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Geometric Fractal

Our first example of a geometric fractal is a shape known as the Koch snowflake , named after the Chapter 12: Swedish mathematician Helge von Koch (1870–1954). Fractal Geometry Like other geometric fractals, the Koch snowflake is constructed by means of a recursive process , a process in which the same procedure is applied repeatedly in an infinite feedback loop–the output at 12.1 The Koch Snowflake one step becomes the input at the next step. and Self-Similarity

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THE KOCH SNOWFLAKE THE KOCH SNOWFLAKE

Start. Step 1. Start with a shaded equilateral triangle. To the middle third of each of the sides of We will refer to this starting triangle as the seed add an equilateral triangle with the seed of the Koch snowflake. The size sides of length 1/3. The result is the 12- of the seed triangle is irrelevant, so for sided “snowflake”. simplicity we will assume that the sides are of length 1.

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THE KOCH SNOWFLAKE THE KOCH SNOWFLAKE Step 2. For ease of reference, we will call the procedure of adding an equilateral To the middle third of each of the 12 triangle to the middle third of each side sides of the “snowflake” in Step 1, add of the figure procedure KS . an equilateral triangle with sides of length one-third the length of that side. The result is a “snowflake” with This will make the rest of the construction 12 × 4 = 48 sides, a lot easier to describe. each of length (1/3) 2 = 1/9. Notice that procedure KS makes each side of the figure “crinkle” into four new sides and that each new side has length one-third the previous side.

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THE KOCH SNOWFLAKE THE KOCH SNOWFLAKE Step 3. Step 4. Apply the procedure KS to the Apply the procedure KS to the “snowflake” in Step 2. This gives the “snowflake” in Step 3. This gives the more elaborate “snowflake” with 48 × following “snowflake.” 4 = 192 sides, each of length (1/3) 3 = 1/27.

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THE KOCH SNOWFLAKE Koch Snowflake At every step of this recursive process procedure KS Steps 5, 6, etc. produces a new “snowflake, ” but after a while it ’s At each step apply the procedure KS to the hard to tell that there are any changes. “snowflake ” obtained in the previous Step. Soon enough, the images become visually stable : To the naked eye there is no difference between one snowflake and the next.

For all practical purposes we are seeing the ultimate destination of this trip: the Koch snowflake itself.

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Koch Snowflake THE KOCH SNOWFLAKE (REPLACEMENT RULE) One advantage of recursive processes is that they allow for very simple and efficient definitions, even when the objects being defined are quite Start complicated. Start with a shaded equilateral triangle. (This is the seed.) The Koch snowflake, for example, is a fairly complicated shape, but we can define it in two lines using a form of shorthand we will call a replacement Replacement rule: rule –a rule that specifies how to substitute one piece Replace each boundary line segment with a for another. “crinkled” version .

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Koch Curve Self-similarity

If we only consider the boundary of the Koch This seemingly remarkable characteristic of the snowflake and forget about the interior, we get an Koch curve of looking the same at different infinitely jagged curve known as the Koch curve , or scales is called self-similarity . sometimes the snowflake curve .

We ’ll just randomly pick a small part of this section and magnify it. The surprise (or not!) is that we see nothing new–the small detail looks just like the rough detail.

Magnifying further is not going to be much help. The figure shows a detail of the Koch curve after magnifying it by a factor of almost 100.

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Self-similarity: Koch Curve Self-similarity: Cauliflower

As the name suggests, self-similarity implies that the There are many objects in nature that exhibit some object is similar to a part of itself, which in turn is similar form of self-similarity, even if in nature self-similarity to an even smaller part, and so on. can never be infinite or exact.

In the case of the Koch curve the self-similarity has A good example of natural self-similarity can be three important properties: found in, of all places, a head of cauliflower. 1. It is infinite (the self-similarity takes place over infinitely many levels of magnification) The florets look very much like the head of cauliflower 2. it is universal (the same self-similarity occurs but are not exact clones of it. In these cases we along every part of the Koch curve) describe the self-similarity as approximate (as 3. it is exact (we see the exact same image at opposed to exact) self-similarity. every level of magnification).

