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         Fractals:     more books (100)
  1. The Fractal Geometry of Nature by Benoit B. Mandelbrot, 1983
  2. Fractal Time: The Secret of 2012 and a New World Age by Gregg Braden, 2010-02-01
  3. Fractals, Googols, and Other Mathematical Tales by Theoni Pappas, 1993-02-16
  4. The Misbehavior of Markets: A Fractal View of Financial Turbulence by Benoit Mandelbrot, Richard L. Hudson, 2006-03-07
  5. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise by Manfred Schroeder, 2009-08-21
  6. Fractal Geometry: Mathematical Foundations and Applications by Kenneth Falconer, 2003-11-14
  7. Introducing Fractals: A Graphic Guide by Nigel Lesmoir-Gordon, 2005-10-15
  8. The Science of Fractal Images
  9. Fractals: The Patterns of Chaos: Discovering a New Aesthetic of Art, Science, and Nature (A Touchstone Book) by John Briggs, 1992-11-01
  10. Chaos and Fractals: New Frontiers of Science by Heinz-Otto Peitgen, Hartmut Jürgens, et all 2004-02-03
  11. The Beauty of Fractals: Images of Complex Dynamical Systems by Heinz-Otto Peitgen, 1986-08
  12. Introducing Fractal Geometry by Nigel Lesmoir-Gordon, 2002-01-26
  13. Fractals Everywhere by Michael F. Barnsley, 2000-04-18
  14. Lectures on Fractal Geometry and Dynamical Systems (Student Mathematical Library) by Yakov Pesin and Vaughn Climenhaga, 2009-10-21

1. Sprott's Fractal Gallery
Computer generated artwork, with thousands of downloadable images and a new fractal every day!
Sprott's Fractal Gallery
Awards Received
MIDI Fractal background music
courtesy of Forrest Fang
Fractal of the Day
Every day at a few minutes past midnight (local Wisconsin time), a new fractal is automatically posted by a variation of the program included with the book Strange Attractors: Creating Patterns in Chaos by Julien C. Sprott . The figure above is today's fractal. Click on it or on any of the cases below to see them at higher (640 x 480) resolution with a code that identifies them according to a scheme described in the book. Older Fractals of the Day are saved in an archive . If your browser supports Java, you might enjoy the applet that creates a new fractal image every five seconds or so. If you would like to place the Fractal of the Day on your Web page, you may do so provided you mention that it is from Sprott's Fractal Gallery and provide a link back to this page. If you want to make your own fractals, I recommend the Chaoscope freeware.

2. Cynthia Lanius' Lessons: A Fractals Lesson - Introduction
Jan 19, 2004 Mathematics lessons for elementary school, middle school fractals, fractals, fractals.
Cynthia Lanius
Pictured: A Famous Fractal - The Mandelbrot Set
A Fractals Unit for Elementary and Middle School Students
That Adults are Free to Enjoy
Table of Contents
Introduction Why study fractals?
What's so hot about

fractals, anyway?
Making fractals
Sierpinski Triangle

Using Java

Math questions

Sierpinski Meets Pascal
Using Java
Fractal Properties

Fractional dimension
Formation by iteration For Teachers Teachers' Notes Teacher-to-Teacher Send mail Fractals on the Web The Math Forum Other Math Lessons by Cynthia Lanius Awards This Site has received
What are Fractals?
They're everywhere, those bright, weird, beautiful shapes called fractals. But what are they, really? Fractals are geometric figures, just like rectangles, circles and squares, but fractals have special properties that those figures do not have. There's lots of information on the Web about fractals, but most of it is either just pretty pictures or very high-level mathematics. So this fractals site is for kids, to help them understand what the weird pictures are all about - that it's math - and that it's fun! Teachers: Every lesson has a print version for classroom use.

3. Fractals - Fractal Recursions
Jock Cooper s gallery of still images and animations.
Order: DVD Amazon or Createspace
Book Blurb Welcome to the ever-evolving exhibition of Jock Cooper's Fractal Art. A noteworthy recent addition to the site is the Zoomable Fractal Gallery; an interactive playground that allows one to zoom through six levels in each quadrant of a given fractal image. As always, the newer images are in the highest gallery numbers, and new images are added several every month or so.
Traditional Gallery
1029 images
Mechanical Gallery
281 images
Miscellaneous Gallery
246 images
Limited Palette Gallery
168 images
Fractal Animations Gallery
30 animations
Zoomables Gallery
6 Zoomable Fractal images
Please send comments to me at Jock.Cooper At Gmail Dot Com
For commercial use please contact Bruststrokes At Comcast Dot Net
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 Unported License You can now view gallery images on your Hires or Lores mobile device. Most recent images are shown first.

