 Image Coding Using Fractals  Signal and Image Processing Group, University of Bath
Graphs show how image compression using fractals compares to other methods. Also includes comparison pictures and a list of articles on fractal compression, some of which may be downloaded in gzipped PostScript format.
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 Images and Mathematics (Mathematics Archives)  University of Tennessee, Knoxville (UTK)
A large collection of images based on or related to mathematical principles, many purely mathematical abstractions, such as fractals; others of natural objects whose shape is explained by equations and mathematical models (for example, a drum vibrating).
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 Images Created by Man & Chaos  Alan Beck
Some personal observations on fractal art and artists. "...the fractal artist does not create but explores, does not manufacture but discovers. Instead of being repelled by computers, such adventurers revel in them, as European mariners of old rejoiced
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 The Institute for Figuring  Margaret and Christine Wertheim, CoDirectors
"If mathematics is a language of pattern, its structures may be seen as the verses of a formal, yet fantastical poetics. Across the globe people have delighted in the harmonies of this language and the patterns discovered therein. These are the songs
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 Integer Sequences and Arrays  Clark Kimberling; Dept. of Mathematics, Univ. of Evansville, Evansville, IN
Certain seemingly simple sequences of integers baffle the best mathematicians. Other sequences, less baffling, exhibit patterns  or absence of patterns  whose appeal shines beyond whatever applications these sequence might find outside mathematics.
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 Interactive Java Applets  Fergus Murray
Dragoric (The animated fractal toy); Toroidia (Adventures in toroidal geometry); Trochor (Fun with trochoids). See also Fergus Murray's Mathematical Graphics Page, and his mathematical animations.
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 Interactive Learning in Calculus and Differential Equations with Applications  Mathematics Dept., Indiana Univ. of Pennsylvania
Mathematica notebooks: Finding Areas with the GaussGreen Formula, Newton's Method and Fractals, Vibrating Drumheads, The Drag Force on a Sphere, and a collection of 26 notebooks from the course Advanced Mathematics for Applications. In order to disseminate
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 The Interactive Mandelbrot Set
A Java applet that allows you to zoom in on portions of the Mandelbrot set.
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 Interactive Mathematics Miscellany and Puzzles  Alexander Bogomolny
"The peculiar beauty of Mathematics lies in deduction, in the dependency of one fact upon another. The less expected a dependency is, the simpler the facts on which the deduction is based  the more beautiful is the result." This awardwinning site offers
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 Interactive Mathematics Online  ThinkQuest 1996
Algebra (Postulates  Things you can do with equations and inequalities, Function Basics and Simple Graphing, Kinds of functions (periodic, odd, even, etc.), Function Squashing); Geometry (an introduction, applications, glossary, sample problems); Trigonometry,
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 Interactive Science Activities on the Web (ISAW)  Richard L. Bowman; Bridgewater College, Bridgewater, VA
Java activities demonstrating fractals and Kepler's laws of planetary orbits. Includes a lab activity using the planetary orbit applet.
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 The InterFACE: Internet Fractal and Chaos Education  ThinkQuest 1997
Lessons, games, activities, demonstrations, and software, to inform users of the science and mathematics behind fractal images and chaos theory. The lessons, identified by subject and level of difficulty and including quizzes, are designed to promote
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 Introduction to Chaos Theory  Andrew Ho
Designed for those with little background in mathematics. Topics covered include bifurcation diagrams; the Lorenz, Rossler, and Henon strange attractors; KolmogorovArnoldMoser orbits; and fractal geometry, including the Sierpinski Triangle, Von Koch
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 Introduction to Fractal Geometry  Michael Frame, Benoit Mandelbrot, and Nial Neger
A collection of pages to support a first course in fractal geometry for students without especially strong mathematical preparation. See examples of fractals in the arts, humanities, and social sciences. Learn about selfsimilarity, plane transformations,
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 An Introduction to Fractals  Jeff Haley
An introduction and an exploration of the von Koch curve. "Without a detailed analysis, fractals are colorful patterns that yield great detail when very tiny sections are observed. There are many different fractals, yet different ones often yield similar
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 Introduction to Lacunarity  Erik Rauch
Lacunarity is a counterpart to the fractal dimension, and describes the degree of gappiness of a fractal. It is strongly related to the size distribution
of the holes on the fractal and to its deviation from translational invariance; roughly speaking,
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 Introduction to the Mandelbrot Set  David Dewey
Images of the Mandelbrot set, with information on complex numbers and on graphing Mandelbrot sets.