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Cauliflower - Koch Snowflake Perimeter and Area: Koch Snowflake

In fact, one could informally say that the Koch One of the most surprising facts about the Koch snowflake is a two-dimensional mathematical snowflake is that it has a relatively small area but an blueprint for the structure of cauliflower. infinite perimeter and an infinitely long boundary–a notion that seems to defy common sense.

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Perimeter of the Koch Snowflake Area of the Koch Snowflake

At each step we replace a side by four sides that are The area of the Koch snowflake is less than the area 1/3 as long. Thus, at any given step the perimeter P is of the circle that circumscribes the seed triangle and 4/3 times the perimeter at the preceding step. This thus, relatively small. implies that the perimeters keep growing with each step, and growing very fast indeed.

Indeed, we can be much more specific: The area of the Koch snowflake is exactly 8/5 (or 1.6) times the After infinitely many steps the perimeter is infinite. area of the seed triangle.

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Nature and Fractals Nature and Fractals

Having a very large boundary packed within a small The vascular system of veins and arteries in the area (or volume) is an important characteristic of human body is a perfect example of the tradeoff that many self-similar shapes in nature (long boundaries nature makes between length and volume: Whereas improve functionality, whereas small volumes keep veins, arteries, and capillaries take up only a small energy costs down). fraction of the body ’s volume, their reach, by necessity, is enormous.

Laid end to end, the veins, arteries, and capillaries of a single human being would extend more than 40,000 miles.

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PROPERTIES OF THE KOCH SNOWFLAKE Chapter 12: ■ It has exact and universal self-similarity. Fractal Geometry

■ It has an infinite perimeter.

■ Its area is 1.6 times the area of the seed triangle. 12.2 The Sierpinski Gasket and the Chaos Game

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Snowflake to Gasket THE SIERPINSKI GASKET With the insight gained by our study of the Koch snowflake, we will now look at another well-known geometric fractal called the Sierpinski gasket , named Start. after the Polish mathematician Waclaw Sierpinski Start with a shaded triangle ABC . (1882–1969). We will call this triangle the seed triangle.

Just like with the Koch snowflake, the construction of the Sierpinski gasket starts with a solid triangle, but this time, instead of repeatedly adding smaller and smaller versions of the original triangle, we will remove them according to the following procedure:

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THE SIERPINSKI GASKET THE SIERPINSKI GASKET Step 1. Step 2. To each of the three shaded triangles Remove the triangle connecting the apply procedure SG ( “removing the midpoints of the sides of the seed triangle. middle ” of a triangle). The result is the This gives the shape shown consisting of “gasket ” consisting of 3 2 = 9 triangles, three shaded triangles, each at one-fourth the scale each a half-scale of the seed triangle, plus three version of the seed small holes of the same size and a hole where and one larger hole in the the middle triangle middle. used to be. We will call this procedure SG.

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THE SIERPINSKI GASKET THE SIERPINSKI GASKET Step 3. To each of the nine shaded triangles apply procedure SG. The result is the “gasket ” Steps 4, 5, etc. consisting of 3 3 = 27 triangles, each at one- Apply procedure SG to each shaded eighth the scale of the triangle in the “gasket ” obtained in the original triangle, nine small previous step. holes of the same size, three medium-sized holes, and one large hole in the middle.

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Sierpinski Gasket Sierpinski Gasket After a few more steps the figure becomes visually stable–the naked eye cannot tell the difference between the gasket obtained at one step and the gasket obtained at the next step.

At this point we have a good rendering of the Sierpinski gasket itself. In your mind ’s eye you can think of the figure on the next slide as a picture of the Sierpinski gasket (in reality, it is the gasket obtained at Step 7 of the construction process).