4. Fractal Geometry
I find the ideas in the fractals, both as a body of knowledge and as a metaphor, an incredibly important way of looking at the world. Vice President and Nobel Laureate Al

5. Fractal -- From Wolfram MathWorld
Weisstein, E. W. Books about fractals. http// Yamaguti, M.; Hata, M.; and Kigami, J. Mathematics of fractals.
Applied Mathematics

Calculus and Analysis

Discrete Mathematics
... Interactive Demonstrations
Fractal A fractal is an object or quantity that displays self-similarity , in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension . The prototypical example for a fractal is the length of a coastline measured with different length rulers . The shorter the ruler , the longer the length measured, a paradox known as the coastline paradox Illustrated above are the fractals known as the Gosper island Koch snowflake box fractal Barnsley's fern , and Mandelbrot set SEE ALSO: Attractor Backtracking Barnsley's Fern Box Fractal ... Zaslavskii Map REFERENCES: Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, 1993. Bogomolny, A. "Fractal Curves and Dimension."

6. Suzanne Alejandre - Fractal Links
Basic fractal information for the math teacher or interested user, with Internet resources to expand your fractal knowledge.
Suzanne's Math Lessons
Suzanne Alejandre
Magic Squares Multicultural Math Fair Polyhedra ... Tessellations
What is a fractal?
From the Fractal FAQ:
    "A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale."
    "There are many mathematical structures that are fractals; e.g. Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor. Fractals also describe many real-world objects, such as clouds, mountains, turbulence, and coastlines, that do not correspond to simple geometric shapes." The Fractal FAQ was created and edited by Ken Shirriff through September 26, 1994. The current editor is Ermel Stepp.
Alan Beck 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." Beck further explains that when we look very closely at patterns that are Euclidean, the shapes look more and more like straight lines, but that when you look at a fractal up close you see more and more details.

7. NOVA | Hunting The Hidden Dimension | PBS
The Most Famous Fractal What exactly is the Mandelbrot set? Find out in this excerpt from the book fractals The Patterns of Chaos.
document.write(unescape("%3Cscript src='" + (document.location.protocol == "https:" ? "https://sb" : "http://b") + "' %3E%3C/script%3E")); A Radical Mind
Benoit Mandelbrot is a true maverick, as his interview reveals. The Most Famous Fractal
What exactly is the Mandelbrot set? Find out in this excerpt from the book Fractals: The Patterns of Chaos Design a Fractal
Create and save your own wildly colorful fractals using our generator. A Sense of Scale
Explore the infinite detail of a Mandelbrot set as you zoom to a magnification of 250,000,000x. Watch a Preview
TV Program Description

Teacher's Guide

e-mail newsletter
... Program Participants This Web site was produced for PBS Online by WGBH.
Funding for NOVA is provided by David H. Koch, the Howard Hughes Medical Institute, the Corporation for Public Broadcasting, and public television viewers.
Major funding for "Hunting the Hidden Dimension" is provided by the Alfred P. Sloan Foundation. Support provided by

8. Cool Math - Fractals - Fractal Art, Lessons And Generators
Coolmath's Fractal Gallery Very cool art that's made from math!
Your browser does not support the IFRAME tag.
Fractal Gallery
Very cool art that's made from math!
Sorry, your browser doesn't support Java. Your browser does not support the IFRAME tag. Your browser does not support the IFRAME tag. Your browser does not support the IFRAME tag. Back in the 1960's, a French mathematician named Benoit Mandelbrot started thinking about simple pretty simple: How long is the coast of Britain? Sure, it seems easy enough... But, think about how you'd really measure it... Would you fly over the coast in an airplane and measure how far you flew? Or would you take a ruler (or meter stick) and get down on your hands and knees and crawl around the edge? It would be different! In 1975, he made up the word "fractal" because he thought these figures would look fractured or broken up. It wasn't until he had a computer that he could SEE a picture of what he'd been thinking about! To learn about what a fractal is

