 Symmetry and Orbifolds (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
Given a symmetric pattern, what happens when you identify equivalent points? You get an object with interesting topological and geometrical properties, called an orbifold... hearts, paper dolls, wave patterns, quotient orbifolds, Moebius bands.
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 Symmetry  Gyorgy Darvas
Organizations, events, publications, projects related to symmetry.
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 Symmetry  Rick Engel
A lesson on the symmetry of the Platonic and Archimedean solids using Polymorf, a math manipulative product. Demonstrates inscription, truncation, sectioning, and space filling with polyhedra.
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 Symmetry  Science Museum of Minnesota
Patterns in Nature  butterflies and moths are symmetrical, with two equal sides. Symmetry means balanced proportions. Mrs. Schmidt's class spotted symmetry everywhere! Make your own butterfly using the pattern and instructions provided.
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 Symmetry Studio 1.01  Binkley, Simon; Artware, Inc.
Symmetry software for mathematics and art, for the Macintosh. A computeraided design tool that allows you to quickly create complex twodimensional symmetrical patterns. It simplifies the study and use of mathematical principles that underlie the regular
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 Tangrams: History, Puzzles, Make It, and Links  Randy Crawford
Follow the "history" link to learn about tangrams through the ages; click on "puzzles" for over 100 silhouettes to reconstruct, with solutions. "Make it" provides instructions for building your own tangram set. With references to more sites, software,
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 Tangrams  Tom Scavo
Tangrams, a puzzle that helps develop spatialvisualization skills, may also be used to introduce or reinforce geometric concepts such as congruence, similarity, symmetry, etc. This unit for grades 4 through 6 uses tangrams to compute the area of polygons
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 Tanzzle™  Franco Cocchini
Software for exploring tangrams. View published patterns and/or submit one of your own. Request a free license for schools and educational programs. Free 30day trial; free viewer download. Choose from four different sets of pieces: Tangram (traditional
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 Teacher Exchange  the Math Forum
Web units, lessons, and activities organized by NCTM Standards and grade band level. Created by Math Forum staff members, educators, and collaborations between the two. Teachers may contribute their own lessons and ideas as well.
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 Teaching Math to Visually Impaired Students  Susan Osterhaus, Texas School for the Blind and Visually Impaired (TSVBI)
Questions and answers offering teaching strategies for improving the mathematical
understanding of visually impaired students. Challenges include solving quadratic equations; geometric constructions; transformations, line symmetry, and tessellations;
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 Technology in the Geometry Classroom  Math 5337, The Geometry Center
Course materials developed at the Geometry Center and used for teaching pre and inservice teachers of high school geometry interested in using technology in their classrooms (see the Summer 1996 syllabus ). Four selfcontained parts: Internet Skills,
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 Tessellated Polyhedra  Jill Britton
Fullpage printable nets of each of the regular polyhedra, decorated with individual surface tessellations in full color. A higherresolution blackandwhite version of each tessellated net appears in Polyhedra Pastimes, by Jill Britton, published by
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 Tessellation Database  Patrick Snels
Find tessellations by subject, author/artist or by plane geometry. A virtual tessellation museum is planned.
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 Tessellation Methods Using MS Paint  Diana Hunter
Lessons and examples in creating tessellations using MS Paint include stepbystep instructions on the translation method, glide method, rotation method, midpoint rotation, and right triangles. Hunter also provides suggestions for making better tessellations,
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 Tessellation  Roger Carthew
Sets of foam shapes for use in making tessellations. Site includes purchasing information, description of shapes and colors provided, and example tessellations.
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 Tessellations: An Application Of Simple Regular Polygons (SMILE)  Mary Racky, Kenwood School
A lesson designed to develop basic skills making and identifying homogeneous
tessellations, both regular and semiregular. From the Patterns and Logic section of a collection of almost 200 single concept lessons by the Science and Mathematics Initiative
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 Tessellations, Lifelike Tilings...  Andrew Crompton
Tessellations, Lifelike Tilings, Grotesque Geometry, Dissection Puzzles, Anamorphic Mirrors, and Escher style drawings: about twenty examples of Crompton's own work as well as the world's first tessellation (not by Escher), by a teacher of geometry to
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 Tessellations.org  David Annal
Overview of the subject of tessellations with practical examples of how to produce them. Background information on tessellations, including symmetry and famous tessellators, and extensive galleries.
