 Formulas for fairness  Ivars Peterson  Science News Online
Applying the math of cake cutting to conflict resolution: researchers turn to the mathematics of cake cutting and point allocation to develop improved methods for people to use to settle disputes.
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 Forum of Geometric Constructor  Yasuyuki Iijima, Aichi University of Education, Japan
A dynamic geometry environment for DOS and Windows used in Japan  see the Tsukuba Journal of Educational Study in Mathematics.
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 Foundations of Greek Geometry  Michael Tirabassi; Tufts University
A linked historical essay outlining the contributions of Thales, Pythagoras, and Plato to the study of geometry.
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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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 Four Corners, Four Faces (MatheMUSEments!)  Ivars Peterson (Math Muse for Kids)
To Arthur Silverman, a sculptor in New Orleans, tetrahedrons, or triangular pyramids, are very special. He's been creating sculptures based on the tetrahedron for more than 20 years. You might see examples on display in plazas and office buildings in
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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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 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  Robert L. Devaney; Dept. of Mathematics, Boston University
A section from the paper "Chaos in the Classroom," describing the concept of fractal dimension, using the Sierpinski triangle as an example.
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 Fractal Expressionism  Richard Taylor, Adam P. Micolich and David Jonas
Can science further our understanding of art? An article discussing the connection between fractals and Jackson Pollock's paintings.
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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 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 Music: Research, Publications, and Compositions by Harlan Brothers  Harlan J. Brothers
In 2003, Benoit Mandelbrot suggested to Harlan Brothers that he undertake a rigorous treatment of the subject of fractal music. Here is an overview of Brothers’ subsequent research, including papers, definitions, and examples.
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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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 Fractions, Cycles, and Time  Ivars Peterson (MathTrek)
In ancient times, people had to confront awkward numbers in astronomical contexts when they compared the motions of the sun and moon. The unfailing, daily passages of the sun across the sky and the corresponding movements of the stars at night represented
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 Fragments of the Past  Ivars Peterson (MathLand)
Historians of mathematics now generally agree that scholars in China, India, and the Islamic world produced remarkably sophisticated mathematics between the fifth and the fifteenth centuries. However, most would probably still argue that Europeans in
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 Frank Garvan
Frank Garvan is a mathematics professor at the University of Florida. His combinatorics research articles are available for download in PostScript and .dvi formats. A Maple package for qseries may also be downloaded. One may also find the syllabi
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 Franklin's Mathematics  Paul C. Pasles, Villanova University
From the author of Benjamin Franklin's Numbers: articles on the founding father's "littleknown role as a mathematician," and examples of Franklin's magic squares and magic circle.
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 Frank's World: Math on Cable  Ivars Peterson (MathLand)
"Hi, you're live on Math Chat!" The question from the caller sounds pretty tough: "Three salesmen visit a shop. One comes every four days, another every five days, and the third every 10 days. If all of them were at the shop today, how long would you
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 Freaky Links  Cut the Knot!, Alexander Bogomolny with William A. McWorter Jr.
Mathematics impinges on itself and the sciences in unexpected and bizarre ways. This page is a small illustration of this phenomenon, taking up recreational mathematics and the game of Chinese checkers, the strategy game called Hex, quasigroups, and block
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 Frederic Chyzak
Frederic Chyzak researches combinatorics and computer algebra, specifically holonomic functions. His thesis and other articles are available as abstracts and as PostScript files. Slides for a series of talks on holonomic functions and computer algebra
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 Fred Rickey's History of Mathematics Page  Fred Rickey, Department of Mathematical Sciences, United States Military Academy
Information about courses Rickey and others have taught about the history of mathematics. Historical notes for calculus teachers cover Torricelli's trumpet,
Clepsydra, the history of the brachistochrone, the bridge and the catenary, Cauchy's famous wrong
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 Fritz Dooley's Mancala Center  Fritz Dooley
Game theory analysis of the board game Mancala, written as a research paper for Harvard Business School.
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 From Constructivism to Active Learning  Siegfried M. Holzer
Summary of an essay defining constructivism and outlining its implications for education: active learning, authentic activities, students' views, authentic assessment, and innovative curriculum. Published in the hypermedia publication of SUCCEED: The
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 From Counting to Writing  Ivars Peterson (MathLand)
Abstract numbers are the product of a long cultural evolution. They also apparently played a crucial role in the development of writing in the Middle East. Indeed, numbers came before letters, contends archaeologist Denise SchmandtBesserat of the University
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 From Galaxies to Electrons (Math Chat)  Frank Morgan, MAA Online
What are the largest and smallest objects anyone has ever seen with the naked eye? heard? felt? (For the first part, the whole object must be seen, though not necessarily in full detail or from every side.)
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 From Lewis Carroll to Archimedes  Cut the Knot!, Alexander Bogomolny
On March 29, 1879, Vanity Fair began offering its subscribers a new weekly puzzle invented by Lewis Carroll... on March 17, 1953 Frank Gray, a research scientist at Bell Labs, filed patent no. 2632058, for the Gray code encoding
the vacuum tube. An ndigit
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 From Linear to Nonlinear Optimization: The Missing Chapter  Prof. Hossein Arsham; University of Baltimore
This presentation in the form of a research paper provides a general solution algorithm for a large class of problems with linear constraints and a continuous objective function. A hypertext companion site for Applied Management Science: Making Good Decisions.
