 Math Magic  Erich Friedman
A site devoted to original mathematical recreations. If you have a math puzzle,
discovery, or observation, you are invited to email the author. You can also submit answers to the problem of the month; questions and solutions are archived. Examples include:
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 Math Projects  Curriculum Research & Development Group (CRDG), Univ. of Hawaii
Innovative curricula that are based on research, promote interaction among students, match NCTM standards, support students' construction of knowledge, develop ideas over time, and involve professional development and support services. Mathematics projects
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 Math Puzzle of the Week  Ed Pegg, Jr.
A site that celebrates math puzzles and mathematical recreations. Ed Pegg is a twentyyear member of the National Puzzler's League, and frequently contributes to the New York Times crossword, Games magazine, and National Public Radio's Sunday Puzzler.
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 Math Quilts  Rebecca Chaky
A showcase for quilters inspired by mathematical themes: geometry, perspective, topology, irrational numbers, feathered star, fractals, and tessellations and tiling, including Escher. Also a link to the Math Quilts Online Bookstore, a Reference Department,
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 Math Recreations  Thinks.com
The brain games, puzzles and pastimes website, proving that mathematics can be fun. Java Kali (explore symmetry groups and make fantastic patterns); The Largest Known Prime (can you beat it?); Ball Drop (demonstrate the normal distribution curve); Tackle
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 Math software demos, freeware, shareware for IBM PCs  Math Forum
DYSyst, Euklid, Fractal Grafics, FractInt, IBP Math tools (unit conversions), Penrose, Polyray, Persistence of Vision (POVRay). Download zipped demos from the Math Forum archives. Read the Index for descriptions of these programs.
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 Maths Outdoors  Juliet Robertson
Blog posts from an education consultant "passionate about getting children outside." Posts on learning math outside with a kinesthetic emphasis, which date back to September, 2008, have included "Leaf Logic," "Investigating Right Angles with Sticks,"
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 Math Trails in Ottawa  Ivars Peterson (MathTrek)
A math field trip to the National Gallery of Canada, where students followed a detailed itinerary with a variety of mathrelated activities including estimation, symmetries, magic squares, and more.
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 M.C.Escher: Artist or Mathematician?  ThinkQuest 1997
Biographical information on Escher, images from his extensive output, a tessellations tutorial (with illustrations from Escher's notebook), and links to other Escher sites.
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 M. C. Escher  François Loup
Des informations pour le monde francophone sur Maurits Cornelis Escher: biographie, bibliographie, diaporama, les thèmes, technique graphique, expositions et musées, les héritiers d'Escher, éspace éducation, aide documentaire,
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 MicroScapes  Lucent Technologies
A series of photographs taken through high power microscopes. Galleries include:
Silicon Dreams, Atoms and Ions, The Edge of Light, Solid States, Symmetry in Motion, and Memories, Glass and Apparitions.
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 MiMa  Museum for Minerals and Mathematics Oberwolfach
This German museum's exhibits about mathematics, which focus on symmetry and crystallography, have included IMAGINARY; SURFER, a realtime ray tracer of algebraic surfaces; Cinderella & Curved Spaces; jReality; Morenaments; Doppelpendel; Penrose Puzzle;
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 Minerals, Crystals and Their Systems  Charles LewtonBrain
An introductory listing of definitions and nomenclature concerning gem materials. Includes geometric and symmetrical properties of some gemstones, with diagrams, and major gemstones listed by crystal system.
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 Mirror Curves  Slavik Jablan, Mathematical Institute, Belgrade
Making art by repeating a curve to make a more complicated design, by symmetry/reflection. This paper illustrates Tamil threshold designs, Lunda and Chokwe sand drawings, and the knots/curves of Leonardo da Vinci.
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 Mirror, Mirror (MatheMUSEments!)  Ivars Peterson (Math Muse for Kids)
If you stand between two mirrors that are parallel to each other, one in front of you and one behind, you see countless images of yourself. Light rays bounce back and forth between the mirrors to create that endlessly repeated pattern.
