Sekino's Fractal Art Gallery

By induction, it follows that

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

Proof: If |p| > 2, (2.1) and (2.2) imply 

|z₂| = |z₁² + p| ≥ |z₁|² - |p| = |p|² - |p| = |p|(|p| - 1) > |p| = max{2, |p|}.

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

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

The Divergence Criterion: 

We now use the divergence criterion and a computer to

We then regard R as a

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 |z_M| > 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 z_n of the parameter p escapes from the circle of radius θ = 2 before taking a long journey toward ∞; see the

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

Of course, an actual computer program based on the divergence scheme can be streamlined in many ways. Probably the most important is to use

The first of the two images in

Coloring Fractals by 24-Bit Colors:  Modern computers show graphics in the "24-bit colors" comprising 2²⁴ ≈ 16 million colors and we can modify the divergence scheme to take advantage of the capacity to plot colorful fractals such as Figures

The question leads to our common practice of

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

   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.

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,

(2) It is not too hard to generalize the divergence criterion and find a threshold like

Figure 2.4.  Seventh Degree mini-Mandelbrot Set

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

§ 3.  The Mandelbrot Set

In 2008, PBS broadcast a NOVA program proclaiming that the

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

In § 2, we noted that the divergence scheme paints the complement of ℳ in full color and ℳ itself by a single color like black or white with a cautionary remark that a computer plot is an approximation which is not 100% accurate. We have also seen that ℳ is

Because the topological dimension of the boundary is known to

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

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

For example, the boundary (perimeter) of the Koch Snowflake in

Another Local Image of ℳ and its Boundary (Right)

For example, it is known that the "neck" of the "snowman" in Figure 2.1 is the point

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

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

Local Images of ℳ in a Neighborhood of p = (0.28212348434375, 0.0110096504375) "Daytime" and "Nighttime Views" of a Fractal:  Shown above on the left is another fractal generated by the Mandelbrot equation (2.1) and the divergence scheme, where the razor-thin filaments of the boundary of the Mandelbrot set ℳ are invisible. When its colors are darkened and the thin filaments are lit up, we get the "nighttime view" of the fractal on the right, vividly showing the presence of the complex boundary of ℳ hidden in the "daytime view" on the left.

Note that the two views may appear totally different, mainly because the daytime view shows the

For the above image on the left, we used whopping 1,500,000 iterations of the Mandelbrot equation for

Shown below is a nighttime view of the fractal on the left that reveals the boundary of the mini Mandelbrot set. In

Figure 3.4.  The Boundary of the mini-Mandelbrot Set

We stated earlier a precise definition of a set in the complex plane being "connected" as "one piece" and now wish to dig into the notion of "

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

In general, if S is any nonempty set of points in the complex plane, a nonempty maximal connected subset of S is called a

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

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

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

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

The set S is said to be

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

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

§ 4.  The Convergence Scheme

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

As we saw in

Figure 4.1.  The Mandelbrot Set with Colorful Atoms

  If |z_1+k - z₁| < ε then color the pixel p col₁,

else if |z_2+k - z₂| < ε then color the pixel p col₂,

else if |z_3+k - z₃| < ε then color the pixel p col₃,

· · ·

else if |z_M+k - z_M| < ε then color the pixel p col_M.

Here, col₁, col₂, ... , col_M are prescribed colors and ε a small positive real number like 10^-6, Δx or Δy; see

We first apply the divergence scheme with

The first image is generated by the

The second image is given by the

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

Example 1 shows that the convergence scheme may mean the one with

In this article, we assume that the conjecture is true. For example, it implies that the critical orbit of every parameter in the interior of ℳ converges to a cycle and it resolves the fundamental

Figure 4.3.  Closeups of the Interior of the Mandelbrot Set

For example, the most visible mini-Mandelbrot set of Figure 4.3 happens to have the base period

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

In fractal art, the "sensitive dependence on a parameter" near the boundary of ℳ causes the drastic changes of patterns in a nighttime view of the fractal in