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 From Number to Formula  Ivars Peterson (MathLand)
Given the number 1.6180339887, how can you find out whether this particular number is special in some way, as the output of a specific formula or the value of a familiar mathematical function? Do you have a favorite number: Pi, the golden ratio, e (Napier's
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 From Starting Line to Ancient Angle  Ivars Peterson (MathTrek)
The ruins of ancient Corinth include a stadium that featured several courses for foot races. In 1980, archaeologists found and excavated the curved starting line for one such racecourse, dating from about 500 B.C. Curved to allow a staggered start, the
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 Frustration Solitaire  Doyle, Grinstead, Snell
A paper on the rankderangement problem, which asks for the number of permutations of a deck of cards such that each card is replaced by a card of a different rank, and which arises when computing the probability of winning the game of 'frustration solitaire'.
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 Fun and Games in Nevada  Ivars Peterson (Mathland)
...in the hotel's casino, the sights and sounds were... the glare of neon lights, the jangle of coins erupting from slot machines, the clink of chips at blackjack tables, and the mutter of avid gamblers testing their luck. It was the middle of April,
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 Functional Analysis  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to functional analysis, which views the big picture in differential equations, for example, thinking of a
differential operator as a linear map on a large set of functions. Thus this area becomes the
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 Functions of a Complex Variable  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to complex variables, studying the effect of assuming differentiability of functions defined on complex
numbers. The effect is markedly different than for real functions; these functions are much more
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 The Fundamental Theorem of Algebra  MacTutor Math History Archives
Linked essay covering from Cardan in the 1500's to the Euler and Argand proofs through the 1800's, with 8 references (books/articles).
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 The Funnel Cosmology  Vladimir Trifonov; American Mathematical Society (AMS)
Haar Measure and Geodesics in Funnel Cosmology  direct computation of the geodesics in the funnel model. Cosmological shape, dark matter, accelerating expansion, etc.
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 Fun with Probability! The Probable Pen in the Cereal Box  Michael Cornell; College of Education, Univ. of Illinois at UrbanaChampaign
A cooperative classroom project for grades K9. Over a threeweek period (24 April  10 May 1996), project participants jointly calculated the expected value of a simple probability problem via experimentation. This joint effort among interested schools
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 The Fun Works  Education Development Center, Inc. (EDC)
A career exploration digital library for middle school youth (ages 1114 yrs), designed to capture imaginations and direct youth toward careers in science, technology, engineering and mathematics (STEM). It places particular emphasis on engaging youth
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 Future of Leap Seconds  Steve Allen
Allen, an astronomer, discusses leap years, leap seconds  and how to avoid their use when redefining Coordinated Universal Time (UTC): three pictures that show the situation; elapsed time (JavaScript that shows POSIX's inability to provide an interface
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 Gallery of Pseudospheres  Robert McLachlan
A short version of an article by Robert McLachlan, "A gallery of constantnegativecurvature surfaces," (Mathematical Intelligencer, Fall 1994, 3137) about "pseudospherical" surfaces, equally "saddleshaped" at each point, extensively studied in the
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 Gallica  Bibliothèque nationale de France
Une sélection de documents numérisés qui montrent la diversité des collections manuscrites de la Bibliothèque nationale de France: les oeuvres de Joseph Liouville, Augustin Cauchy, Joseph Fourier, Henri Poincaré, Janos
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 The Galois Story  Ivars Peterson (MathTrek)
The life and death, myth and reality, of Evariste Galois, with a short bibliography.
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 The Game of Nim Explained  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
A Java simulation of the famous game, with a theoretical explanation and extension to other similar games: see also similar Java games: Nimble, Northcott's game, Plainim, the Scoring game, and Turning Turtles.
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 Game Theory in the News  Mike Shor
A regularly updated archive of news articles about game theory or probability. Hundreds of articles currently archived are indexed by mathematical theme or by area of application, including evolutionary biology, computer science, politics, and economics.
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 Gardner Index  Lee, Kluepfel
An index to fifteen books containing collections of Martin Gardner's articles from Scientific American.
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 Gateways to Advanced Mathematical Thinking (GAMT)  Michelle Manes; Education Development Center, Inc. (EDC)
The goal of this project is to research the ways in which high school and college mathematics students come to acquire flexible understandings of essential concepts needed for analytic and algebraic thinking. Integral to this research effort is the development
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 Gauss's Orbits  Ivars Peterson (MathTrek)
During the latter part of 1801, Gauss brought his formidable powers to bear on celestial mechanics. Like a skillful mechanic, he systematically disassembled the creaky, ponderous engine that had long been used for determining approximate orbits and rebuilt
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 Gödel and the Nature of Mathematical Truth  Edge interview with Rebecca Goldstein
Interview with Rebecca Goldstein, the philosopher and novelist, on her most recent book, Incompleteness: The Proof and Paradox of Kurt Gödel.
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 Gödel's Theorem and Information  Gregory J. Chaitin
Gödel's theorem may be demonstrated using arguments having an informationtheoretic flavor arguing that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast
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