What happens when
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 Mirrors (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
How do you hold two mirrors so as to get an integral number of images of yourself? Discuss the handedness of the images. Set up two mirrors so as to make perfect kaleidoscopic patterns. How can you use them to make a snowflake?...
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 Miscellaneous Sketches: Conics, Pythagorean Theorem...  Key Curriculum Press
Sketches: describing the relationships of parabola, hyperbola, and ellipse from an eccentricity point of view; four ways to demonstrate (manipulate) Pythagorean theorem  shear & rotation, shear & translation, similarity, and tessellation; studying
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 Miscellaneous Sketches: Linear Sketch, MidRhombus...  Key Curriculum Press
Sketches: a dynamic display of the equation of a line as you move it around; an illustration of the midpoint figure of a quadrilateral; a tessellation made in the pattern of a Navajo rug; a sketch that draws the regular polygon with given internal angle;
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 Modern Math  Taoufik Nadji, Malek Physix Inc.
Mr. Taoufik Nadji's precalculus students at Interlochen Arts Academy introduce modern mathematical topics: fair division (methods: dividerchooser, lone divider, lone chooser, last diminsher, markers, sealed bids); graph theory (applications, terms,
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 Modularity in Art  Slavik Jablan, Mathematical Institute, Belgrade
Jablan uses the word "modularity" to refer to the process of making art by repeating a simple block, tile, or motif to make a more complicated design, by symmetry/reflection and/or tessellation. This site includes a paper on such art and its math, linked
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 More Papercutting Patterns (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
Conical, cylindrical patterns, Moebius, and other symmetrical patterns to experiment with.
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 Mr. Myers' Classroom  Nathan Myers
Read the puzzle of the week and then submit your answers online. Other "brain busters" include matrix logic puzzles, featuring readymade charts. Explore and identify fraction bars with JavaScript, or geometry with Java applets. Also, fact sheets and
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 Multicultural Math Fair  Suzanne Alejandre, Frisbie Middle School
Twentynine percent of Frisbie's student population is Limited English Proficient; Arabic, Cambodian, Chinese, German, Indonesian, Korean, Philipino, Polish, Portuguese, Romanian, Samoan, Spanish, Taiwanese, Vietnamese, and Urdu are represented. In 199394
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 My Puzzle Note  WenShan Kao
The 18 ProTangram is an interesting and excellent puzzle game for the
purpose of education at school and of recreation at home. Players of
different ages can practice and enhance their concept of reflective or
rotational symmetry, and of the congruent
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 Names for features of symmetrical patterns (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
Names for features that may occur in symmetrical patterns. To each such feature of the pattern, there is a corresponding feature of the quotient orbifold. Mirrors and mirror strings; mirror boundaries; gyration points; cone points.
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 Names for symmetry groups and orbifolds (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
A symmetry group is the collection of all symmetry operations of a pattern. We give the same names to symmetry groups as to the corresponding quotient orbifolds, regarding every orbifold as obtained from a sphere by adding conepoints, mirror boundaries,
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 Nearest Neighbors and Voronoi Diagrams (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material on Delaunay triangulations and Voronoi Diagrams.
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 New Geometric Tilings  Adrian Fisher Maze Design
A page that celebrates the beauty of 4, 5, 6, and 7fold geometric tilings, grouped according to the regular polygons that they fill. Where possible, actual dates and designers are stated. Nearly all these tiling examples are based on sides of the
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 New Zealand Mathematics  Ministry of Education, New Zealand
Units and lesson plans organised by level and curriculum strand, in Maori and English, focusing on solving problems. Topics address levels 16 of the New Zealand mathematics curriculum and include probability, statistics, measurement, and geometry.