The picture shown below on the left is essentially the same as

Figure 4.7.  Periodicity Diagram of ℳ

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

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

§ 5.  Julia Sets and the Fundamental Dichotomy

(5.1)  f(z) = c_m z^m + c_m-1 z^m-1 +  · · · + c₂ z² + c₁ z + c₀,

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

(5.2)  z_n+1 = f(z_n) = c_m z_n^m + c_m-1 z_n^m-1 +  · · · + c₂ z_n² + c₁ z_n + c₀,  

consisting of all orbits of z₀ belonging to the

Define the

Julia and Fatou independently established the following powerful theorem in

Figure 5.0(A).  "Cloisonné Lion"

Henceforth in this section, we deal exclusively with the Mandelbrot equation with a fixed parameter p comprising all orbits of z₀ taken from the

As in the case of ℳ, therefore, the image other than "the Cloisonné metal wires" shows the

Figure 5.0(B) is given by highlighting the Julia set (the Cloisonné metal wires) in

Just like the boundary of the Mandelbrot set, a Julia set was found to be

Recall that the center of the

It is another fascinating fact about the Mandelbrot set that the period of the parameter p is always reflected in the shape of the filled-in Julia set of p, as in the number of "snake-shaped lion's manes" although why it is so is not completely understood. The image in

Figure 5.0(C) "Medusa Lion"

Figure 5.0(D) "Medusa Lion"

Born from an Atom of Period 10

Born from an Atom of Period 21

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

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

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

He knew how to compute ℳ as the Fatou-Julia theorem clearly implies that the Julia set of p is connected if and only if the critical orbit of p does not diverge to ∞; see the computer-friendly definition of

Henceforth, we call (†) the

Figure 5.0(E)

"Twin Lions"

Figure 5.0(F)

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

Today the Mandelbrot set ℳ as a "field guide map" is considerably more detailed because of the

The two "lions" are painted by the convergence scheme with period index 85 and the background by the divergence scheme with the threshold

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

Figure 5.0(G).  "Ruby Lion"

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

An area around the atom of period 4 painted purple and located directly above the cusp of the main cardioid of ℳ in the periodicity diagram is a "

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

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

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

Figure 5.2(C). "Hydra's Ash"

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

Again, if p is near but not in an atom, the Julia set of p still resembles the shape of a Julia set born from the atom as shown in

Figure 5.2(C). As p gets further away from the atom, the "gravity" of the atom on p naturally wanes and the Julia set of p gradually loses the resemblance. Thus, the "field guide map" based on the Mandelbrot set is not confined to ℳ and instead stretches all over the complex plane.

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

All other Julia sets

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

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

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.

Figure 5.3(B). "Run for the Sun"

Many other shapes of Julia sets that can be found on the Mandelbrot set are well documented and shown on the Internet. The locations for the "lion sanctuary," etc. relative to the Mandelbrot set can be translated to the locations relative to any mini Mandelbrot set and provide us with Julia sets often with dazzling backgrounds. The image shown below is a Julia set born in the "hydra territory" of the mini Mandelbrot set in

Figure 5.4(A). "Hidden Hydra"

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

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

Figure 5.5(B). "Partying Cuttlefish"

If we zoom in on the center of the global image between the eyes of the central cuttlefish using different colors but without the eyeball effect, we get another local image of

Figure 5.5(A) revealing the cuttlefish's mouth painted black. The black mouth is a part of the interior of the filled-in Julia set, showing that the Julia set is indeed not a Cantor set.

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

A parameter p is called a

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

Because it is not a cycle but becomes a

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.

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

This probably explains why the local images like Figures

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

§ 6.  The Logistic Equation

We have seen so far many wonders of the Mandelbrot set ℳ. Despite the fact that it is generated by a simple quadratic equation

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

Figure 6.1(A).  "Spring Reflection"

Figure 6.1(B).  "Bowl and Apple"

Possible Projects for Multivariable Calculus Students

In this section, we discuss the eye-catching simplicity of

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

Rather than showing the "conjugacy" in full generality, we will verify it using a special quadratic dynamical system called the

Figure 6.1(C). "Circus Elephants"

(6.1)   z_n+1 = f_p(z_n) = p(1 - z_n) z_n .

