 Pennies in a Tray  Ivars Peterson (MathLand)
What's the largest number of pennies that you can pack inside a circular tray to form a carpet of nonoverlapping coins? What about inside a square or triangular tray? What if you could expand or contract the size of all your pennies to fit a required
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 Pennsylvania Education Network: Link To Learn  Pennsylvania Department Of Education
A threeyear (199699), $132 million initiative aimed at expanding the use of technology in the classroom, including links to related technology initiatives, agencies, and grant programs, as well as Internet resources for students and parents. Resources
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 Pennsylvannia Governor's School For The Sciences
A summer enrichment program for high school students, in session for five weeks, seven days per week. It is an intense program in which the students take lecture courses in biology, chemistry, physics, mathematics and computer science. It emphasizes handson
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 A Penny Surprise  Ivars Peterson (MathTrek)
Flipping a coin in the air, catching it, then determining whether it has come up heads or tails is to many people a prototypical random process. In some sense, however, coin tossing isn't really random at all. A mechanical gadget can flip a properly positioned
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 The Penrose Puzzle Room  World of Escher
Mathematically based Pentaplex Puzzles designed by Sir Roger Penrose, the wellknown English mathematician. The original 45 piece Gamebirds, 190 piece Perplexing Poultry  2 designs, 500 piece Jigsaw Puzzles  4 designs.
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 Pens and Crystals (Math Chat)  Frank Morgan, MAA Online
Answering the challenge: How should you fence in two adjacent rectangular pens of given areas A, B with the least amount of fencing (including the fencing separating them)? Ms. Mattuck's 8th graders at the Richmond School use the Geometer's Sketchpad
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 Pentagonal Geometry and the Golden Ratio (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material on geometric problems defined on regular pentagons, involving pentagonal angles, or based on the golden ratio (the ratio of diagonal to side length in a regular pentagon).
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 The Pentagon Universe (Penrose tiles and more)  Kadon Enterprises, Inc.
Manipulatives based on Penrose tiles, golden triangles and other tilings based on the golden ratio.
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 Pentangle  U.K.
To buy: wooden, mechanical, interlocking, mathematical and logical thinking puzzles; brainteasers, games, toys, gifts, threedimensional puzzles, promotional items, and collectors' puzzles,
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 Pentium Bug Revisited  Ivars Peterson (MathLand)
The incident that triggered a barrage of ridicule occurred in the fall of 1994, a few months after Intel had introduced its Pentium microprocessor. The furor started with an email message from Thomas R. Nicely, a mathematician at Lynchburg College in
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 Pentominoes  Robert Simms; Dept. of Mathematical Sciences, Clemson University
A Java applet spatial relations game. Try to place the pieces in the box with no overlapping. With detailed instructions, source code, development stages of the applet, and related links.
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 Pentomino  K.S.O. Glorieux Ronse, Belgium
An introduction to polyominoes, with a sitewide focus on pentominoes. The site features puzzles and contests, as well as galleries of pentomino constructions and problems, such as how to form letters of the alphabet and silhouettes of zoo animals. Available
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 Pentomino Relationships  Adrian Smith
The site describes how pentominoes (twelve pieces of wood, each a different shape and each made of five cubes) fit together into rectangles or cuboids, as well as the subsequent challenge of finding as many other solutions as possible from the one just
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 Percent  David Giordano
For Win95/98/ME/NT4/2000/XP, or for Pocket PC/Windows CE: calculate the percentage of any number out of any number. Free trial download; shareware program.
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 Perfect, Amicable and Sociable Numbers  David Moews
Definitions, descriptions, and discussions of perfect, amicable, and sociable numbers; aliquot sequences, related problems, and other definitions of divisibility, with a short bibliography.
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 A Perfect Collaboration  Ivars Peterson (MathLand)
Euclid and Euler seem an unlikely pairing. One was the most prominent mathematician of antiquity, best known for his treatise on geometry, the Elements. The other was the most prolific mathematician in history, the man whom his 18century contemporaries
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 Perfection  Greg Lobanov
This mobile game for iPhone and Android handheld devices presents convex and concave polygonal shapes for players to cut into the outlines indicated. Available directly from the iTunes app store and Google Play.
