 Foundations of Infinitesimal Calculus  Henle and Keisler
Free download, PDF format.
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 Four Coloring Maps  Mario Stefanutti
Personal notes and ideas from a computer software engineer in pursuit of a "very easy pencil and paper proof of the four color problem." Posts, which date back to January, 2011, have included "Four color theorem: slow motion maps" and "Counting maps."
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 Four Colors Suffice: How the Map Problem Was Solved  Robin Wilson; Princeton University Press
"On October 23, 1852, Professor Augustus De Morgan wrote a letter to a colleague, unaware that he was launching one of the most famous mathematical conundrums in historyone that would confound thousands of puzzlers for more than a century. This is the
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 The Four Color Theorem  Robertson, Sanders, Seymour, Thomas; Georgia Tech
A brief summary of a new proof of the Four Color Theorem, with a fourcoloring algorithm found by Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas, illustrated using a map of the United States. Contents: History; Why a new proof?; Outline
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 The Four Colour Theorem  MacTutor Math History Archives
Linked essay describing work on the theorem from its posing in 1852 through its solution in 1976, with two other web sites and 9 references (books/articles).
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 Four Construction Problems  Cut the Knot!, Alexander Bogomolny
Construct a triangle, given the three points in the plane that are the outer vertices of equilateral triangles constructed outward on the sides of the desired triangle. Construct a triangle, given the three points in the plane that are the centers of
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 Fourier Analysis  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to Fourier analysis, which studies approximations and decompositions of functions using trigonometric
polynomials. Of incalculable value in many applications of analysis, this field has grown to include
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 Fourier Synthesis  Manfred Thole
A periodic signal can be described by a Fourier decomposition as a Fourier series, i.e. as a sum of sinusoidal and cosinusoidal oscillations. By reversing this procedure a periodic signal can be generated by superimposing sinusoidal and cosinusoidal waves.
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 The Four Learning Styles in the DVC Survey  Suzanne Miller, Diablo Valley College
Descriptions of and strategies for four learning styles: visual/verbal, visual/nonverbal, tactile/kinesthetic, and auditory/verbal style. With tips for studying mathematics and a link to the Learning Styles Survey online.
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 Four Numerical Triangles in Nexus  Len Smiley
Three wellknown combinatorial number triangles are tightly related to each other as coefficients in transforms of a seminal sequence of exponential generating functions (for n^{n+m}). A fourth triangle is a part of the picture in much the same way that
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 Four Travelers Problem  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
Four roads on a plane, each a straight line, are oriented so that no two are parallel and no three pass through the same point. A traveler walks along each road at a constant speed. The speeds, however, may not be the same. Traveler 1 meets Travelers
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 FoV
Puzzles and pointers on math and programming from a future secondary school math teacher in England. Blog posts, which date to March, 2010, have included "Make 42," "Find a, b and c," "Programming Mahjong Scoring," "Broken Geometry," and "Using Mersenne
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 Fractal Alhambra  Philip Northover
Galleries of fractal art, in an array of sizes and colors. Fractal prints for sale, and MP3s of fractal music.
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 FractalArt.com  Fractal Art Designs, LLC
Purchase original color digital prints of ultrahigh resolution fractals by Andrew Ryan or Bryan Berman from this fractal art gallery.
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 Fractal Art, etc.  Linda
Fractal fantasies produced using Flarium or KPT Fractal Explorer2.0 and KPT filters, postprocessed in Photoshop.
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 Fractal Art Gallery  Fractal Image Wallpaper  Victoria B Brago
Extensive collection of fractal images which can be downloaded as desktop
wallpaper, technical explanation of fractal geometry, animated fractal
zooms.
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 Fractal Clouds  Robert F. Cahalan
Information on the fantastic variety of cloud forms and structures, and their implications for climate. Albedo (abstract of a short technical paper for downloading); authors of papers on fractal clouds; cloud types and associated geometry, processes,
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 Fractal Dimension Calculator (FDC) User Manual  Paul Bourke
FDC takes a black and white image in the form of a Macintosh PICT file or clipboard contents and uses the boxcounting algorithm to estimate the fractal dimension of the object represented in the image. The "object" is assumed to be the black part of
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 frActAles  Philippe de Courcy
450 fractal images, sorted into categories. Site is available in English and French.
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 Fractal Explorer  Yuri Alkin
A Java applet that allows the user to explore infinite fractal surfaces in real time. Features unique algorithms and multiple fractal types and color schemes. Images are not based on Mandelbrot or Julia sets.
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 Fractal FAQ  Shirriff, Stepp
Basic frequently asked questions, suggestions for fractal resources and sites, software, archives, tutorials, from the newsgroups sci.fractals, sci.answers, news.answers. Sample questions: What is a fractal? What are some examples of fractals? What is
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 Fractal Gallery  Karen McCormack
Images, 19982000. Quaternions, Symmetric Icons, Flame Fractals, Secant City, Ikenaga's Garden, Fractals with Russell, Fractal Eggs, Filatura.
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 Fractal Gallery  Phil Bodor
A gallery of fractal thumbnail sketches that can be expanded for full viewing. Contents: About Fractals; fractals created by Bodor; fractals downloaded from Usenet; and an overview of WAN (wide area networks) Technologies.
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 Fractal Gallery  Rico Wack
An online fractal exhibition with images computed by a variety of software packages, raw and postprocessed, 2D and 3D, updated weekly. In English and French.
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 Fractal Geometry  Introduction  Harold Brochmann
Links to a series of articles exploring: The notion of dimension; The notion of selfsimilarity; The length of a coastline; The Sierpinsky Carpet; The Mandelbrot Set  the icon of chaos; and Applications of Fractal Geometry.