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Sierpinski Gasket THE SIERPINSKI GASKET We might think that the Sierpinski gasket is made of a (REPLACEMENT RULE) huge number of tiny triangles, but this is just an optical illusion–there are no solid triangles in the Sierpinski Start gasket, just specks of the original triangle surrounded Start with a shaded seed triangle. by a sea of white triangular holes.

If we were to magnify any one of those small specks, Replacement rule: we would see more of the same–specks surrounded by Whenever you see a white triangles.

replace it with a . As with the Koch curve, the self-similarity of the Sierpinski gasket is (1) infinite, (2) universal, and (3) exact.

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Sierpinski Gasket Perimeter and Area: Sierpinski Gasket

Magnification of the Sierpinski gasket. As a geometric object existing in the plane, the Sierpinski gasket should have an area, but it turns out that its area is infinitesimally small, smaller than any positive quantity.

Paradoxical as it may first seem, the mathematical formulation of this fact is that the Sierpinski gasket has zero area.

At the same time, the boundary of the “gaskets ” obtained at each step of the construction grows without bound, which implies that the Sierpinski gasket has an infinitely long boundary.

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Application of Sierpinski Gasket

PROPERTIES OF There is a surprising cutting-edge application of the THE SIERPINSKI Sierpinski gasket to one of the most vexing problems of modern life: the inconsistent reception on your cell GASKET phone. ■ It has exact and universal self-similarity. ■ It has infinitely long boundary. A really efficient cell phone antenna must have a ■ It has zero area. very large boundary (more boundary means stronger reception), a very small area (less area means more energy efficiency), and exact self-similarity (self- similarity means that the antenna works equally well at every frequency of the radio spectrum).

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Application of Sierpinski Gasket The Chaos Game

What better choice than a design based on a Now we will introduce a recursive construction rule Sierpinski gasket? that involves the element of chance.

Fractal antennas based on this concept are now The construction is known as the chaos game and is used not only in cell phones but also in wireless modems and GPS receivers. attributed to Michael Barnsley, a mathematician at the Australian National University.

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The Chaos Game THE CHAOS GAME We start with an arbitrary triangle with vertices A, B, and C and an honest die. Before we start we Start. assign two of the six possible outcomes of rolling Roll the die and mark the chosen vertex. the die to each of the vertices of the triangle. For Say we roll a 5. This puts us at vertex C. example, if we roll a 1 or a 2, then A is the chosen vertex; if we roll a 3 or a 4, then B is the chosen vertex; and if we roll a 5 or a 6, then C is the chosen vertex. We are now ready to play the game.

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THE CHAOS GAME THE CHAOS GAME Step 2. Step 1. Roll the die again, and move to the point Roll the die again. Say we roll a 2, so the halfway between the last position M 1 and the new chosen vertex is A. We now move to new chosen vertex A . Say we roll a 3–the the point M 1 halfway between the previous move then is to M halfway between M position C and the winning vertex A . Mark 2 1 and B. the new Mark the new position M1. position M2.

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THE CHAOS GAME Chaos Game after 5000 rolls

Steps 3, 4, etc. What happens after you roll the die many times? Continue rolling the die, each time moving to a point halfway between the The figure shows the pattern of points after 50 rolls of last position and the chosen vertex and the die–just a bunch of scattered dots–then after 500, marking that point. and 5000 rolls.

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Chaos Game into Sierpinski Gasket Sierpinski Gasket The last figure is unmistakable: a Sierpinski gasket! After a few more steps the figure becomes visually stable– The longer we play the chaos game, the closer we the naked eye cannot get to a Sierpinski gasket. tell the difference between the gasket After 100,000 rolls of the die, it would be impossible to obtained at one tell the difference between the two. step and the gasket obtained at the next step.

At this point we have a good rendering of the Sierpinski gasket itself.

