and, yes, the boundary of the Mandelbrot set is incredibly intricate.
In 2008, PBS broadcast a NOVA program proclaiming that the Mandelbrot set had become "the most famous object in modern mathematics." Naturally then, we begin our stories with the Mandelbrot set and devote a large part of the article to a myriad of its attributes that fascinate mathematicians and artists alike. The Mandelbrot set popularized the fractal plotting by computers and has been the gold standard for all types of fractals.
In § 2, we state the notion of an orbit that diverges to ∞ and define the Mandelbrot set, denoted by ℳ, using the Mandelbrot equation (2.1) and its critical point. In § 4 we also introduce the notion of an orbit that converges to a cycle of period k for some positive integer k and in conjunction with § 2 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 two distinct algorithms, while Figure 1.4(C) shows a local image of ℳ together with its complex boundary. As we have seen, ℳ contains its boundary as its subset.
By far the biggest attraction of ℳ is the fact that it conceals a greater number of varying local images than the number of stars in the universe, together with the simplicity of the divergence scheme (aka the "escape time algorithm") that allows even amateurs to find many of them and turn them into beautiful art pieces. Thus, its appeals have extended way beyond mathematical communities, contributing ℳ to gain its monumental popularity―which is unheard of for an object in mathematics.
In § 3, we discuss some of the important mathematical properties of ℳ, focusing on its boundary. Note that in Figure 1.4(C) the goldish boundary comprising razor thin filaments gets increasingly more intricate and space-filling as it gets nearer the mini-Mandelbrot set until it gets an appearance of being solidly painted. It makes the painted area look two-dimensional. Shishikura's theorem formally explains a phenomenon like that in terms of "fractal dimensions," and shows, in effect, that no figures in the plane are more complex than the boundary of ℳ. It boosts the Mandelbrot set ℳ to be one of the most complex figures ever plotted on a plane.
In § 4, we turn our attention to the interior of ℳ and use Mandelbrot's idea as a cue to define an atom to be a connected component of the interior. Then atoms include all of the disks and cardioids visible in ℳ, and as per the "density of hyperbolicity" conjecture, all atoms are associated with periods of certain cycles. A part of the association between atoms and periods is illustrated by the meticulously aligned numerical structure shown above in the periodicity diagram.
In spite of all these astounding complexities of ℳ, Adrien Douady and John H. Hubbard established earlier that ℳ is topologically tidy as it is compact, connected in one piece and without holes. The connectedness of ℳ is particularly important and called the Douady-Hubbard Theorem. The next-step problems of whether ℳ is pathwise connected and whether ℳ is locally connected are conjectured affirmatively but still remain unsolved, despite the efforts by some of the world's brightest minds.
In § 5, we define, given a parameter p in the complex plane, the filled-in Julia set of p generated by the Mandelbrot equation and show how to plot it on a z-canvas using the divergence and possibly convergence schemes. If p belongs to an atom of ℳ, then the period of the atom affects the shape of the filled-in Julia set in an amazing and sometimes amusing way, although the exact cause is shrouded in mystery. It makes the periodicity diagram all the more important in plotting filled-in Julia sets. For example, the image shown below is the filled-in Julia set of a parameter chosen from an atom of period 9 (near the "neck of the warty snowman"), while Figure 1.4(A) shows two filled-in Julia sets born from the same atom of period 17 × 5 (which is near the cusp of the cardioid).
Note that the Julia set of Figure 1.6(B) is connected as its parameter belongs to ℳ. but the Julia set of Figure 1.6(C) shown above is totally disconnected. It is interesting and important to note in the latter that the period of the nearby atom, namely k = 11, still shows in the shape of the Julia set as the number of the "hydra's heads."
To further consolidate the close-knit relations between ℳ and the Julia sets given by (†), we will also discuss Tan Lei's Theorem in § 5, which explains why we frequently observe striking similarities in appearance between local images of ℳ and Julia sets. The phenomenon is so prevalent that the Mandelbrot set ℳ is often called an "index" to all Julia sets.