9. Fractals
Easier A fractal is a shape, often drawn by a computer, that repeats itself in a pattern. The design shapes usually reoccur
The Topic:
Fractals Easier - A fractal is a shape, often drawn by a computer, that repeats itself in a pattern. The design shapes usually reoccur in different sizes. Harder - Fractals are endlessly repeating patterns that vary according to a set formula, a mixture of art and geometry. Fractals are any pattern that reveals greater complexity as it is enlarged A real-life example of fractals is ice crystals freezing on a glass window. You can see countless variations of the same pattern emerge in the crystals over time.
Exploring Fractals (Grades 9-12) by M.A. Connors This website explores fractal dimensions of strictly self-similar fractals, from Cantor Dust to the Fractal Skewed Web. Includes a teacher information section. Related Websites: 2) Fract-ED (Grades 8 and Above)

10. The Fractals On Myspace Music - Free Streaming MP3s, Pictures & Music Downloads
Myspace Music profile for The fractals. Download The fractals Rock / Rockabilly / Surf music singles, watch music videos, listen to free streaming mp3s, read The fractals's blog.

11. The Beauty Of Fractal Magic Transformation
Four galleries of fractals, black and white art by Isidoros Printezis. Isidoros Printezis - Afterzed FRACTALS GALLERY 1
Imagination GALLERY 2

Fractal Photoshop Works GALLERY 5 NEW
Latest Fractal Works PHOTOCUBE
Fractal Java Photcube BIO EMAIL
Site created 11/1/1999
Isidoros Printezis - AfterZed

12. Fractal Geometry - Crystalinks
Fractal Geometry A Fractal is generally a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reducedsize copy of
Fractal Geometry
A Fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractious meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. A fractal often has the following features:
  • It has a fine structure at arbitrarily small scales.
  • It is too irregular to be easily described in traditional Euclidean geometric language.
  • It is self-similar (at least approximately or stochastically).
  • It has a simple and recursive definition.
  • It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals - for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.

13. Fractal Zoom Animations By Eric Bigas
Collection of short fractal animations in MPG format.
20 high-quality fractal zoom movies. Zooms are 1 to 4 minutes each. Free downloads (MPG, XviD HD, H.264 HD).
Click on the small pictures

14. FractalNet Home Page
The Fractal Images Company 268 Arlington Road Kensington, CA 94707 . Voice 510528-0258 Fax 510-528-0243 e-mail
Best experienced with
Click here to start.
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ChiaraMail Version 2.0 is now available for download. Check out the coolest and most effective spam filter on the planet, at
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Version 3.0 of WebReader(tm) includes a number of significant new features and benefits; for an overview of functional enhancements in Version 3.0, check out new features and enhancements . Beginning December 7, 1999, you can download WebReader 3.0 at no cost. Before downloading WebReader, make sure you meet the minimal system requirements
FractalNet Commercial Links
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Kensington, CA 94707 Voice: 510-528-0258
Fax: 510-528-0243

15. Concepts: Fractals
fractals are entities that look the same under magnification, they are selfsimilar. More specifically, a geometric fractal is formed of parts, which, when magnified, are the
Concepts in Complex Systems Yaneer Bar-Yam Fractals Fractals are entities that look the same under magnification, they are "self-similar." More specifically, a geometric fractal is formed of parts, which, when magnified, are the same as the original shape. Fractals can also involve randomness, so that the similarity of parts to the whole can be of a statistical or average property. An example of a geometric fractal is the Koch curve (an approximate version is shown below) which can be made using a simple algorithm: Take away the middle third of a line segment and insert two segments that are the same length as the one that was removed. Place them so they would make an equilateral triangle with the removed segment. Repeat this procedure with each of the line segments that now make up the curve. Fractals get their name because they can be seen to have fractional dimension. Length, area and volume measure the size of one, two and three dimensional shapes respectively. However, a Koch curve can be shown to be infinite in length and to cover no area. It has a dimension between one and two (strictly its dimension is d=ln(4)/ln(3)~1.2619). This means that if we use a ruler to measure the length of the curve, the length we find depends on how long the ruler is. The shorter the ruler, the longer is the length we measure. This is because a longer ruler cannot "get inside" smaller bumps. An example of a fractal-like shape found in nature is a coastline. How long is the coastline? Just as with the Koch curve, the length of the coastline depends on how long the ruler is used to measure it because there are inlets and peninsulas of many different sizes. A picture of the coastline of Cape Cod is shown below.