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 Tessellations Project  Mrs. Sulik's Fifth Grade Math Class
"M. C. Escher was a Dutch artist who lived from 18981971. Many of his drawings were tessellations. Instead of using the usual polygons, like triangles and squares to tessellate, he liked to create pictures of horses, fish, salamanders, and birds that
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 Tessellations  Robert Fathauer
A company that makes Puzzellations, Colorful Connectables, and Perpetual Patterns puzzles, plus a variety of other products inspired in part by the Dutch graphic artist M.C. Escher, but also by mathematicians such as Roger Penrose and Benoit Mandelbrot.
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 Tess  Pedagoguery Software
A shareware program for creating symmetric planar illustrations. With Tess, while you draw you can automatically maintain the symmetry group you have chosen; 11 rosette, all 7 frieze, and all 17 wallpaper groups are included. Illustrations may be printed
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 Texture and Colour  Paul Bourke
In many texturing applications it is necessary to be able to "tile" the texture over a larger region than the texture segment covers. Bourke offers a discussion with examples of tiling formats, together with links to tiles: Tile; Brick, Tile; Stone tile;
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 TileDreams  ZayanteCreek Solutions
Explore geometry, color, symmetry, and design by creating mosaics, tessellations, mandalas, and other designs. View galleries and submit your own TileDreams designs; read the user manual and software reviews; and download a trial demo, or purchase a license
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 Tiletris  Oktay Haracci
A tiling program for aperiodic or periodic tiling. Freeware, for PCs only, running Win 9x/2000. Help, including instructions, available on the site.
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 Tiling: A Java Program for Exploring Plane Tilings  Tom White
A Java application for experimenting with tilings and tessellations through scripting. Construct tilings interactively using a mouse  "like doing a jigsaw." Alhambra comes with many tile shapes already supplied; the site offers images created from the
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 Tiling and Packing results  Torsten Sillke
Packing problems, mostly, in English, with some in German. Packing Polycubes and Polyspheres; Tilings with Matching Conditions; links.
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 Tiling Plane and Fancy  Steve Edwards
The 17 wallpaper groups identify the plane symmetry groups and link to background patterns illustrating them. Tiling Definitions define, with examples, the terms fundamental region, isohedral tiling (isohedral and nonisohedral), isogonal tiling, lattice,
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 Tilings and Geometric Ornament  Kaplan, Salesin
A project designed to explore the intersection between computer graphics, geometry, and ornamental design. Papers and illustrations on Escherization, Islamic Star Patterns, Parquet Deformations, Symmetrohedra, and Voronoi Diagrams.
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 Tilings (Graphics Archive, Special Topics)  The Geometry Center
Regular subdivisions of the plane or space.
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 Tilings (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material on tilings. One way to define a tiling is a partition of an infinite space (usually Euclidean) into pieces having a finite number of distinct shapes. Tilings can be divided into two types, periodic and
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 Tiling with Polyominoes  Ivars Peterson (MathTrek)
"Mathematicians have proved that the general question of whether it's possible to cover the plane with identical copies of a given finite set of tiles is, in principle, computationally undecidable. In other words, there's no cookbook recipe or handbook
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Tour: M.C. Escher  Life and Work  National Gallery of Art, Washington, DC
A tour of four virtual rooms devoted to the art of M. C. Escher. For each graphic a larger picture with accompanying commentary is available by clicking on a small picture. The gallery has more than 400 works by Escher with many images available online.
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 Traffic Signs  Richard C. Moeur
Graphics of the most commonly used traffic signs in the United States, useful for challenging students to find symmetrical signs (in addition to polygons).
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 Transformational Geometry  Leslie Howe
Java applets to illustrate translation, rotation, glide reflection, etc. Directions for tessellations by hand and using dot paper. Links to sites of artists that illustrate these ideas.
