101. Math Forum: Suzanne Alejandre - MandelBrot Activity Alan Beck in What Is a Fractal? And who is this guy mandelbrot? writes. Basically, a fractal is any pattern that reveals greater complexity as it is http://mathforum.org/alejandre/applet.mandlebrot.html

Studying Mandelbrot Fractals Fractals NOTE: Use of Internet Explorer 5.0 is recommended. What is a fractal? Alan Beck in What Is a Fractal? And who is this guy Mandelbrot? writes: "Basically, a fractal is any pattern that reveals greater complexity as it is enlarged. Thus, fractals graphically portray the notion of 'worlds within worlds' which has obsessed Western culture from its tenth-century beginnings." 1. Click on the button Col+ or Col- to change the colors of the fractal image. 2. Now that you have the colors set to your liking, it is time to investigate the fractal itself! 3. Using the mouse, draw a small rectangle on the fractal image. Click on Go and watch as the smaller section of the image is redrawn to fill the fractal screen. 4. What do you notice? How do the images compare? Click on the Out button to revisit the first image and the In button to return to the enlarged image. 5. Continue going into the fractal image. What do you observe? 6. It has been stated that fractals have finite areas but infinite perimeters . Do you agree? Why?/Why not?

102. Realtime Mandelbrot Fractal Generation CGI Fractal is a term coined by Benoit mandelbrot to refer to a structure bearing statistically similar details over a wide range of scales. http://mandelbrot.collettivamente.com/mandel.cgi?cmap=volcano.map

103. The Fractal Microscope A popular representation of fractal geometry lies within the mandelbrot set, named after its creator Benoit B. mandelbrot who coined the name fractal in http://archive.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html

The Fractal Microscope A Distributed Computing Approach to Mathematics in Education The Fractal Microscope is an interactive tool designed by the Education Group at the National Center for Supercomputing Applications (NCSA) for exploring the Mandelbrot set and other fractal patterns. By combining supercomputing and networks with the simple interface of a Macintosh or X-Windows workstation, students and teachers from all grade levels can engage in discovery-based exploration. The program is designed to run in conjunction with NCSA imaging tools such as DataScope and Collage. With this program students can enjoy the art of mathematics as they master the science of mathematics . This focus can help one address a wide variety of topics in the K-12 curriculum including scientific notation, coordinate systems and graphing, number systems, convergence, divergence, and self-similarity. Why Fractals? Many people are immediately drawn to the bizarrely beautiful images known as fractals . Extending beyond the typical perception of mathematics as a body of sterile formulas, fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers. With fractal geometry we can visually model much of what we witness in nature, the most recognized being coastlines and mountains. Fractals are used to model soil erosion and to analyze seismic patterns as well. But beyond potential applications for describing complex natural patterns, with their visual beauty fractals can help alter students' beliefs that mathematics is dry and inaccessible and may help to motivate mathematical discovery in the classroom.

104. Introduction To The Mandelbrot Set A simple explanation of how the mandelbrot set fractal is calculated and graphed. Includes a progression of images to demonstrate zooming. http://www.olympus.net/personal/dewey/mandelbrot.html

This page has been moved to http://www.ddewey.net/mandelbrot/ . The version here will no longer be maintained or updated. Please update your links. Introduction to the Mandelbrot Set A guide for people with little math experience. By David Dewey According to Web-Counter you are visitor number since November 02, 1996. The Mandelbrot set, named after Benoit Mandelbrot, is a fractal . Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to the large-scale characteristics. Although the Mandelbrot set is self-similar at magnified scales, the small scale details are not identical to the whole. In fact, the Mandelbrot set is infinitely complex. Yet the process of generating it is based on an extremely simple equation involving complex numbers. Understanding complex numbers The Mandelbrot set is a mathematical set, a collection of numbers. These numbers are different than the real numbers that you use in everyday life. They are complex numbers . Complex numbers have a real part plus an imaginary part . The real part is an ordinary number, for example, -2. The imaginary part is a real number times a special number called

105. Index Of All Pictures Of The Journey Translate this page Index of All Pictures of This Journey. Select a picture or return to the title page. All pictures available on www with their original size are marked with http://i30www.ira.uka.de/~ukrueger/fractals/Index.html

Index of All Pictures of This Journey Select a picture or return to the title page . All pictures available on www with their original size are marked with (*). To get one of these just click on the name and then select a www source. All pictures of this index are available now on CD-ROM (frac0001) a (frac0003) ad (frac0004) b ... ukrueger@ira.uka.de