16. Exploring Fractals
Jan 17, 2009 It is is based on a curriculum, entitled Exploring Fractal Dimension, developed by Mary Ann Connors and Anna Rose Haralampus at an NSF
Exploring Fractals
Mary Ann Connors
publications and photos
Department of Mathematics and Statistics
University of Massachusetts Amherst
This World Wide Web project commenced in July 1994. It is is based on a curriculum, entitled "Exploring Fractal Dimension," developed by Mary Ann Connors and Anna Rose Haralampus at an NSF funded Institute for High School Mathematics Teachers at Georgetown University July - August, 1991. Its revision entitled "Exploring Fractals: From Cantor Dust to The Fractal Skewed Web" has been edited by Mary Ann Connors 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, and 2009.
Exploring Fractal Dimensions of Strictly Self-Similar Fractals:
Cantor Dust to the Fractal Skewed Web
What is a fractal? A fractal is a geometric shape which
  • is self-similar and has fractional (fractal) dimension.
  • An introduction
    So, what are fractals?
    What does it mean for a shape to be self-similar
    Strictly Self-similar shapes
    What is dimension?
    How do we assign dimension to an object intuitively?
    What is the dimension of a geometric object?
    Mathematical Interpretation
    Dimension of a Fractal
    Non-integer and integer dimensions
    Questions and Answers
    Generating Fractals
    Strictly Self-similar Fractals
    Teacher Information
    Sources Sources for Exploring Fractals Some Interesting Links for Further Exploration If you wish to use any of the text or images in Exploring Fractals please contact its author Mary Ann Connors at the following address.

    17. Fractals_by_Nous
    Several galleries generated using Ultra Fractal, Tierazon, Grafzvizion, and Fractint.
    A Member of the Infinite Fractal Loop Fractint Gallerys I II III IV ... Technical and Acknowledgments UltraFractal UltraFractal Gallery 1 UltraFractal Gallery 2 Tierazon Tierazon Gallery 1 GrafZViZion GrafZViZion Gallery 1

    18. Linda Bucklin S Fractals
    Galleries of Ultra Fractal, Frax Flames, and collage.
    NEW! Gallery One Gallery Four Frax Flame Gallery Click on a picture to enter a gallery Three NEW Galleries added 6/24/04 NEW! GalleryTwo Gallery Five Frax Flame 2 NEW! Gallery Three Apophysis One Fractal Collage Visit ‘Linda’s Universe’ - My Main Page, with links to all of my sites, including ‘Cats! Cats! Cats! and ‘Linda’s World’. Visit Linda’s World to see my paintings, drawings and digital images. There are also two galleries that exhibit mental illness - related art. My photos, paintings, drawings and digital images of cats. Visit my Renderosity Gallery to see a wide selection of my artwork. Lots of links to some gorgeous fractal galleries! Contact Linda Bucklin Linda's Fractals Gallery One Frax Flame1 ... Frax Flame 2

    19. Fractals
    fractals 1 I am doing fractals next because that is how we usually see nonlinear systems. Fractal is a name coined by a fellow by the name of Mandelbrot in the 1960's.
    I am doing fractals next because that is how we usually see non-linear systems. Fractal is a name coined by a fellow by the name of Mandelbrot in the 1960's. Fractals are the branch of geometry dealing with 'broken curves'. Remember that non-linear problems have graphs that have strange and unusual lines or ... 'broken curves'. There are two approaches to fractals: the natural lines (Has anyone ever seen a truly straight line or perfect curve in any natural object?) and the theoretical mathematical constructions. I will start with the lines occurring in nature since everyone here should be familiar with them and finally work our way into the mathematical curves.
    An object in nature repeats itself. Consider a mountain. A peak on the shoulder of the mountain looks very much the same as the mountain. In fact a rock can be mistaken for a mountain if there is nothing next to it to show its size. A branch of a tree is a miniature of the tree. Some leaves are miniatures of branches. Each tributary of a river is a repetition of the whole river at a smaller scale. This repetition of structure with a change of scale is a characteristic of fractals.
    If you seem to have any trouble understanding this idea of repetition of structure across scale, rent an old Godzilla movie from the video store. The effect is crude, but the repetition of structure between a branch and a tree, or hill and a mountain can easily be seen. You can also see another feature of natural curves. Although the imaging of structure across scale is very close to being a direct duplication it isn't exact. Waves in a bathtub are not quite the waves in an ocean, although they are very close.

    20. Wonders Of Math
    Build your own fractals and learn about the math behind the images. Mandelbrot and Julia Set Explorer Zoom into fractals. Fractal Galleries

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