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 Perfect Numbers  MacTutor Math History Archives
Linked essay describing the work on perfect numbers from Pythagoras through Euclid, Nicomachus, the Arabs, Cataldi, Descartes, Mersenne, Fermat, Euler, Barlow, Catalan, Cole, and others; with other web sites and 20 references (books/articles).
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 Perfect Numbers  Math Forum, Ask Dr. Math FAQ
What is a perfect number? How many are there? How can you calculate a perfect number from a Mersenne prime?
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 Perimeter Efficiency  William Blackwell
Perimeter Efficiency, also known as Isoperimetric Inequalities, compares the actual perimeter of a shape with the circumference of the circle that would be required to enclose the same area. Since a circle will enclose any given area with the least amount
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 Perimeter Institute for Theoretical Physics
An independent, nonprofit, scientific research organization working to advance our understanding of physical laws and develop new ideas about the essence of space, time, matter, and information. Located in Waterloo, Ontario, Canada, Perimeter also provides
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 Periodic Points of Quadratic Polynomials  Cut the Knot!, Alexander Bogomolny
The main purpose of the Mandelbrot set is to index Julia sets corresponding to various values of the parameter c... iterations inside M evolve differently depending on the value of c. (You may explore some iterations via the applet supplied.) An interactive
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 Permadi.com  F. Permadi
Java and Flash applets: logic games, drawing and patternmaking programs, Conway's Life, and more arcadestyle offerings. Also includes tutorials in using Flash and a graphics gallery.
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 Permutations and Combinations  Math Forum, Ask Dr. Math FAQ
How to calculate all possible groupings of objects where order does  or doesn't  matter.
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 The perplexing mathematics of presidential elections  Keith Devlin (Devlin's Angle)
It's not the idea of one person one vote that's the problem, it's that math that is used to turn those votes into a final decision. Ideally, that math should reflect the wishes of the electorate. But does it? The answer usually comes as a surprise to
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 Perplexus  flooble
A large archive of mathematically inclined puzzles and problems displayed by category. Includes logic, number theory, geometry, probability and other problems. All are rated by difficulty and quality and discussed in depth by visitors. Most with solutions.
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 Personal Algebra Tutor (PAT)  CyberEd, Inc.
An algebra problem solving program for high school and college algebra. Students can enter a wide variety of algebra problems and PAT will solve them stepbystep, with detailed explanations. PAT solves, simplifies, graphs, etc. Solutions with sidebyside
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 Personalized Programming Service Inc.
Educational software for teachers and students. Topical software in Math, English, Social Studies, and Science, as well as Quizmaker and Testmaker programs. Download demos and upgrades, browse the products, or order on site.
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 Personal Tutor  Memory Banks Pty. Ltd., Australia
Maths and science software designed to complement the secondary school math syllabus. A comprehensive learning system usually used by students and schools as a supplement and aid to coursework, but thorough enough that it can be used by individuals without
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 Perspective Drawing  Jan Garner
An examination of geometry in art: exercises in drawing, pictures and paintings in perspective on the Web (Albrecht Dürer, the Ancient City of Athens exhibit).
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 Perspective Images  H. B. Meyer
The computation of image points of perspective images by the use of similar triangles ("numerical perspective") is shown. By inversion of the corresponding mapping equations the determination of proportions of various distances in the real scene becomes
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 Perspective Study  Hop David
David suggests a mapping from half of 3D space onto a plane, feeling that traditional perspective drawings are similar to the image projected by a pinhole camera. Perspective illustrations of arches, looking up or down at a skyscraper, peripheral vision,
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 Peter Ash's Thoughts on Math and Education  Peter Ash
Musings on doing and teaching mathematics, book reviews, and math problems both elementary and advanced from the creator of Cambridge Math Learning. Blog posts, which date back to August of 2007, include "Very Proper Fractions"; "Physical Models for NonEuclidean
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 Peter Gustav Lejeune Dirichlet  Australian Mathematics Trust
A biography of the mathematician that discusses his theorem, introduction of the function notation y = f(x), boundary problems, minimization principle, and the Pigeonhole Principle.
