 Don Hatch's Home Page  Don Hatch
Hyperbolic tessellations shown in various stages of truncation, and represented by their Schlafli symbols. Java applets explore hyperbolic tessellations, a magic Rubik'slike cube in four dimensions, polygon reciprocation (dual), modern portfolio theory
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 Dueling Pinwheels  A Geometer's Sketchpad Activity  Cynthia Lanius
Construction directions for rotations, translation, and reflection, with questions to explore and a demonstration sketch to download.
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 Dynamic Mathematics Visualization for Young Learners
Sketchpad in Grades 38  a working conference, held at the University of Illinois at Chicago in February, 2003. Researchers, curriculum developers, and educators began developing a curriculum and software structure around The Geometer's Sketchpad technology
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 Ed Dickey's Home Page  Ed Dickey
Syllabi for courses on teaching in middle/high school (mathematics), teaching mathematics with manipulatives (grades 712), graphics calculators in high school mathematics, computers in mathematics education, technology to support instruction, technology
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 Electronic Games for Education in Math and Science (EGEMS)  British Columbia, Canada
Archives of a collaborative effort by scientists, educators, and professional video game and educational software developers to do research on and develop teaching materials that integrate video games and computerbased explorations with existing classroom
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 Elementary Math Enrichment  Beth Schaubroeck
Free mathematics enrichment materials for accelerated 4th and 5th graders. The 70 lessons, grouped into 12 different units, include solutions and teacher guides: Ancient Mathematics (Mayan math, in base 20; and Egyptian math, which lacks zero), Baseball
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 Elementary Thoughts: A Collection of Gail's Responses  Gail Englert
A collection of Gail Englert's responses to questions from elementary and middle school teachers, parents, and students, in the Teacher2Teacher service. Covers a wide range of elementary mathematics topics, as well as teaching points.
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 The Elverson Puzzle Co., Inc.
The creators of Jacob's Revenge, the classic wine bottle puzzle. Elverson Puzzle also makes a Tangram Puzzle, the eleven tiles of which form a 6 inch diameter circle; Farkle, a game played with extra large wooden dice or online, with the dice roller that
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 Erich's Packing Center  Erich Friedman
A huge collection of packing problems, each explained with graphics. 2D examples include Triangles in Squares, Circles in Squares, Squares in Squares, Triangles in Triangles, Circles in Triangles, Squares in Triangles, Triangles in Circles, Circles in
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 Escher Tiles  Hop David
Tessellations that resemble recognizable objects, created by the author in the style of M. C. Escher.
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 Escher Web Sketch  Wes Hardaker, Gervais Chapuis; Univ. of Lausanne, Switzerland
A Java applet that allows you to draw repeating patterns. With documentation on how to use the applet, a pattern archive, the authors' favorite patterns, a link to download Escher Web Sketch, the program's source code, and interesting crystallographic
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 The Euler Characteristic of an Orbifold (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
Suppose we have a symmetric pattern in the plane. We can make a symmetric map by subdividing the quotient orbifold into polygons, and then 'unrolling it' or 'unfolding it' to get a map in the plane...
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 Exercices de Géometrie en Troisième  T. Touré
Révisions de vacances: de nombreux exercices de géométrie. Thalès/Droite des milieux; Pythagore; Sinus, cosinus, et tangente; Cercle et triangle rectangle; Droites remarquables; Distance et aire; Volume; Symétrie, rotation
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 A Field Guide to the Orbifolds (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
Tongueincheek hints for learning how to classify the 17 types of symmetry patterns in the Euclidean plane. "...information presented in this section has been gleaned from a cryptic manuscript discovered among the personal papers of John Conway after
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 Flags  Jim Croft
Links to information about and images of flags of the world: National Flags, International Maritime Signal Flags, Motor Racing Flags, the Semaphore Flag Signaling System, and others, good for exploring symmetry in flag design.
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 Folding Shapes: Are the Sides the Same?  Janna Moore
Folding Shapes gives students practice in identifying typical shapes (circles, squares, triangles, etc.), in the process of teaching about lines of symmetry. Students find symmetry in alphabet cutouts by folding them in half in various ways.
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 The Four Types of Symmetry in the Plane  Susan Addington
Rotation, translation, reflection, and glide reflection, with illustrations and problems for consideration. Written by Dr. Susan Addington for the California Math Show; presented on the Web by Suzanne Alejandre as part of her Tessellation Tutorials.