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 Investigating Patterns: Number Patterns Fun with Curves & Topology  Jill Britton
Annotated list of links within various number pattern, curve, and topology topics. Linked items feature activities for students. Topics include: prime numbers/magic squares, clock or modular arithmetic, the golden ratio, Fibonacci numbers, binary numbers/Pascal's
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 Iterated Function Systems  David Arnold; Mathematics Dept., College of the Redwoods, CA
What will the final image look like after you have plotted the infinite collection of points X1, X2, X3, ... generated by the chaos game? What if you plotted the infinite set of points generated by Barnsley's rules? This page provides a program that you
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 Iteration and Recursion  Harold Brochmann
New classes of mathematical objects have been uncovered in the last couple of decades, with names like Strange Attractors and Fractals. What they share is that they can only be revealed through computerbased mathematical operations called iterations
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 Jackson Pollock's Fractals  Ivars Peterson (MathTrek)
How Pollock's approach was "more systematic than the chaotic results might suggest... Physicist Richard P. Taylor of the University of New South Wales in Sydney, Australia, who is also trained as an artist, has taken a mathematical look at Pollock's splatter
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 Java Boutique Educational Applets  internet.com Corp.
Investigate or download various Java applets, including 3D viewers, calculators, applets to graph functions or data, and simulations, such as the bell curve and fractals. The source code for many applets may also be downloaded; a few applets are shareware
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 Java Examples  Richard Koch; Dept. of Mathematics, Univ. of Oregon
Fractal and FourD, two Java programs with source code and standalone versions available. "Fractal" can display various selfsimilar objects. "Four Dimensions" displays regular solids in three and fourdimensional space. The solids can be rotated and
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 Java Fractals  James Henstridge
Java applets to draw different kinds of fractals: Iterated Function Systems (Sierpinski Triangle, fern, etc.), complex number fractals, and orbit fractals. Includes explanations of the algorithms used, and code for each applet.
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 JAVA Gallery of Interactive OnLine Geometry  The Geometry Center
Applets illustrating geometry concepts. WebEQ, Hyperbolic Triangles, Lorenz Simulation, OneDimensional Iteration, Tetris, Leap Fractal, Fractals from Iterated Function Systems, Kali: Wallpaper Patterns. Created by The Geometry Center.
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 Java Simulations of Cellular Automata Models  Ed Young and L. Charles Biehl
The Forest Fires simulation applet models the spread of fire as a function of the probability that a burning tree will set its neighbor afire, and can include factors of wind and obstacles. The Percolating Oil Spill simulation applet models the percolation
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 John Allen Paulos
John Allen Paulos is a mathematician and author of the books Mathematics and Humor, I Think, Therefore I Laugh, Innumeracy  Mathematical Illiteracy and Its
Consequences, Beyond Numeracy  Ruminations of a Numbers Man, A Mathematician Reads the Newspaper,and
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 John and Betty's Journey into Complex Numbers  Matt Bower
A children's bookontheWeb from Australia, designed to introduce complex numbers in story form in a way that is intuitive and enjoyable for students. John, Betty, and their dog Trevor solve a series of problems designed to introduce integers, fractions,
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 Journal of Computational Analysis and Applications  George Anastassiou, Ed. Raymond A. Clapsadle
A peerreviewed journal that publishes research articles from all subareas of Computational Mathematical Analysis, including: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential
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 k12.ed.math  Math Forum
A moderated discussion forum for K12 mathematics issues such as curriculum, assessment, classroom practices, suggested assignments, help on particular problems, grading practices, requests for advice, history of mathematics, etc. Messages are reviewed
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 K12 Teaching Materials (Mathematics Archives)  University of Tennessee, Knoxville (UTK)
An extensive list of Internet sites with significant collections of materials that can be used in the teaching of mathematics at the K12 level. Organized into categories: lesson plans, schools, software and other K12 Internet math sites.
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 Ken Brakke's home page
Home page of the Susquehanna University mathematics professor who coauthored The Physics of Microdroplets and developed Surface Evolver, a free interactive program for modeling liquid surfaces shaped by various forces and constraints. Brakke also offers
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 Ken Monks's Software  Ken Monks; University of Scranton
Links to some programs written by the author, as well as by others, that have been useful in his mathematics courses. Ken Monks's software includes RevSolid, a Maple program for making animated plots of solids of revolution; STEENROD, a Maple program
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 Ken Shirriff
A researcher at Sun Microsystems. Annotated bibliographies on fractals and cryptography, and related papers, which may be downloaded in PostScript format. Graphics of Voronoi fractals (dividing the plane into regions closest to a given point) and fractals
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 Keplerian Fractals  Hop David
Just as a Star of David can be transformed into a Koch Snowflake, Kepler's solids can be made into fractals by placing inside the pyramids smaller versions of the original solid. See also Escher's rendering of a dodecahedron and his Double Planet.