106. Julia And Mandelbrot Sets The boundary between the mandelbrot set and its complement is often called the mandelbrot The mandelbrot set is the black shape in the picture. http://aleph0.clarku.edu/~djoyce/julia/julia.html

Julia and Mandelbrot Sets David E. Joyce August, 1994. Last updated May, 2003. Function Iteration and Julia Sets Gaston Julia studied the iteration of polynomials and rational functions in the early twentieth century. If f x ) is a function, various behaviors can arise when f is iterated. Let's take, for example, the function f x x We will iterate this function when initially applied to an initial value of x , say x a . Let a denote the first iterate f a ), let a denote the second iterate f a ), which equals f f a )), and so forth. Then we'll consider the infinite sequence of iterates a a f a a f a a f a It may happen that these values stay small or perhaps they don't, depending on the initial value a . For instance, if we iterate our sample function f x x a = 1.0, we'll get the following sequence of iterates (easily computed with a handheld calculator) a a f a f a f a f a f a It helps to see what's going on graphically. In the diagram above, the graph y x y x is drawn in green. Then the values a a a a , and a are shown grapically, starting with our first value of a , namely, 1.0. To find

107. Mandelbrot And Julia Sets DR Hofstadter, Metamagical Themas, Basic Books, Inc., 1985, Chapter 16. B. mandelbrot, The Fractal Geometry of Nature, WH Freeman and Co., NY, 1977. http://www.cut-the-knot.org/blue/julia.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Mandelbrot and Julia sets (An index may have a content of its own.) Given a function f(x) and a starting value x one can construct a new value x =f(x ). With some persistence, the next value x is obtained by another application of f: x =f(x )=f(f(x )). This is an iterative process that, generally speaking, generates a sequence x , x , ... x k , ..., where x k is the k th iterate obtained by applying the function f to x k times. The sequence is known as an orbit of its starting point x . Gaston Julia (1893-1978) and Pierre Fatou (1878-1929) made a fundamental contribution to the study of iterative processes. Their contribution (Ref [3]), although regarded as a masterpiece, was largely ignored by the mathematical community until a revival in the late 1970s spawned by the discovery of fractals by Benoit Mandelbrot. For a given function f, behavior of an orbit very much depends on the selection of the starting point x . Following is a rough classification of possible behaviors: k =x p so that the sequence repeats itself Chaos k The set of points with chaotic orbits is called the Julia set for a given function f. Until quite recently the study of iterations and Julia sets has been in a prolonged limbo. B.Mandelbrot has the following to say on the development of the theory

108. Fractal Images These images are of the mandelbrot set, a fractal generated from a simple mathematical formula. They were produced by DRAWING Librarian Professional (DP). http://www.softsource.com/softsource/fractal.html

Fractal Images These images are of the Mandelbrot set, a fractal generated from a simple mathematical formula. They were produced by DRAWING Librarian Professional (DP) . If you press (Alt-M) in DP, it will generate the main view of the Mandelbrot set. From that point, you can zoom in and examine all the detail of the set. The numbers next to each image indicate the coordinates of the location of the picture - the lower left corner in parentheses followed by the length of a side. If you pass these three numbers to DRAWING Librarian Professional's MAND script command, it will generate the same view. For example, MAND -2 -1.25 2.5 will generate the main set. Here's another site with Mandelbrot pictures . If you have questions regarding fractal images in general, you can take a look at the Fractal FAQ Related pages... Info on DRAWING Librarian Return to Softsource Home Page scotts@softsource.com

109. Richard Dickerson's Lab The familiar 2nd order fractal image of Benoit mandelbrot is produced by iterating the expression z = z2 + c. Iteration can be expanded to include higher http://www.doe-mbi.ucla.edu/People/Dickerson/