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 Peter J. Lu  Department of Physics, Harvard University
Lu's mathematical and historical research has revealed new insights in medieval Islamic quasicrystalline tilings and the earliest machines, such as the one he suggests precisely interconverted the linear and rotational motion responsible for the distinctive
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 peterlu.org  Peter Lu
Publications and research into medieval Islamic geometry (e.g., decagonal and quasicrystalline tilings in its architecture), softmatter and colloid physics, highperformance imaging, ancient Chinese technology (e.g., first use of diamond, early precision
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 Peterson's Education Center
A leading provider of information on U.S.accredited educational institutions in the U.S. and abroad. Topics include: Summer Programs for Kids and Teens; Public Schools K12; Private Day and Boarding Schools; VocationalTechnical Schools; Undergraduate
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 Peter Winton's Radical Home Page
Math topics include puzzles, a gallery of fractals, and Visual Basic programs to download (count the grains of rice on a chessboard, or graph different curves).
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 Petits Textes sur l'éducation  Laurent Lafforgue
Essais écrit par le laureat de la medaille Fields 2002, incluent "De l'école et de ce qui fonde la valeur de la culture et du savoir" et "De la culture générale des futurs instituteurs et professeurs" et "Les mathématiques
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 Página Web de Fibonacci, Humor, Matemáticas  José Mari López Goitia
Página personal, con secciones para humor y matemática, información sobre HTML, manual de C, buscadores de todo el mundo, y más.
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 PhD! ... plus epsilon  Adriana Salerno
Having earned her math doctorate in 2009, Salerno writes "about some of the challenges I've encountered while teaching, attempting to do research, and the lesser known aspect of 'service'" on institutional committees. In addition to blogging about "publish
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 The 'PhiNest'  Gary Meisner
This source for the golden section, golden mean, divine proportion, Fibonacci Series, and phi discusses phi "for neo'phi'tes"; credit cards; energy; geometry; history; life; the human hand; the human face; the human body; human beauty; development; the
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 Phi's Fascinating Figures  Ron Knott
More properties of the number Phi, its multiples and powers, continued fractions, formulae, and numerical relations between Phi and its powers.
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 Phi: That Golden Number  Mark Freitag, Univ. of Georgia
Phi can be found in many geometric shapes as an approximation of the Golden Ratio. A geometric figure commonly associated with Phi is the Golden Rectangle, which has sides A and B that are in proportion to the Golden Ratio. It has been said that the Golden
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 Photographic Polyhedron (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
As you stand in one place and look around, up, and down, there is a sphere's worth of directions you can look. One way to record what you see would be to construct a big sphere, with the image painted on the inside surface. To see the world as viewed
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 PhotoMath  microblink
"Smart camera calculator" apps that use a cell phone's builtin camera to read mathematical expressions in real time, then output stepbystep solutions.
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 Photos by Paul Halmos  Mathematical Association of America (MAA)
Snapshots taken by Paul R. Halmos of notable contemporaries, academics, and other intellectuals of the 20th century. Search or browse the tagged collection of pictures, each of which includes a brief note about the subjects photographed in Halmos' own
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 Phyllotaxis  Pau Atela, Chris Golé; Smith College
An interactive site for the mathematical study of plant pattern formation. Sections of the site include: Introduction to Phyllotaxis (classification, Fibonacci/golden angle, microscope, mathematical model, and lattices); history; glossary; references;
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 The Physics Classroom  Tom Henderson, hosted by StudyWorks! Online
A physics tutorial for high school students, including explanations, animations, quizzes, and practice problems.
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 Physics Forums
Discussions emphasize physics, but "PF" also includes an entire directory on mathematics: General Math; Calculus & Analysis; Differential Equations; Linear & Abstract Algebra; Topology & Geometry; Set Theory, Logic, Probability, and Statistics;
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