 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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 Penn State University Mathematics Department
Course home pages and instructional material; seminars, colloquia, and conferences; and subject area pages for Penn State research centers, with preprints and links to other resources on the Web: Algebra and Number Theory; Dynamical Systems; Mathematical
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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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 Penrose Tiling Generator  Stephen Collins
A software package for Windows NT, for generating rhombic Penrose tilings and investigating symmetrical topological chains within them.
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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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 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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 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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 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 Symmetry Number Theory  Derek Nalls
A fundamental, educational exposition of a nonstandard number theory based upon perfect symmetry (in contrast to broken symmetry) modelled throughout arithmetic, algebra, analytic geometry, analytic trigonometry and calculus.
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 Performance Evaluation  Keith Devlin (Devlin's Angle)
"Dear Professor G,..." A critique of current professorial evaluation schemes in the college/university setting.
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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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 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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 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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 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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 PEST: ModelIndependent Parameter Estimation  Watermark Computing
PEST is a package that can be used with any preexisting model for data interpretation or model calibration. Use of nonlinear parameter estimation techniques allows you to exercise greater control. The site includes PEST, Parallel PEST, SENSAN, PEST MODFLOW/MT3D
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 Pete Casazza
Articles by this University of Missouri professor, some of them scanned from journals, include "A Mathematician's Survival Guide," "Classes of Finite Equal Norm Parseval Frames," "Dear Pythagoras: See What You Started," "Equivalents of the KadisonSinger
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 Pete L. Clark
Course notes, open problems, publications and preprints by Clark, who teaches at the University of Georgia, and does research primarily in number theory and arithmetic geometry. See, in particular, Clark's freely downloadable PDF expositions on commutative
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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 Doyle  Mathematics, Univ. of California at San Diego
Advanced tutorials, proofs and papers, mostly statistics, probability, geometry, and graph theory, but with examples from other mathematics areas as well. Don't miss Dr. Doyle's Flip Flops.
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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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 Peter Olver's Home Page  Peter Olver
Publications, seminar and conference talks, and more resources from the Head of the University of Minnesota's School of Mathematics. Download notes of Olver's lectures on applied mathematics, numerical analysis, orthogonal bases and the QR algorithm,
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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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 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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 Pfaff's Method (III): Comparison with the WZ Method (Organic Mathematics Proceedings)  George Andrews
In the 1990's, the WZ method has been the method of choice in resolving new conjectures for hypergeometric identities. The object here is to compare the WZ method with Pfaff's method. Such a comparison should (it is hoped) provide some suggestions for
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 Phalanstère  Équipe de Combinatoire Algébrique, Université de MarnelaVallée
Research into the relations between algebra and combinatorics. Membres; Publications; Logiciels; Séminaire. In French.
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 Phase Portraits for ODE's and Autonomous Systems  Mansfield, Beukers; Mathematics Dept., Penn State University
A Java applet that allows the user to draw the direction field and integral curves of an ODE. You may also plot a single first order equation by letting x'=1. Or better yet, since dy/dx = y'/x', to solve dy/dx = f(x,y), choose any two functions whose
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 Phase State Diagrams  Harold Brochmann
Phase diagrams reveal much information about physical phenomena. Brochmann illustrates with a variety of graphs created by computer programs. Phase state diagrams of unstable orbits reveal 'strange attractors', useful in a variety of fields including
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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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 Philosophy of Mathematics Education Journal  Paul Ernest; School of Education, University of Exeter, UK
A page with links to past issues of the journal, formerly the Philosophy of Mathematics Education Newsletter, along with links to information on the Falmer Book Series 'Studies in Mathematics Education', the Masters program in mathematics education at
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 Philosophy of Science  David Banach
A syllabus and collected resources for a course in the philosophy of science, including excerpts from various books. Topics include ancient Greek science and mathematics: the Golden Section, Pythagoras, infinity and continuity, and Plato and Aristotle,
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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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 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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 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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 Physics 109N: Galileo and Einstein  Michael Fowler; Dept. of Physics, University of Virginia
A course that explores two revolutions in our perception of the universe. The first, in which Galileo played the leading role, was the realization that what we see in the heavens  the moon, the planets, the sun and stars  are physical objects. The second
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 Physics and Math Help Online  Bryan Gmyrek
Tutoring available through email. Previously answered questions are archived on the site, and a tutorial on The Plank Radiation Law  Blackbody Spectrum is available. The author also provides a physics help newsletter and a list of sites for further
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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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 Physics (MathPages)  Kevin Brown
More than 60 "informal notes" by Kevin Brown on mathematical physics: Doppler for sound and light, optical rays in heterogeneous media, radial orbitgs in Schwarzschild geometry, the formation and growth of black holes, can Schrodinger's cat factor numbers?,
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 The Physics of Baseball  Alan Nathan
Read about the benefits (or not) of corked bats, the differences between aluminum and wood bats, the issues involved with characterizing the performance of bats from a physics point of view, and a description of the collision between a wooden bat and
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