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Chaos Game into Sierpinski Gasket Chaos Game on the Web

This is a truly surprising turn of events. Clearly, the best way to see the chaos game in action is with a computer. The chaos game is ruled by the laws of chance; thus, we would expect that essentially a random pattern of There are many good Web sites for the chaos game points would be generated. and its variations, and a partial list of them is given in the references at the end of the chapter in the text Instead, we get an approximation to the Sierpinski book. gasket, and the longer we play the chaos game, the better the approximation gets.

An important implication of this is that it is possible to generate geometric fractals using simple rules based on the laws of chance.

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The Twisted Sierpinski Gasket

Our next construction is a variation of the original Chapter 12: Sierpinski gasket. For lack of a better name, we will call it the twisted Sierpinski gasket. The construction Fractal Geometry starts out exactly like the one for the regular Sierpinski gasket, with a seed triangle.

12.3 The Twisted Sierpinski Gasket

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The Twisted Sierpinski Gasket The Twisted Sierpinski Gasket

We cut out the middle triangle, whose vertices we will call M, N, and L. The next move (which we will call the One possible resulting shape is shown. “twist ”) is new. Each of the points M, N, and L is moved a small amount in a random direction–as if jolted by an earthquake–to new positions M´, N´, and L´.

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Procedure TSG THE TWISTED For convenience, we will use the term procedure TSG SIERPINSKI GASKET to describe the combination of the two moves ( “cut ” and then “twist ”). Start. Start with a shaded seed triangle. ■ Cut. Remove the “middle” of the triangle.

■ Twist. Translate each of the midpoints of the sides by a small random amount and in a random direction.

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THE TWISTED THE TWISTED SIERPINSKI GASKET SIERPINSKI GASKET Step 1. Step 2. Apply procedure TSG to the seed triangle. To each of the three shaded triangles in This gives the “twisted gasket” with three twisted apply procedure TSG . The result is the triangles and a (twisted) hole in the middle. “twisted gasket” consisting of nine twisted triangles and four holes of various sizes.

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THE TWISTED Twisted Sierpeinski Gasket - Step 7 SIERPINSKI GASKET This figure shows Step 7 of the construction of a twisted Sierpinski gasket. Even in this basic form it is remarkable how much the figure resembles a snow- Steps 3, 4, etc. covered mountain. Apply procedure TSG to each shaded triangle in the “twisted gasket” obtained in the previous step.

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Real versus Computer Generated THE TWISTED SIERPINSKI Top is real. GASKET (REPLACEMENT RULE) Bottom is CGI.

Add a few of the Start. standard tools of Start with a shaded seed triangle. computer graphics–color, lighting, and shading–and we Replacement rule. can get a very Whenever you see a shaded triangle, realistic-looking mountain. apply procedure TSG to it.

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Twisted Sierpinski Gasket Twisted Sierpinski Gasket

The twisted Sierpinski gasket has infinite and universal It is the approximate self-similarity that gives this self-similarity, but due to the randomness of the shape that natural look that the original Sierpinski twisting process the self-similarity is only approximate– gasket lacks, and it illustrates why randomness is an important element in creating artificial imitations of when we magnify any part of the gasket we see natural-looking shapes. similar but not identical images.

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The Mandelbrot Set

In this section we will introduce one of the most interesting and beautiful geometric fractals ever Chapter 12: created, an object called the Mandelbrot set after Fractal Geometry Benoit Mandelbrot, a mathematician at Yale University.

Some of the mathematics behind the Mandelbrot set goes a bit beyond the level of this book, so we will 12.4 The Mandelbrot Set describe the overall idea and try not to get bogged down in the details.

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The Mandelbrot Set The Mandelbrot Set

We will start this section with a brief visual tour. The In the wild imagination of some, the Mandelbrot set Mandelbrot set is the shape shown, a strange-looking looks like some sort of bug–an exotic extraterrestrial blob of black. flea. The “flea” is made up of a heart-shaped body (called a cardioid ), a head, and an antenna coming out of the middle of the head.