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 Transformations and Tessellations  Lynda Kozicz Leahey
A unit introducing the concepts of transformations and tessellations. Students will explore the attributes of transformations: reflections, translations, rotations and dilations by using pattern blocks, algebraic notation and computer design. They will
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 Transformations  David A. Reid
A transformation is a onetoone mapping on a set of points. The most common transformations map the points of the plane onto themselves, in a way which keeps all lengths the same. These transformations are called isometries. Another common sort of transformation
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 Trisecting the Angle  Steven Dutch; University of Wisconsin  Green Bay
Why is trisecting an angle with a straightedge and compass impossible? Discussion includes several other proofs of impossibility (the largest prime number, the square root of 2, repeating patterns in the plane) and some alternate methods of trisection
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 Truncated Octahedron Blocks  Hop David
A discussion of a threedimensional tessellation that may make a good bridge between the rectangular solid tessellation and the octa/tetrahedral tessellation.
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 Tumblehome Learning
Makers of educational products such as Lost in Tangrams and  through a partnership with TERC, with funding from the National Science Foundation (NSF)  Mixing in Math (MiM) cards, gameboards, and books.
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 Undergraduate Research Project in Random Tilings  Propp, Wilson
This project investigates different regions of random tilings of nxn squares with 1x2 dominoes: square, hexagon, fortress, dungeon, and Aztec diamond (the set of lattice squares completely inside a square tilted 45 degrees, discovering that the dominoes
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 Unfolding Orbifolds  Heidi Burgiel; The Geometry Center
A study of how a pattern can be reconstructed from an orbifold, using both handson exploration and the computer program TesselMania!. These activities are a good introduction to the movie Shape of Space. From Math 5337, Technology in the Geometry Classroom,
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 The Uniform Polyhedra  Roman Maeder
A shortened version of the notebook Polyhedra.ma from Illustrated Mathematics [Gloor/Amrhein/Maeder95]. Uniform polyhedra have regular faces and congruent vertices. Allowing for nonconvex faces and vertex figures, there are 75 such polyhedra, as well
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 Unifpack (Web version 2.0)  Carlos O'Ryan Lira
A set of programs and C libraries for studying Riemann Surfaces with symmetry. Unifpack accepts a presentation of a finite group and calculates the Cayley graph of the group for that presentation. If the presentation is "admissible," Unifpack can calculate
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 Using Symmetry to Create A Community Quilt (SMILE)  Beverly Brown, Hedges West
A lesson for grades 412 designed to teach students about line and reflectional symmetry through cooperative work on a quilt with four lines of symmetry. From the Patterns and Logic section of a collection of almost 200 single concept lessons by the Science
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 The Virtual Mathematics Center  Martindale's Graduate & Undergraduate Center
Preprints and online journals from a wide variety of math and science organizations; Online math dictionaries and encylopedias; mathematics courses, tutorials and databases; courses by subject; related mathematics information and databases.
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 Visual Mathematics (VisMath): Art and Science Electronic Journal of ISISSymmetry  Nagy, Jablan
The electronic journal of ISISSymmetry (International Society for Interdisciplinary Studies of Symmetry), published by the Mathematical Institute, Belgrade, Yugoslavia. Papers, conferences, the Nexus Network Journal of Architecture and Mathematics Online,
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 Wallpaper Groups (Plane Symmetry Groups)  David Joyce; Dept. of Mathematics & Computer Science, Clark University
Mathematicians classify plane patterns such as wallpaper patterns by means of symmetry groups  groups of transformations of the plane that leave the pattern invariant. A transformation slides, rotates, and reflects the plane into another position. Such
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 Wallpaper Groups: The Discontinuous Groups of Rotation and Translation in the Plane  Xah Lee
An undergraduate tutorial on wallpaper groups, with illustrations of the 17 wallpapers groups and a wallpaper gallery. Technically, the derivation of all discontinuous groups of rotation and translation in the plane. Contents include: Some Theorems on
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 Welcome to PERTH: PEerless on EaRTH  Dave Watson
Dave Watson's main interest is computational geometry, and he specializes in spatial interpolation. Books/software on Contouring (a survey of published bivariate contouring and interpolation methods), Tessellations, Spherical Tessellations, Isolines,
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 A Whale of a Tale  Dive into Math  INNOVA Multimedia Ltd.
Three online math lessons (Addition, Super Sea Sets, Symmetry) for ages 59, from INNOVA Multimedia's "A Whale of a Tale" series. Shockwave (required) activities offer ways for children to visualize and manipulate concepts.
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