As we have just noted, the Mandelbrot set is uniquely determined by the critical pointz0 = 0 of the Mandelbrot equation. So, what happens if we deal with a dynamical system with multiple critical points?
In § 6, we extend the Mandelbrot equation to more general dynamical systems, and, in particular, consider the relatively simple cubic dynamical system (6.2) with two critical points z0 = ± i / √3 = ± (0, 1 / √3).
If we define and generate the "Mandelbrot sets" using (6.2) and the two critical points, we get rather comical fractals called the Speared Mandelbrot Sets ℳ1 and ℳ2; see Figure 1.7(A). To be precise, ℳ1, for instance, corresponds to z0 = i / √3 and is the portion not including the dark green background. It does include, however, the mini-Mandelbrot set seen near the bottom of Figure 6.1 like an isolated island, but it is omitted from Figure 1.7(A) for simplicity. The tip of the "spearhead" turned out to be the origin (0, 0) of the complex plane and ℳ2 is the mirror image of ℳ1 through the real axis of the complex plane.
Atoms of ℳ1, for instance, are again defined to be connected components of the interior of ℳ1 and are in a wide variety of shapes that include the interior of a blue disk as well as the interior of the red spearhead.
In addition to numerous local images such as Figure 1.2(C) that are similar to those from the Mandelbrot set, ℳ1 and ℳ2 have numerous others including Figure 1.7(B) and the first image of Figure 1.7(C) shown below, where some of the atoms appear to break into fragments. Interestingly, all of the fragments disappear in the third image of Figure 1.7(A), which we call "Atomic Fusion," given by superimposing ℳ1 and ℳ2.
As we will see in § 6, "Atomic Fusion" has a handy application in the pictorial interpretation of the Fatou-Julia Theorem and the classification of the Julia sets given by the dynamical system (6.2). There we learn that the Julia set of a parameter p is connected if p belongs to ℳ1 ∩ ℳ2 and it is totally disconnected if p belongs to the complement of ℳ1 ∪ ℳ2. Thus, if the Julia set of a parameter p defies the dichotomy, then p belongs to the "symmetric difference"
ℳ1 Δ ℳ2 = (ℳ1 ∪ ℳ2) - (ℳ1 ∩ ℳ2).
Computer experiments show that the converse of the last statement seems to hold but it is not known for sure. Here are examples:
In § 7, we go back to the Mandelbrot set ℳ and discuss possibly the most charming feature it possesses, which is the striking simplicity of the Mandelbrot equation (1.1) generating all the wonders of ℳ we have witnessed. Does it come at a cost? The answer turned out to be "no" in the sense that the "Mandelbrot set" ℳ ' defined by any quadratic dynamical system and (†) is essentially the same as ℳ.
Rather than showing this fact in full generality (which can be done by high school algebra), we will verify it using a special quadratic dynamical system called the logistic equation. The process involves the important idea of the "conjugacy" between two dynamical systems. As mentioned in Introduction, the logistic equation became famous with the advent of chaos and is interesting in its own right.
We will also show several examples of fractals from the logistic equation, even though similar fractals can be found by the Mandelbrot equation theoretically. Art involves more factors than mathematics such as colors and textures, etc. making it impossible to reproduce exactly the same images by any means. In art, we can also deviate from standard procedues such as using critical points and conduct computer experiments. Here's an example:
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:
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.
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).
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 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."
Recall that the Mandelbrot set is denoted by ℳ and is closed so it contains its boundary as its subset. It is also 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 one of the most complex objects ever plotted on a plane.
Shishikura's Theorem (1998): The fractal dimension of the boundary of the Mandelbrot set is 2 (which is the topological dimension of the plane).
Let's pause for a moment 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:
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 of indefinitely repetitive patterns, 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 topological dimension of the plane.
Fractal dimensions in fact measure complexities and space-filling capacities of any curves (i.e., objecs of topological dimension 1) by fractional scales between 1and 2, and the boundary of ℳ attains the maximum value of 2 per Shishikura's theorem. By comparison, the fractal dimension of the boundary of the "Koch Snowflake" shown in Figure 0.4 is 1.2619. It should be noted in the aforementioned Mandelbrot's definition of fractal that the fractal dimension of the boundary of ℳ also exceeds its topological dimension by the largest possible value.