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 Fractals, Art, Pattern, Symmetry...  Clifford A. Pickover
Melding art, science, mathematics and other seemingly disparate areas of human endeavor. Links to detailed information on Pickover's books (of particular math interest are: Chaos and Fractals: A Computer Graphical Journey; The Loom of God; Fractal Horizons
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 Frieze Patterns  David A. Reid
A frieze is a pattern which repeats in one direction. Friezes are often seen as ornaments in architecture. A mathematical analysis reveals that there are seven different frieze patterns possible. These resources were designed for a standardsbased project
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 Frieze Patterns in Cast Iron  University of Cambridge
The 'Sydney Lace' of Australia and the rest of the world, from Paris and New Orleans to Brisbane and Melbourne: examples of all seven frieze pattern types in cast iron decorative work.
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 FroZenLight  Zoltan Palmer
Interrelating line arts, mathematics, and cryptography, this free Java application simulates experiments in classic optics. Use FroZenLight to see how beaming a light ray onto circular mirrors would produce reflection patterns ranging from the symmetric
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 The FUNKtion  Appreciating Math through Art  ThinkQuest 1996
How to do really hard graphic stuff. Tilings, Fractals, Sierpinski Gaskets, animations, pictures, links. Math "crash courses" on Complex Numbers; Magnitude and the Pythagorean Theorum; Functions; and 3D.
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 GANG Symmetry Library  James M. Lawrence; Dept. of Mathematics & Statistics, Univ. of Massachusetts, Amherst
Symmetry detection software. Input file, sample output, graphs, and download. From the Center for Geometry Analysis Numerics and Graphics.
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 Gecko Stone  John August
Tessellated, interlocking concrete pavers and molds. Like Escher's reptile, the tessellating shapes are modified hexagons.
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 The Generalized Theory of Life  Vyacheslav L. Kalmykov; Institute of Cell Biophysics, Russian Academy of Sciences
A paper that presents an integrally generalized theory or conceptualization of life that includes elements of an axiomatic approach and is physically interpretable, formulated as the result of attempts to invent a holistic system of creative synthetic
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 The Geometer's Sketchpad and Tessellations  Cathleen V. Sanders
A tessellation tutorial that provides stepbystep instructions for tiling the plane using the Geometer's Sketchpad. Linked to it are examples of student tessellations created by Cathi Sanders' pupils at Punahou School in Honolulu, Hawaii.
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 Geometric Models (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material describing physical objects corresponding to geometric constructions (and methods for creating such objects), particularly polytopes.
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 Geometric Topology (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material in an area of mathematics that concerns the assignment of geometric structures to topological spaces, so
that they "look like" geometric spaces. For instance, compact twodimensional surfaces can have
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 The Geometry Junkyard  David Eppstein, Theory Group, ICS, Univ. of California at Irvine
A collection of usenet clippings, web pointers, lecture notes, research excerpts, papers, abstracts, programs, problems, and other stuff related to discrete and computational geometry  some serious and much also entertaining. Junk sorted into piles (Topics):
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 Geometry of Fullerenes  Slavik Jablan, Mathematical Institute, Belgrade
Fullerenes (Buckminster Fuller) are considered from the points of view of geometry, graph theory and knot theory. Contents: Introduction; General fullerenes, graphs, symmetry and isomers; 5/6 fullerenes; Knot theory and fullerenes; "Perfect" fullerenes;
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 Geometry Out of Africa  Ivars Peterson (MathTrek)
Two recent books that introduce African geometry. Geometry from Africa, by Paulus Gerdes, shows how one can discover or derive the Pythagorean theorem from African designs, explores the role of symmetry (with suggestions for home and classroom crafts
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 Geometry (Seeing is Believing)  De Silva, D'Andreti; ThinkQuest 1997
A geometry tutorial, comprising: the basic building blocks of geometry; segments and rays; angles; parallel lines; triangles; polygons; right triangles (Pythagorean theorem); geometric inequalities; ratios, proportions and similarities; trigonometry;
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 Geometry Web Page  Ephraim Fithian; Kutztown University
Sixteen geometry activities: Tangrams; Golden Ratio; Geoboard; Polygons; Symmetry; Constructions; Polyominoes and Polyiamonds; Star Polygons; Special Points in Polygons; Polygon Tessellations; Kaleidoscope Tessellations; Escher Tessellations; Space Cubes;
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 Great Math Programs  Xah Lee
A listing about 40 excellent recreational math programs for Macintosh that do: polyhedra and Rubic cubes, curves and surfaces, fractals and Lsystems, tilings and symmetry, game of hex and game of life, chess and fiveinarow, peg solitare and polyominos,
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 Greg Egan's Home Page
Information, illustrations, and Java applets that supplement some of the works of the science fiction author. The Applets Gallery includes groups of rotations in three dimensions and in four dimensions; Escher, inspired by the artist's conflicting orientation
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 Heureka: páginas lúdicomatemáticas  Rodolfo Valeiras Reina
Home page of a Spanish math enthusiast from Cádiz, España. Calidociclos (a kaleidocycle is a ring of tetrahedra joined edge to edge; as you twist it inward or outward, the tessellations on the faces of the tetrahedra meet in new ways); Problemas
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 Highland Tessellations  Karen M. Stagge, Highland Middle School, Libertyville, IL
Tessellations A La Escher: an introductory unit based on MECC's Tesselmania! software and the graphics of M.C. Escher, to help students realize the importance of mathematics in the art world. Also links to work done by students at other schools.