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 Numeracy Works  David Hartmann, Numeracy Works LLC
Numeracy Works is a commercial software company which sells Tabula, a new tool for math education. Use Tabula to create, construct, cut, tape, fold, and transform. Make fold patterns, tiling patterns, tessellations, and perspective drawings. Also on
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 nUniform Tilings  Brian Galebach
Illustrates the edgetoedge tilings of regular polygons having n transitivity classes of vertices for 1<=n<=6. Prior to my doing so, no enumeration had ever been completed of the nuniform tilings for any n>3.
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 On the Puzzles with polyhedra and numbers  Jorge Rezende
Several articles in PDF format, with related links: one on a set of didactic puzzles published by the Portuguese Mathematical Society (SPM); "Jogos com poliedros e permutações", an article in Portuguese describing puzzles with polyhedra
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 Optique & Cabri  Charrier et Cortial
A collection of classic diagrams created using Cabri Géometre, reworked in Paint Shop Pro, and some have animated GIFs. Optique Géométrique (reflections and mirrors, refractions and diopters, thin lenses, optical instruments); Optique
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 Orbifold Pinball  Yevgeniy Dodis; The Geometry Center
A game in which you roll a ball on an unusual playing field. The game board is like ordinary pinball in that it features 'bumpers' off which the ball can bounce. But in this game, the board is 'curved' in such a way that a ball barely missing the bumper
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 The Orbifold Shop (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
A symmetry project: the Orbifold Shop offers a free coupon for $2.00 worth of parts, installation included, to anyone acquiring a new orbifold. There are only a few kinds of features for twodimensional orbifolds, but they can be used in combinations:
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 Paper Mosaics  Alex Bateman
Origami tessellations consist of a simple crease pattern unit that gets repeated across a sheet of paper. Download Tess, a computer program that aids in the design of such tilings; and browse a gallery of works created with the application.
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 Papers  Chaim GoodmanStrauss, Dept. of Mathematics, Univ. of Arkansas
On Composite Twisted Unknots; Matching Rules and Substitution Tilings; Aperiodic Hierarchical Tilings; Bounding the Results of Arithmetic Operations on Random Variables with Unknown Dependency; A Small Set of Planar Aperiodic Tiles; A Pair of Aperiodic
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 Parcelles d'infiniPromenade au jardin d'Escher  Alain Nicolas
Book in French, largely composed of pictures, on methods of figurative tiling. Site includes links to related sites in French and English.
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 Pattern  Drew, Wallace; Mathematics Across the Curriculum, Dartmouth College
An interdisciplinary course in textile design and elementary group theory. Syllabus, course pages (notes, figures, questions, group work and homework), final project and class artworks from a 1996 math and art crosscurricular class at Dartmouth College.
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 Patterns, Patterns, Everywhere  Nina Heal
Three lesson plans: an internetbased research project on patterns; connections between the arts of M. C. Escher and quilters; and a technologyenriched tessellations lesson. Includes a teacher resource page with rubrics.
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 Peanut Software for Windows  Rick Parris
Abstracts and links for downloading free mathematics software: Wingeom (highprecision geometric constructions); Winplot (generalpurpose plotting utility); Winstats (scatter plots, curve fitting, histograms, statistical data, and standard theoretical
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 Pearson Learning  Dale Seymour Publications
A leading publisher of educational materials, emphasizing math, science, and the arts. The site contains: an online catalog; math curricula include: CMP (the Connected Mathematics Project) and Investigations (Investigations in Number, Data, and Space,
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 Peeze  Patrick Snels
Tessellations and Escherlike perspectives, and tessellations of names, with Peeze, a tessellation program (Windows, PC) to download.
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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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 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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 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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 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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 Plane Patterns  Ivars Peterson (MathTrek)
Exploring the link between mathematical symmetry and artistic effort. One and twodimensional repeating patterns appear ubiquitously on surfaces, from quilts and colored fabrics to pottery and ceramic tiles... Analyzing the symmetries that underlie such
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 PolygonFlux  John Miller
An iPad app for exploring the path taken by a point bouncing around inside a regular polygon of 3 to 14 sides. Set the initial angle and starting location, the number of bounces, and the number of sides for the polygon; collect an album of resulting "fluxagons"
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