The boundary image of Figure 3.1 also shows that it works like the basic monochromatic line art of the multicolored ukiyo-e woodblock prints, which are completed by the additional steps of coloring between the lines. The finer the line art, the more elaborate the final ukiyo-e print; see Figure 0.6(A).
One of the most important topological properties in fractal geometry is "connectedness" of a set and Figure 3.1 appears 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 3.3. 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.
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.
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 Figure 3.1 shows, 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. An atom and the interior of a mini-Mandelbrot set 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).
Example 1 (The Mandelbrot Set): Start with the p-canvasR, 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.
where k is the period of the corresponding atom of ℳ. We call λ the period of ℳ '.
For example, the most visible mini-Mandelbrot set of Figure 4.3 happens to have period λ = 4 and is shown in Figure 4.4 and the inset of the periodicity diagram. There, it is painted by the convergence scheme with period indices λ k = 4, 8, 12, ..., 100 and with the colors of the Mandelbrot set of Figure 4.2 so as to emphasize the one-to-one correspondence between the atoms of ℳ and the atoms of the mini-Mandelbrot set.
Here's an additional technical detail: For the convergence scheme with period indices λ 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.
Figure 4.4. The the mini-Mandelbrot Set of Period λ = 4 and Its Boundary
Periodicity Diagram: If we label the atoms of the Mandelbrot set in Figures 4.2 and 4.4 by their periods instead of colors, we get the following periodicity diagram. The periods in the diagram show meticulously aligned numerical patterns that are easy to recognize and will play an important role in plotting many of the "Julia sets" in the next section. The numerical patterns are yet another amazing property of the Mandelbrot set ℳ.
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 setℳ, which possesses the dazzling features we have witnessed so far.
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. Doing so also reveals, as we'll see, the root of the famed Mandelbrot set and explains why the critical orbits are used in its computation.
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 that the orbit of p with an initial value z0 is also called the orbit of z0 with the parameter value p. Define the filled-in Julia set of p to be the set of all initial values z0 in the complex plane whose orbits (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 ℳ, 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 as shown in the preceding section, so it makes sense to say that the filled-in Julia set of p is "born" from such an atom of ℳ.
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.
Remark on Plotting the Filled-in Julia Set: 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 critical 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. The shape of the filled-in Julia set reveals interesting dynamics of the orbits affected by the behavior of the critical orbit. Here are additional examples:
By a Cantor set or Cantor dust we mean 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.
Example 2: The filled-in Julia set of p = (-0.6891, 0.27896) called "Hydra's Ash" in Figure 1.6(C) is a Cantor set. The parameter p happens to be near the aforementioned atom of period 11 but lies outside of the Mandelbrot set. We'll see the reason why it caused the set to be totally disconnected.
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 shape and complexity of the filled-in Julia set and has a strong association with chaos. It is named after Gaston Julia, who was one of the pioneers of fractals generated by dynamical systems along with Pierre Fatou.
In fractal art, we call a fractal painted on a z-canvas a Julia fractal and it is usually given by a monochromatic Julia set and additional coloring that fills the space defined by the Julia set; see Figure 0.6. For example, the "hydra" of Figure 5.1(A) is a Julia fractal comprising the Julia set that dictates the shape of the "hydra," the background painted by the divergence scheme and the interior of the "hydra" painted by the convergence scheme.
Example 3: Figure 5.1(D) shows a highly chaotic Julia set of p belonging to the interior of the mini-Mandelbrot set of Figure 3.3. Because the period of p is large and unknown, Figure 5.1(E) is painted by the divergence scheme alone on a z-canvas centered at the critical point z0 = 0 and shows the complement of the filled-in Julia set; see the daytime view of the fractal. To our eyes, therefore, the filled-in Julia set is identical to the Julia set of Figure 5.1(D) as its interior is invisible in either figure. If we zoom in on a microscopic neighborhood of the center of Figure 5.1(E), we see a part of the interior of the filled-in Julia set; see the "black hole" of Figure Figure 5.9.