People Dickerson Gallery Current Projects Structures Solved Publications ... Richard Dickerson 's Lab We use X-ray crystal structure analysis to study the way that base sequence affects the local structure and deformability of the DNA double helix, in a manner that can be recognized by antitumor drugs and by DNA-binding proteins such as repressors, other control proteins, restriction enzymes, and the reverse transcriptase (RT) of HIV. We have found that the sequence G-G-C-C has a persistent bend under quite varied crystal condition, and believe it to be an important player when DNA wraps around a repressor or the histone core of nucleosomes. Other sequences of DNA 10-mers are being sequenced and solved to look for additional structure elements. What we learn is being applied to the design of minor groove binding antitumor drugs based on netropsin, distamycin, and anthramycin. The Hin recombinase acts as a control element in Salmonella by inverting a 1 kb section of DNA, assisted by the Fis enhancer-binding protein. Fis is unique among DNA-binding proteins in that it binds specifically to more than 20 different sequences, of no obvious consensus. We have solved the structure of the Fis protein alone, and of the DNA-binding domain of Hin when complexed to its DNA site. We want to examine the structures of the entire Hin protein bound to DNA, and also the structures of Fis complexed with several of its binding sites, in order to understand how this complex recognition process takes place. We have found that some drugs that bind to the minor groove of DNA inhibit strand synthesis by HIV RT, hence blocking incorporation of the AIDS provirus into the human genome. This inhibition is being investigated both structurally and kinetically, and we are looking for drugs whose inhibition of RT falls in a pharmaceutically useful range.

110. Fractal EXtreme Fractal Theory The mandelbrot set has been proven to have a fractal dimension of two. It also means that the mandelbrot set is as complicated as a fractal can get. http://www.cygnus-software.com/theory/theory.htm

Sample Code Imaginary Numbers Complex Numbers The Mandelbrot Set ... Connectedness Introduction What is the Mandelbrot set? A mathematician might say it was the locus of points, C, for which the series Zn+1 = Zn * Zn + C, Z0 = (0,0) is bounded by a circle of radius two, centered on the origin. But most of us aren't mathematicians. It's a pretty picture. It's a mathematical wonder that we can appreciate, and to some extent understand, even if we don't understand the first paragraph. It's just one example of an amazing new science with applications as far ranging as weather forecasting, population biology, and computerized plant creation. It's a floor wax and a dessert topping! It's all of these and more. Sample Code To demonstrate just how simple it is to generate pictures of the Mandelbrot set, we have included a small program written in "C". If you have a C compiler, try it out. It is a complete working program. For those of you who aren't programmers, we have excerpted the code which actually does all of the calculations. Here it is, all eleven lines of it: That's all it takes to do a rudimentary exploration of the Mandelbrot set. Slowly.

111. Mandelbrot Fractal - Java Zoomable mandelbrot Fractal. Java (tm) version More information. Some screenshots from this applet About the mandelbrot fractal http://www.thorsen.priv.no/services/mandelbrot/

Zoomable Mandelbrot Fractal Java (tm) version '; Since your browser does not have Java-support you are not able to see the fractal. Here is how it may look like: How to use: Press the left mousebutton ro "Z" where you want to zoom in, press the right mouse button key or "z" when you want to zoom back out again. You could enter coordinates yourself. More information Some screenshots from this applet About the Mandelbrot fractal Java is a registered Trademark of Sun Back to Main Page Back to Steffen Thorsen's homepage Guestbook iv.no

112. Mu-Ency -- The Encyclopedia Of The Mandelbrot Set At MROB the mandelbrot Set, one of the most wellknown fractal images in the world. The mandelbrot Set is one of my hobbies, and I have collected a large http://www.mrob.com/pub/muency.html

Mu-Ency - The Encyclopedia of the Mandelbrot Set A second-order embedded Julia set This is a picture from the Mandelbrot Set, one of the most well-known fractal images in the world. (Click it for a larger version). The Mandelbrot Set is one of my hobbies, and I have collected a large amount of information about it. To organize that information I have created Mu-Ency, a large collection of text files linked to each other. Here are some entries from Mu-Ency: Mandelbrot Set: The mathematical definition. History: How the Mandelbrot Set was discovered, how it became popular, etc. Exploring: The many things you can expect to find when you explore on your own. Area: I have been involved in finding the area of the Mandelbrot Set. Here are the latest results. Algorithms: How to compute the Mandelbrot Set and how to draw it. R2 Naming System: I have also developed a rather precise (and complex) naming system for features of the Mandelbrot Set. Mu-Ency presents many examples of this naming system. Some entries with pictures of parts of the Mandelbrot Set are: Cusp Paramecia Delta Hausdorff Dimension You can also look up specific terms in the index Image coordinates: Center: -1.769 110 375 463 767 385 + 0.009 020 388 228 023 440 i Width (and height): 0.000 000 000 000 000 160 Algorithm: distance estimator Iterations: 10000