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 Fractal Image Encoding  Yuval Fisher; Institute for Nonlinear Science, University of California at San Diego
A page with links to a variety of information and resources on fractal image encoding and related topics: Bibliographies and Other References; Books  including Fisher's own Fractal Image Compression: Theory and Application to Digital Images; Conferences
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 Fractal Images  Sharon Webb
Fractal images made for the most part using a software application called Flarium24. Galleries contain about 15 images each and should be viewed in hicolor or truecolor settings. Tilable images that can be downloaded for wallpaper are also available.
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 Fractal Images  Softsource CAD
Images of the Mandelbrot set, a fractal generated from a simple mathematical formula, produced by DRAWING Librarian Professional (DP).
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 Fractal Landscapes  Adam Brown
Results from the author's fractal landscape generation rendering computer program. Galleries, with an introduction to the ideas behind fractal landscape generation, and links to other fractal sites on the Internet.
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 Fractal Metaphysics Home Page  Christopher Sunami
A clearinghouse of (serious) materials on the web focused on the relationship between fractal geometry, chaos theory, and metaphysics. Essays, related sites, galleries, and recommended reading.
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 Fractal Mountains  Matthew Caryl
Fractal systems repeat the same type of features on smaller and smaller scales when you look closer and closer. Mountains are a natural example of this  a mountain range is rugged  a mountain is rugged  a valley on the mountain is rugged  and the
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 Fractal Music  G. Forrest Cook
Audio excerpts from a compilation of fractal music put together by the author.
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 Fractal Page  Frode Gill
Tutorials on Mandelbrot and Julia Sets and Quaternions; JavaScript for generating Mandelbrot sets; downloadable Quaternion generator (v1.2) with or without Borland.DLL files, and its Borland C++/OWL source (program requires Windows 3.1 or above, and the
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 Fractal Past, Fractal Future  Ivars Peterson  Science News Online
Fractals have invaded the popular imagination. Calendars, computer screens, and books feature vivid, phantasmagorical images of weirdly branched, wildly swirling structures. Some fractal history, and an essay about the geometric complexity of natural
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 Fractals and Circle Inversions  Michael Frame, Barbara Bemis, Colleen Clancy, Tatiana Cogevina; Union College, Schenectady, NY
Inversion across a circle was introduced by Appolonius of Perga, whose definition of inversion was synthetic, as was all geometry at that time. This article gives an analytic characterization, with an explanation of inversion limit sets, and three animations
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 Fractals, Art, Pattern, Symmetry...  Clifford A. Pickover
Melding art, science, mathematics and other seemingly disparate areas of human endeavor. Links to detailed information on Pickover's books (of particular math interest are: Chaos and Fractals: A Computer Graphical Journey; The Loom of God; Fractal Horizons
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 Fractals as Art  Melissa Binde, Swarthmore College
"...fractals are an art form. The mathematics provides the form, but it is up to the individual creator (artist) to choose the boundaries, colors, and manner of presentation of each fractal." Three categories: "standard" fractal fare, reproducible with
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 Fractals by Nous  Stephen A. Karl
A collection of fractal art generated using UltraFractal, Tierazon, Grafzvizion and Fractint 19.6.
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 The Fractals Calendar  Daryl Hepting, University of Regina, Saskatchewan, Canada
In each year since 1991, a calendar of computergenerated images has been published, based on the fractal geometry of Benoit B. Mandelbrot. Previously, the calendar was first done as a custom edition and then as a trade edition in the following year.
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 Fractals @ chaotic nspace network  Jon Camp, Benjamin Martin
A short introduction to fractals, fractal galleries, a link to a small fractal software collection, and links to other fractal galleries.
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 Fractal Science Kit  Ross Hilbert
A Windows program for exploring fractals including: Mandelbrot, Julia, Convergent, Newton, Orbit Traps, Sierpinski Triangle, IFS, Strange Attractors, RepN Tiles, Symmetric Icons, Apollonian Gasket, Circle Inversion, Schottky Group, Kleinian Group, LSystem
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 Fractals from Ojai  Art Baker
A fractal gallery and homepage in one.
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 Fractals in Science: Simulations  Center for Polymer Studies, Boston University
Handson activities, laboratory experiments, and interactive visualization programs that encourage students to explore how fundamentally random microscopic events can give rise to fractal macroscopic patterns. An image gallery and fractal related software
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 Fractals  JeanPierre Louvet
Information and more than 250 images created by the author using Fractint, GrafZViZion, etc.
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 Fractals (Mathematics Archives)  University of Tennessee, Knoxville (UTK)
An extensive list of links to sites with information about fractals, classified by level and type of resource.
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 Fractal Software  Thinks.com Ltd.
Fractal software, including many freeware and shareware programs, for PCs and Macs, for learning about fractals and for creating and exploring fractal images.
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 Fractals on the Web  Chaffey High School
An extensive searchable collection of sites dealing with fractals, in the following categories: programs, galleries, fractal music, fractal compression, fractal theory, and fractal newsgroups.
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 Fractals (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material on fractals.
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 Fractals with Mathematica  Robert M. Dickau; Math Forum
Some simple fractal images and Mathematica code for Cantor dust; Julia sets; Mandelbrot (and related) sets; Randelbrot set; Menger sponge; Sierpinski tetrahedron; Newton's method; variation on Newton's method; Sierpinski carpet; 2D and 3D Lsystems (also
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 Fractalus Entrance  Damien Jones et al.
Galleries: a site designed to display the best in fractal art.
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