Suppose (6.1) is a dynamical system comprising all functions f_p of a complex variable z where p varies through the

As the standard procedure, let

Figure 6.2(A). The Logistic Set

The origin

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

As briefly mentioned in

Figure 6.3(A). "Birth of Seahorses"

Figure 6.3(B).  "Pearls"

The reason is that finding a certain fractal is, to put it in a nutshell, a chance encounter and nobody can find the parameter p of, say, Figure 6.4(A), from scratch. It means that it is impossible for someone to come up with the Julia set of Figure 6.4(A) using the logistic equation, let alone using the Mandelbrot equation. It is always a good idea to retain our curiosity and try all kinds of venues as we never know when an artistic masterpiece suddenly turns up.

Figure 6.4(A) shows the filled-in Julia set of the parameter

Figure 6.4(A). "Circus Elephants"

The Filled-In Julia Set of p = (2.994915, 0.1) by the Logistic Equation (6.1)

Figure 6.4(B). "Circus Elephants"

The Julia Set of p = (2.994915, 0.1) by the Logistic Equation (6.1)

Figure 6.4(C) shows the Julia set of the parameter

p = (3.0014564, 0.08) belonging to the

Figure 6.4(C). "Pearly Elephants"

The Julia Set of p = (3.0014564, 0.08) Generated by the Logistic Equation (6.1)

The parameter

p = (3.0237615, 0.1) that generates "Dancing Seahorses" shown below belongs to the orange atom of period 2 in the

Figure 6.5(A). "Dancing Seahorses"

Figure 6.5(B). "Cloisonné Elephants"

(6.2)  ζ_n+1 = q(1 - ζ_n) ζ_n ,

and reserve z_n and p for the Mandelbrot equation

(6.3)   z_n+1 = z_n² + p ,

where p and

It is not particularly difficult to show that the transformation (6.4) with

Now, without assuming conjugacy, we wish to show that (6.2) can be written in the form

(6.5)   a ζ_n+1 + b = (a ζ_n + b)² + p,

which is the result of applying (6.4) on (6.3). Rewrite (6.2) as

   -q ζ_n+1 = q² ζ_n² - q² ζ_n ,

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

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

   a ζ_n+1 = (a ζ_n + b)² - b²,

which is equivalent to (6.5) with

(6.6)   p = q(2 - q)/4.

Since q ≠ 0, it follows that (6.4) is defined by a = -q and b = q/2 and transforms (6.3) to (6.2), as was to be shown, provided (6.6) is true. Summarizing, we have:

Theorem: If p = q(2 - q)/4 then the logistic equation (6.2) and the Mandelbrot equation (6.3) are conjugate to each other and the Julia set of q by (6.2) and the Julia set of p by (6.3) are

As we have seen, the logistic set and the Mandelbrot set generated by (6.2) and (6.3) lie in two different complex planes, the

Figure 6.7(A). Relationship
Between the Logistic Set and the Mandelbrot Set

The corresponding parameters generate geometrically similar Julia sets.

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

q-values

(6.7)  

q = 1 ± √ (1 - 4 p)

that correspond to a single

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

Because of the symmetric shape of the logistic set and the two-to-one correspondence shown in Figure 6.7(A), the "

Figure 6.8(A).  "Cloisonné Lion"

A rather amazing result coming out of the above computation from (6.2) to (6.6) is another way of writing the logistic equation (6.2), namely,

(6.8)   z_n+1 = z_n² + p = z_n² + q(2 - q)/4 ;  cf. (6.6).

If we plot the Mandelbrot set ℳ(z₀) of (6.8) using its critical point

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

§ 7.  Cubic Equations

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

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

In this section, we consider, as a starting point for a generalization of what we have seen, a simple cubic equation with two critical points and show that even such a simple extension significantly increases the varieties of fractals. At the end of the section, we'll briefly discuss a quartic equation of a similar form with three critical points.