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 Kids' Tech University  Virginia Tech
Kids' Tech University (KTU) sparks kids' interest in science, technology, engineering, and mathematics (STEM) through a handson activity, virtual lab, and interactive session, held once a month from January to April, in a lecture hall on the campus of
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 Launch into Math  C. Halter
A project created to further student understanding about the Internet, its resources, and the NASA Space Shuttle program. SpaceMath activities are provided for geometry (measure area and distance in space photographs); algebra (use the math that keeps
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 Álbumes Ilustrados Sobre Ciencia  Verónica Navarro
Álbumes ilustrados de geometría y arte abstracto como lo de Joan Miró, Paul Klee, y Wassily Kandinsky: La rebelión de las formas hace un itinerario por la geometría, la física, y el arte en un intento de reflexionar
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 Lessons and Curriculum  MSTE, University of Illinois at UrbanaChampaign (UIUC)
Browse lessons predominantly for middle and high school students, from the Office for Mathematics, Science, and Technology Education (MSTE) and from other sources.
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 Lesson Study: A PCMI Experience  Tony Anderson, Jennifer Kumi Burkett, Gail Burrill, Joyce Frost, Jerry Gribble, Jill Ryerson, and Celeste Williams
An article written by the Japanese Lesson Study group from the 2003 Park City Mathematics Institute (PCMI) about their experience developing the lesson, "Studying Dimensionality Through a StairStep Fractal." With an explanation of the Japanese lesson
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 Levitated  Jared Tarbell
"Visual poetry and science fun narrated in an object oriented graphic environment." See, in particular, the generators in sketches and applications; and "open source," a library of fractals, cellular automata, spacefilling algorithms, "autofixing words,"
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 Living Graphs  Living Graphs Inc.
Makers of ChaosLab, an interactive math package for visualizing and understanding fractal geometry and chaos theory. This PC software title covers bifurcation diagrams, the chaos game, coin tosses, diffusion limited aggregation, Levy flight, Pascal's
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 lofi paranoid ambient fractal music  Dave Sag
Selfsimilarity is the essence of fractal geometry: this is selfsimilar music, where each part has the same structure as the whole. Download an example in Real Audio, Shockwave, or aiff format. Another version of this page, What do I mean by 'fractal
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 LowComplexity Art  Jürgen Schmidhuber
Lowcomplexity art is the computerage equivalent of minimal art (Leonardo, MIT press, 1997). A lowcomplexity artwork can be specified by a short computer algorithm. It should 'look right', its algorithmic complexity should be small, and a typical observer
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 Magic Carpet Graphics
Computergenerated artwork by a Londonbased team of artists, available for sale as highquality photographic prints at A6 A4 sizes.
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 The Magic Sierpinski Triangle: (Serendip: Complex Systems)  Bryn Mawr College
Order dependent on randomness: the Chaos Game. Construct Sierpinski's Triangle using Java interactively on the Web and focusing on one particularly interesting question: what is the right way of choosing the vertices? In the construction, the elegant
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 Making Order Out of Chaos  Andrew Davenport, Shane Kraynak, and Brian Timko  ThinkQuest 97
An introduction to chaos and fractals: history, theory, and a gallery of fractal images. Includes a review of complex numbers and a Java applet to investigate sensitive dependence on initial conditions.
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 Makin' Magic Programs, Fractals, Freeware, and Graphics  Dave Makin
Fractal freeware and galleries of examples. The fractal program MMFrac2.4 includes solid 3D hypercomplex and quaternion fractals and realtime 3D landscapes. An older version of the page can be found at http://skyscraper.fortunecity.com/terabyte/966/.
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 Mandelbrot and Julia Set Explorer  David Joyce; Dept. of Mathematics & Computer Science, Clark University
Modify Mandelbrot and Julia set images by adjusting the (x,y) sets of the pictures; then explore what you have created. Background information and detailed instructions for how to modify the image is provided, in addition to links to other images, reference
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 Mandelbrot Exhibition  J. P. Bowen; Virtual Museum of Computing
Links to to material on Mandelbrot sets and related fractals. Historical background; introduction to the Julia and Mandelbrot sets; interactive explorers and generators of images; movies and galleries; and an Image Archive.
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