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 High School Projects (COMAP)  Consortium for Mathematics and Its Applications
Products developed in print, video, and multimedia formats, supporting the NCTM Standards, to bring mathematical modeling into high schools around the country, and present mathematics in contexts that are relevant to students. Curriculum: Mathematics:
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 HOT Math  Pauline Vos
Handson tasks in the mathematics classroom, a resource site for teachers of students around 14 years of age. Among the topics are angles, symmetry, scale drawing, balance algebra, tables, combinations, statistics, probability, reasoning, finding rules
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 Hufnagel's Software  Hank Hufnagel
Math games that reinforce basic operations and symmetry: Mouse Math, SpiroMath, and SymmeToy.
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 Hyperbolic Tessellations  David Joyce; Dept. of Mathematics & Computer Science, Clark University
The hyperbolic plane has more space in it then the Euclidean plane. There are only three regular tessellations of the Euclidean plane, namely {3,6} in which equilateral triangles meet six at each vertex, {4,4} in which squares meet four at each vertex,
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 Hyperbolic Tilings  Sascha Rogmann
A link to Rogmann's thesis (in German) and to pages on shells around the dodecahedron; 1+110+3458 cubes; hyperbolic buckyballs; hyperbolic animations; triangletessellations; quadrangletessellations; twoholed cloth torus; Subsets of PSL_2(Z[w]), D=3,
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 IMATH  Cathleen V. Sanders
A professional development course for teachers designed to help them integrate technology and the NCTM Standards into their teaching. Although this course is only available for credit to Hawaii teachers, teachers from other states may participate in the
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 Interactive and animated Geometric Algebra with Cinderella  Eckhard Hitzer
More than 90 JAVA applets that range from the elementary concepts of vector, bivector, outer product, and rotations, to triangle relationships, oscillations, and polarized waves. A special group of 21 applets illustrates three geometrically different
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 Interactive Math  Christopher Henrich
Topics in recreational mathematics, including polyhedra, kaleidoscopes, magic squares, and trochoids, presented with interactive applets.
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 Intriguing Tessellations  Marjorie Rice
Famous for discovering four hitherto unknown tilings of pentagons, Marjorie Rice got her start in tessellations reading an article by Martin Gardner. This site features mathematical explanations and examples of several tessellations using pentagons, as
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 Introduction to Geometry for Primary Students  Varnelle Moore
A Web unit designed to introduce young children (K2) to beginning concepts in geometry: describing shapes, making patterns, building shapes, rotational symmetry, and line symmetry. Each lesson includes an interactive, manipulativebased project, technology,
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 An Introduction to Orbifolds  John Conway, Heidi Burgiel; The Geometry Center
Defining the word and computing the orbifolds of wallpaper patterns and finite objects in 3space. After compiling a list of all the features that can appear in an orbifold, these pages provide the tools needed to discover all the symmetry groups of the
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 Introduction to Symmetries  Chaim GoodmanStrauss; The Geometry Center
Studying symmetries called isometries, which preserve distances. Pattern and symmetry appear in virtually every human culture. Symmetry surrounds us, both in the natural world and in the world of human artifact... the arrangement of needles on a pine
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 Introduction to Wallpaper Patterns  Chaim GoodmanStrauss; The Geometry Center
A unit on about plane patterns whose symmetries are translations, glide reflections, and rotations. Learn how to use a computer program called Kali which generates plane patterns; create and save a design, then learn the definitions of some features of
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