Remarks: We call the aforementioned Mandelbrot's definition the alternative definition of the Mandelbrot set. The fact that it is equivalent to our earlier definition explains why the critical point of (5.1) is indispensable in computing ℳ. The Julia sets (and filled-in Julia sets) of Figures 5.1(A)(B)(C)(D) are connected as their generating parameters belong to ℳ while the Julia set (and the filled-in Julia set) of Figure 1.6(C) is a Cantor set.
The two "lions" are painted by the convergence scheme with period index 85 and the 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.
Figures 5.3 & 5.4. "Twin Lions" born from the same atom of period 17 × 5
Example 5 (Continued): "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.
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
It is a local image of the Julia fractal shown in Figure 5.1(E) 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.
The Julia fractal of Figure 5.1(E) is based on the global Julia set of the parameter p = (0.25000316374967, -0.00000000895972) shown in Figure 5.1(D). If we zoom in on the center of Figure 5.1(E) between the eyes of the central cuttlefish, we get another local image:
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 5.9 and 5.10 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 5.11 shows we can zoom out from Figures 5.9 and 5.10 while retaining some degree of similarity.
Figure 5.11. "Cuttlefish" Swimming in the Mandelbrot Set (Left) and in the Julia Set (Right)
Soon after Mandelbrot published its computer plot generated by the simple process 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 art 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.
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 dark green background comprising the parameters whose orbits diverge to ∞.
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"
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 6.1 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."
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"
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.
Recall that a critical orbit of (6.1) means an orbit whose initial value z0 is a critical point of the function fp. Here is a tremendously useful theorem Gaston Julia and Pierre Fatou independently proved in 1918-1919 when computer-generated images of Julia sets were not available!
The Fatou-Julia Theorem: Consider a polynomial dynamical system
where m ≥ 2, cm, cm-1, · · ·, c1, c0 are complex constants with cm ≠ 0, and one of the coefficients, cj , assumes the role of the parameter p. Then we have:
(1) If all critical orbits of p stay within a finite bound, the Julia set of p is connected;
(2) if no critical orbits of p stay within a finite bound, the Julia set of p is a Cantor set.
An immediate corollary is that if (6.3) has exactly one critical point like the Mandelbrot equation and the Multibrot equation then the fundamental dichotomy holds for the dynamical system.
Thus, like in the case of the Mandelbrot set, we may define the "Mandelbrot-like" set of the dynamical system using the dichotomy and compute it by the divergence scheme. It is not particularly difficult to prove the divergence criterion for (6.3) with the threshold possibly different from 2.
with two critical points i/√3 and -i/√3 of fp. Let ℳ1 be the set of parameters p whose critical orbits with the initial value i/√3 stay within a finite bound and ℳ2 the same with the initial value -i/√3. ℳ1 is precisely the "Speared Mandelbrot Set" depicted by Figure 6.1 and ℳ2 the mirror image of ℳ1 through the real axis. Figure 1.7(A) shows interesting relations between ℳ1 and ℳ2 when they are superimposed.
Figure 6.7 shows a pseudo Venn Diagram of ℳ1∪ℳ2 and ℳ1∩ℳ2 which we call the "Venn Diagram." Here, the union ℳ1∪ℳ2 is painted by colors other than black and the intersection ℳ1∩ℳ2 by colors other than black and green. Thus, the black zone is the complement of ℳ1∪ℳ2 denoted by [ℳ1∪ℳ2]c and the green zone is the symmetric difference
ℳ1Δℳ2 = ℳ1∪ℳ2 - ℳ1∩ℳ2.
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 ℳ1∩ℳ2 then the Julia set of p is connected;
(2) if p belongs to [ℳ1∪ℳ2]c then the Julia set of p is a Cantor set.
(1) and (2) imply:
(3) If the Julia set of p is disconnected but not a Cantor set, then p belongs to ℳ1Δℳ2.