A careful look shows that the flea has many “smaller fleas that prey on it. ”

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The Mandelbrot Set The Mandelbrot Set

We can only begin to understand the full extent of When we magnify the view around the the infestation when we look at the finely detailed boundary even further, we can see that these close-up of the boundary of the Mandelbrot set. secondary fleas have fleas that “prey on The Mandelbrot set has some strange form of them .” At the same time, we can see that infinite and approximate self-similarity–at infinitely there is tremendous variation in the regions many levels of surrounding the individual fleas. magnification we see the same theme– The images we see are a similar but never peek into a psychedelic identical fleas surrounded by coral reef–a world of similar smaller fleas. strange “urchins” and “seahorse tails. ”

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The Mandelbrot Set The Mandelbrot Set

When we magnify the view around the Further magnification shows an even more boundary even further, we view a virtual sea of exotic and beautiful landscape of one of the “anemone” and “starfish. ” seahorse tails.

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The Mandelbrot Set The Mandelbrot Set

A further close-up of a section from the previous slide. An even further magnification of it is seen in revealing a tiny copy of the Mandelbrot set surrounded by a beautiful arrangement of swirls, spirals, and seahorse tails.

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The Mandelbrot Set Mandelbrot Set - Complex Numbers

Anywhere we choose to look at these pictures, How does this magnificent mix of beauty and we will find (if we magnify enough) copies of the complexity called the Mandelbrot set come about? original Mandelbrot set, always surrounded by Incredibly, the Mandelbrot set itself can be described an infinitely changing, but always stunning, mathematically by a recursive process involving simple background. computations with complex numbers .

The infinite, approximate self-similarity of the The basic building block for complex numbers is the number Mandelbrot set manages to blend infinite = − repetition and infinite variety, creating a i 1. landscape as consistently exotic and diverse as nature itself. Starting with i we can build all other complex numbers: 3 + 2 i, –0.125 + 0.75 i, 1 + 0 i = 1

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Complex Numbers in the Plane Mandelbrot Sequence For our purposes, the most important fact about The key concept in the construction of the complex numbers is that they have a geometric Mandelbrot set is that of a Mandelbrot sequence . interpretation: The complex number ( a + bi ) can be identified with the point ( a, b) in a Cartesian A Mandelbrot sequence is an infinite sequence of coordinate system. complex numbers that starts with an arbitrary complex number s we call the seed, and then This identification means that each successive term in the sequence is obtained recursively by adding the seed s to the square of every complex number can the previous term . be thought of as a point in the plane and that operations with complex numbers have geometric interpretations.

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MANDELBROT SEQUENCE The Mandelbrot Sequence (REPLACEMENT RULE) Each point in the Cartesian plane is a complex Start. number and thus the seed of some Mandelbrot Choose an arbitrary complex number s. sequence. We will call s the seed of the Mandelbrot sequence. Set the seed s to be the initial The pattern of growth of that Mandelbrot sequence term of the sequence ( s0 = s). determines whether the seed is inside the Mandelbrot set (black point) or outside (nonblack point). Recursive Rule: To find the next term in the sequence, In the case of nonblack points, the color assigned to square the preceding term and add the the point is also determined by the pattern of growth 2 of the corresponding Mandelbrot sequence. seed [ sN + 1 = ( sN) + s].

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Example: Escaping Mandelbrot Sequences Example: Escaping Mandelbrot Sequences

The figure shows the first few terms of the Mandelbrot The pattern of growth of this Mandelbrot sequence is sequence with seed s = 1. Since integers and decimals clear–the terms are getting larger and larger, and they are also complex numbers, they make perfectly are doing so very quickly. acceptable seeds. Geometrically, it means that the points in the Cartesian plane that represent the numbers in this sequence are getting farther and farther away from the origin.

For this reason, we call this sequence an escaping Mandelbrot sequence.