113. FRACTINT The Mandelbrot Set Fractint documentation on the classic mandelbrot fractal type. The mandelbrot Set. Nice Image Basic mandelbrot type. (type=mandel) http://spanky.triumf.ca/www/fractint/mandelbrot_type.html

The Mandelbrot Set Basic Mandelbrot type (type=mandel) This set is the classic: the only one implemented in many plotting programs, and the source of most of the printed fractal images published in recent years. Like most of the other types in Fractint, it is simply a graph: the x (horizontal) and y (vertical) coordinate axes represent ranges of two independent quantities, with various colors used to symbolize levels of a third quantity which depends on the first two. So far, so good: basic analytic geometry. Now things get a bit hairier. The x axis is ordinary, vanilla real numbers. The y axis is an imaginary number, i.e. a real number times i, where i is the square root of -1. Every point on the plane in this case, your PC's display screen represents a complex number of the form: x-coordinate + i * y-coordinate OK, now pick any complex number any point on the complex plane and call it C, a constant. Pick another, this time one which can vary, and call it Z. Starting with Z=0 (i.e., at the origin, where the real and imaginary axes cross), calculate the value of the expression: Z^2 + C [X] Bailout Test Some features of interest: The upper and lower halves of the first image are symmetric (a fact that Fractint makes use of here and in some other fractal types to speed plotting). But notice that the same general features lobed discs, spirals, starbursts tend to repeat themselves (although never exactly) at smaller and smaller scales, so that it can be impossible to judge by eye the scale of a given image.

114. Edge: A THEORY OF ROUGHNESS: A Talk With Benoit Mandelbrot mandelbrot is best known as the founder of fractal geometry which impacts mathematics, diverse sciences, and arts, and is best appreciated as being the http://www.edge.org/3rd_culture/mandelbrot04/mandelbrot04_index.html

Home About Edge Features Edge Editions ... Edge Search A THEORY OF ROUGHNESS A Talk with Benoit Mandelbrot Introduction During the 1980s Benoit Mandelbrot accepted my invitation to give a talk before The Reality Club. The evening was the toughest ticket in the 10 year history of live Reality Club events during that decade: it seemed like every artist in New York had heard about it and wanted to attend. It was an exciting, magical evening. I've stayed in touch with Mandelbrot and shared an occasional meal with him every few years, always interested in what he has to say. Recently, we got together prior to his 80th birthday. Mandelbrot is best known as the founder of fractal geometry which impacts mathematics, diverse sciences, and arts, and is best appreciated as being the first broad attempt to investigate quantitatively the ubiquitous notion of roughness. And he continues to push the envelope with his theory of roughness. "There is a joke that your hammer will always find nails to hit," he says. "I find that perfectly acceptable. The hammer I crafted is the first effective tool for all kinds of roughness and nobody will deny that there is at least some roughness everywhere."

115. Fractal Geometry - A Gallery Of Monsters Benoit mandelbrot, The Fractal Geometry of Nature, 1977, Ch 1 Magnet fractal Set The formula for the mandelbrot set is only one of a vast number of http://www.calresco.org/fractal.htm

Fractal Geometry - A Gallery of Monsters by Chris Lucas "Why is geometry often described as 'cold' and 'dry' ? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, a tree..." Benoit Mandelbrot, The Fractal Geometry of Nature, 1977, Ch 1 "A man may love a paradox without either losing his wit or his honesty." Ralph Waldo Emerson, Uncollected Prose, 1841 Introduction What do the following have in common ? A galaxy, a lung, a coastline, a tree A figure that has more than two and less than three dimensions A figure with an infinite perimeter and zero area A solid that contains only two dimensions A figure which changes its shape the closer you look at it A figure that looks the same at any scale ? All of these are related to the same thing, the magic of fractals ! Paradoxical Coastlines Impossible ? In this strange world nothing is quite what it seems... Take the coastline of Britain for example. How long is it ? Nobody knows. Of course they do you say ! Ah, but they know roughly the area of the country so they must, by Euclid, know the minimum boundary surely, an equivalent circle ? Yes, but the actual boundary is almost infinite ! To see this, go in your mind to the seaside with a metre rule and measure a section of rock. You will skip over a few crevices will you not ? Now use a kilometre ruler instead - this skips over a lot more resulting in a different, lower reading. Take now a 1 cm measurement, this will go around most irregularities and give a much bigger total. So, the length is variable isn't it ? But not infinite surely ?

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