Our computer experiments show that the converse of (3) seems true, but the Fatou-Julia Theorem does not confirm it. Although the "Venn Diagram" works well for our purpose of finding examples that illustrate the Fatou-Julia Theorem, we need to be a little careful as the diagram does not include the hairy boundaries of ℳ1 and ℳ2 seen in Figure 6.1. Also omitted in the diagram are the Toddler Mandelbrot Set in ℳ1 and its mirror image in ℳ2. Both of them belong to the green zone ℳ1Δℳ2.
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 ℳ1∩ℳ2; 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."
It can be seen near the bottom of the "Venn diagram" that it intersects both ℳ1∩ℳ2 and the symmetric difference ℳ1Δℳ2 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 ℳ1∩ℳ2, so the numerous "roses" seen in the image are connected by the "stems." " We can clearly see the number 7 in the picture but where do we see the number 3 ?
The disconnected "Roses" of Figure 6.9(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 ℳ1Δℳ2, 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 ?
Example 3: The "Toddler Mandelbrot Set" seen near the bottom edge of Figure 5.1 belongs to ℳ1Δℳ2 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 ℳ1Δℳ2. 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.
There is one remaining and possibly the biggest selling point of the Mandelbrot set ℳ we would like to discuss and it is the striking simplicity of the Mandelbrot equation (1.1) from which all the wonders on fractals we have witnessed are generated. Because a canvas comprises millions of pixels on average and coloring each pixel easily requires thousands of orbit evaluations, one extra multiplication in a formula like (1.1) can make a significant difference in a computer's runtime. So, the simplicity is not only aesthetically eye-catching but also important from a practical viewpoint. But does it come at cost? The answer happens to be "no" at least mathematically.
Recall that ℳ was initially given by (1.1) and the alternative definition based on the dichotomy. Suppose (6.3) is any quadratic dynamical system (with a unique critical point), so the dichotomy also holds for its Julia sets. Let ℳ ' be the "Mandelbrot set" defined by (6.3) in exactly the same way as ℳ. Then it turns out additionally that (6.3) is "conjugate" to (1.1), guaranteeing that any Julia set generated by (6.3) is geometrically similar to a Julia set generated by (1.1) and vice versa. It follows then that ℳ ' and ℳ are essentially the same set.
Rather than showing the "conjugacy" in full generality (which can be done by high school algebra), we will verify it using a special quadratic dynamical system called the logistic equation. As mentioned in Introduction, the logistic equation became famous with the advent of chaos and is interesting in its own right.
What is the logistic equation? In 1838 Pierre Verhulst introduced a differential equation called the "logistic equation" widely used to describe the population dynamics with self-limiting growth. If we replace the derivative in the differential equation by its approximating difference quotient and do some algebra, we get the following "difference equation," which is more suitable for computer applications and again called the logistic equation:
(7.1)
zn+1 = fp(zn) = p(1 - zn) zn .
Expand its variables and parameters to complex numbers and let ℳ ' be the aforementioned "Mandelbrot set" defined by (7.1). Then by the reason shown in the remark, we get ℳ ' shown below by applying the divergence and convergence schemes on the critical orbits of p with the fixed initial value z0 = 0.5. For convenience, we call the entire moleculeℳ ' comprising the atoms together with its hairy boundary the logistic set, short for the "logistic equation's Mandelbrot set."
As we'll see later, there is a two-to-one function F mapping the logistic set onto the Mandelbrot set such that for any parameter p, the Julia sets of p and F(p) are geometrically similar; see Figure 7.4. The function F becomes a one-to-one correspondence if it is restricted to the left half or the right half of the logistic set. Thus, F actually fails to be two-to-one at the intersection point (1, 0), which works as a "double cusp" of the Mandelbrot set.
Julia Sets by the Logistic Equation: We now show a few (filled-in) Julia sets generated by the logistic equation. Figure 7.8 shows the filled-in Julia set of the parameter p = (2.994915, 0.1) belonging to the red atom of period 1 in the logistic set 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 a circle.
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."
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.
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
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
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 7.15. "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 equationzn+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
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 bothg 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.
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"
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 8.3through 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.
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