The full generalization of the Mandelbrot and Julia sets entails a

In the preceding section, we saw that all quadratic equations are

(7.0)  ζ_n+1 = α ζ_n³ + β ζ_n² + γ ζ_n + δ ,   with 

We wish to show that (7.0) is conjugate to a simpler cubic dynamical system of the form

(7.1)   z_n+1 = z_n³ + r z_n + s,

i.e., if there are complex constants

(7.1.1)   z_n = a ζ_n + b,

transforms (7.1) to (7.0) for all n ≥ 0. Substituting z_n+1 and z_n in (7.1) by (7.1.1), multiplying out the right-hand side and solving the result for ζ_n+1, we get

(7.1.2)  ζ_n+1 = a²ζ_n³ + 3ab ζ_n^{2 + (3 b + r) ζ_n + (b3 + br - b + s) / a .}

Comparison of (7.0) and (7.1.2) followed by careful algebra gives us

(7.1.3)  a = √α ,  b = β/(3√α),  r = γ - β²/(3α)  and  s = √αδ + β(9α - 9αγ + 2β²)/(27α√α),

where √α is any of the two square roots. The first two identities of (7.1.3) show that (7.0) and (7.1) are indeed conjugate to each other. One of the advantages in the reduced formula (7.1) is that finding its critical points and hence plotting its "Mandelbrot set" is easier. The computation illustrates that a similar reduction—or dropping the "second term"—is possible for quartic equations but it requires considerably greater patience.

(7.1.4)  ζ_n+1 = ζ_n³ + q ζ_n² + (q²/3) ζ_n.

Comparing (7.1.4) and (7.0), we get  α = 1,  β = q,  γ = q²/3  and  δ = 0.  Hence, (7.1.3) implies r = 0 and s = q(9 - q²)/27, and the required conjugate equation is,

(7.1.5)  z_n+1 = z_n³ + p = z_n³ + q(9 - q²)/27,

using letter p in place of s.  Note that (7.1.5) is a multibrot equation we saw in

Consider now as a more typical example of (7.1), the cubic equation

Recall that an

The origin (0. 0) of the complex plane is at the tip of the Spearhead, which coincides with the upper left corner of the closeup image shown below. We call the area "Spearhead Bay." Like the Mandelbrot set, the Giant contains infinitely many circular atoms that satisfy the numerical pattern of the periodicity diagram. These circular atoms include the largest and the second largest blue atoms shown in Spearhead Bay, whose periods happened to be

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

The boundary of the Giant near the circular atom of

The next image, "Cheetah," is a local image of the Speared Giant painted on a

Figure 7.3(A).  "Cheetah"

The image shown below is again a local image of the Speared Giant and is painted on a

Figure 7.4(A).  "Mini Mandelbrot Set with Roses"

We have noted that ℳ₁ contains "irregularities" not seen in the Mandelbrot set from the earlier sections such as the jagged edges of the spearhead. Let

Figure 7.5.
"Atomic Fusion" of the Mandelbrot Sets ℳ₁ and ℳ₂ of the Two Critical Points.

So, what would be the Mandelbrot set of the dynamical system (7.2) with two critical points?

We first recall that the famed Mandelbrot set ℳ was born as a visualization of the

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

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

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

Figure 7.6 shows a simplified version of the "Atomic Fusion" in

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

We now consider the Fatou-Julia Theorem with

It is now straightforward to prove that the Fatou-Julia Theorem applied on (7.2) can be written in terms of ℳ₁ and ℳ₂ as follows:

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

(1) and (2) imply:

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

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

Note that if ℳ₁ = ℳ₂ then (1), (2) and (3) coincide with the earlier (1) and (2). Note also that we have just shown that the "Atomic Fusion" comprising ℳ₁∪ℳ₂ and ℳ₁∩ℳ₂ illustrates the Fatou-Julia theorem applied on (7.2) with exactly two critical points. It is natural, therefore, that we call ℳ₁∪ℳ₂ together with ℳ₁∩ℳ₂ the