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Example: Escaping Mandelbrot Sequences Example: Escaping Mandelbrot Sequences

In general, when the points that represent the terms of While there is no specific rule that tells us what color a Mandelbrot sequence move farther and farther away should be assigned, the overall color palette is based from the origin, we will say that the Mandelbrot on how fast the sequence is escaping. sequence is escaping . The typical approach is to use “hot” colors such as reds, The basic rule that defines the Mandelbrot set is that yellows, and oranges for seeds that escape slowly and seeds of escaping Mandelbrot sequences are not in the “cool” colors such as blues and purples for seeds that Mandelbrot set and must be assigned some color other escape quickly. than black.

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Example: Escaping Mandelbrot Sequences Example: Periodic Mandelbrot Sequences

The seed for s = 1 example, escapes very The figure shows the first few terms of the Mandelbrot quickly, and the corresponding point in the sequence with seed s = –1. The pattern that emerges Cartesian plane is painted blues seed is here is also clear–the numbers in the sequence assigned a “cool ” color (blue). alternate between 0 and –1. In this case, we say that the Mandelbrot sequence is periodic.

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Example: Periodic Mandelbrot Sequences Example: Attached Mandelbrot Sequences

In general, a Mandelbrot sequence is said to be This figure shows the first few terms in the Mandelbrot periodic if at some point the numbers in the sequence sequence with seed s = –0.75. start repeating themselves in a cycle. When the Mandelbrot sequence is periodic, the seed is a point of the Mandelbrot set and thus is assigned the color black.

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Example: Attached Mandelbrot Sequences Example: Attached Mandelbrot Sequences In general, when the terms in a Mandelbrot Here the growth pattern is not obvious, and additional sequence get closer and closer to a fixed complex terms of the sequence are needed. number a, we say that a is an attractor for the sequence or, equivalently, that the sequence is attracted to a. Further computation (a calculator will definitely come in handy) shows that as we go farther and farther out in this sequence, the terms get closer and closer to the Just as with periodic value –0.5. In this case, we will say that the sequence is sequences, when a attracted to the value –0.5. Mandelbrot sequence is attracted, the seed s is in the Mandelbrot set and is colored black.

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Example: A Periodic Mandelbrot Example: A Periodic Mandelbrot Sequence with Complex Terms Sequence with Complex Terms In this example we will examine the growth of the Mandelbrot sequence with seed s = i. At this point we notice that s3 = s1, which implies s4 = s2, s5 = s1, and so on. This, of course, means that this Mandelbrot sequence is periodic, with its terms alternating between the complex numbers –1 + i (odd terms) and –i (even terms).

The key conclusion from the preceding computations is that the seed i is a black point inside the Mandelbrot set.

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Example: A Mandelbrot Sequence with Example: A Mandelbrot Sequence with Three Complex Attractors Three Complex Attractors

We will examine the growth of the Mandelbrot We can see that the terms in this Mandelbrot sequence with seed s = –0.125 + 0.75 i. sequence are complex numbers that essentially cycle around in sets of three and are approaching three different attractors. s0 = –0.125 + 0.75 i s1 = –0.671875 + 0.5625 i s = 0.0100098 – 0.00585938 i Since this Mandelbrot sequence is attracted, the 2 seed s = –0.125 + 0.75 i represents another point in s3 = –0.124934 + 0.749883 i the Mandelbrot set. s4 = –0.671716 + 0.562628 i s5 = 0.00965136 – 0.00585206 i s6 = –0.124941 + 0.749887 i

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Definition: The Mandelbrot Set Definition: The Mandelbrot Set

If the Mandelbrot sequence is periodic or There are a few technical details that we omitted, attracted , the seed is a point of the but essentially these are the key ideas behind the Mandelbrot set and assigned the color black; amazing pictures that we saw in the beginning of if the Mandelbrot sequence is escaping, the this section. seed is a point outside the Mandelbrot set and assigned a color that depends on the Because the Mandelbrot set provides a bounty of speed at which the sequence is escaping aesthetic returns for a relatively small mathematical (hot colors for slowly escaping sequences, investment, it has become one of the most popular cool colors for quickly escaping sequences). mathematical playthings of our time.

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