In terms of pictures, the fusion diagrams of

Figure 7.7(A).  "Twin Dragons"

p = (0.2011575, 0.00002) in ℳ₁∩ℳ₂

p = (0.21828, -0.00230) in [ℳ1∪ℳ2]c

p = (0.2176, 0.0128) in ℳ1∇ℳ2

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: The third image of

The next two Jula sets are born from one of the molecules that can be also seen near the bottom of Figure 7.6. Note that it intersects with both ℳ₁∩ℳ₂ (the red zone) and ℳ₁∇ℳ₂ (the yellow zone).

The connected "Roses" of Figure 7.8(A) is the Julia set of the parameter

The disconnected "Roses" of Figure 7.8(B) shows the Julia set of the parameter

Figure 7.8(B).  "Disconnected Roses"

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

Figure 7.8(C).  "Cantor Elephants"

p = (0.087, -1.1848)

p = (0.092, -1.1728)

Example 3: The Toddler seen near the bottom edge of

Figure 7.9(A).  "Lernaean Hydra with Thirteen Heads and Offsprings"

The image shown below is a Julia set born from the Toddler's cardioid atom of period

Figure 7.9(B). "Flying Lion"

Figure 7.10(A).  "Pearly Dragon"

Figure 7.11(B).  The Mandelbrot Set of the Dynamical System (7.3)

It is again easy to show that the

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

Thus, the second image of Figure 7.11(B) illustrates the historical significance of the Mandelbrot set tied with the Fatou-Julia theorem and the first image the modern role of the Mandelbrot set as a "map" on the complex plane to show where we can find a variety of the Julia sets given by (7.3).

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

    ℳ₁∇ℳ₂∇ℳ₃ =

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

Figure 7.12(A). "Cactus"

Figure 7.12(B). "Clover"

The Julia Set of p = (0, 0.111) in ℳ1∩ℳ2∩ℳ3

The Julia Set of p = (0, 0) in ℳ1∩ℳ2∩ℳ3

Figure 7.12(C).  "Line Dancing Seahorses"

The Julia Set of p = (-0.39985, 0.02025) belonging to ℳ₁∇ℳ₂∇ℳ₃

Figure 7.12(D).  "Dancing Seahorses"

A Local Image of the Julia Set of p = (-0.39985, 0.02025) belonging to ℳ₁∇ℳ₂∇ℳ₃

Figure 7.12(E).  "Unicorns"

The Julia Set of p = (-0.39674, 0.01012) belonging to [ℳ₁∪ℳ₂∪ℳ₃]^c

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

§ 8.  Newton Fractals

The idea of the

(8.1)   z_n+1 = z_n - g(z_n)/g'(z_n),

which generates, like in

As a benefit of using Newton's method, each orbit of (8.1) converges to a root of g quickly more often than not, and it allows us to plot most of the Newton fractals by the convergence scheme (with period index

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

    g(z)  =  z ⁿ -  1 ,

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

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

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

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."

    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  

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

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

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

Newton's method for finding all roots of g(z) requires us to choose (almost blindly) an intial value z₀ to (8.1) hoping it belongs to the basin of attraction to one of the roots, say r₁. Once r₁ is found, we use the aforementioned Horner's method (aka synthetic division) to divide g(z) by

z - r₁ and repeat the process on the "deflated" polynimial, starting again with a new initial value z₀ to find a second root r₂. The process is repeated until we find all roots, but it is likely to fail if any of the initial values hits the Julia set.

So, what should we do if we don't know the roots of the input polynomial? One way is to use Muller's Method instead of Newton's method; e.g., see

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.

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 12^th 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 8.9(A). "Butterflies"

Figure 8.9(B). "Spiders"

Figure 8.9(C). "Dragonflies"

Figure 8.10(A). "Newton Garden"

Figure 8.10(B). "Firefly Forest"

Newton Fractals on a z-Canvas with Different Colorings

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

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