Investigating Patterns


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These links are coordinated with the chapters and activities of
"Investigating Patterns: Symmetry and Tessellations"

ACTIVITY 1 (What is Mathematics?)

Title:  What is Mathematics?
Comment:  Observations on the the nature of mathematics from Pathways to School Improvement. Includes: "Mathematics should include experiences that help students to shift their thinking about mathematics and define mathematics as a study of patterns and relationships." This perspective is mirrored in the anecdote by American physicist and Nobel laureate Richard Feyman (1918-1988).
Title:  Pythagoras: Music and Space
Comment:  A brief introduction to Pythagoras' observations on the relationship between music and number. Graphics show music intervals expressed as mathematical ratios. If you have a plug-in that can play sound (like Real Player), you can pluck the strings to hear the notes.
Title:  How to Make Ukrainian Easter Eggs
Comment:  Tells you everything you need to know about the basics of making pysanky (Ukrainian Easter Eggs). Click on any design and you'll be taken to the instructions for it. Challenge students to explain why the egg at the right from Pysanky Designs (Advanced Designs) is different from the others. [Click on the egg for the solution.] Symbols and Colours can be used to investigate the geometric nature of the designs. There's even a Colouring Book for kids. 
Title:  Decorate an Easter Egg
Comment:  Decorate an Easter egg by dragging and dropping the items with your mouse. When done, print and cut it out for your own decoration. Game uses Javascript. Also available in Flash Format (requires Macromedia Flash Player).
Title:  Donald in Mathmagic Land
Comment:  Donald Duck discovers the fascinating world of mathematics, including the pentagon, the golden rectangle, and the spiral. Students see how mathematical principles influence science, art, music, architecture, and even sports.
Special:  View the entire film.


ACTIVITY 2 (Introduction to Symmetry)

Title:  ETA/Cuisenaire: Products - Plastic Mirrors
Comment:  On-line source of the 2¼" x 3½" plastic mirrors using in this and subsequent activities of Investigating Patterns: Symmetry and Tessellations. Mirrors come in sets of four. They are also available from the author [e-mail address below] in sets of three (the maximum required for all activities), two of which have been hinged with cloth tape, for $3 US including shipping.
Title:  Photo Activity for Symmetry
Comment:  The left side of a human face is close to being the mirror image of the right side - but not exactly. Includes details on an activity requiring a digital camera and a simple paint program, such as Windows 7 Paint or Ultimate Paint. Visit Symmetrical Jill Britton for another example.
Title:  SymFace
Comment:  SymFace is a simple and fun way to see how your face would look if it was perfectly symmetrical. Just follow the steps using an appropriate digital picture of yourself. [If necessary, use Windows 7 Paint or Ultimate Paint to crop and/or resize your picture. Visit Ultimate Paint Pointers for details.] The results can be saved and/or printed. Requires a JAVA-capable browser.
Title:  The Butterfly Web Site
Comment:  The most complete information on butterfly gardening, farming, ecology, and education. Includes a link to a Photo Gallery containing butterfly and moth images. Look for images with a symmetrical configuration. In most cases, a larger picture is available by clicking on a small picture.
Title:  Manual of Traffic Signs
Comment:  Contains graphics of the most commonly used traffic signs in the United States. You can use the site to challenge students to find symmetrical signs.
Title:  SymmeToy - Symmetric Art Program
Comment:  Permits downloading of a trial version of SymmeToy, a Windows program for creating paint patterns, symmetry roses, tessellating art, and so forth.


ACTIVITY 3 (Symmetry in the Alphabet)

Title:  On Reflection
Comment:  Details how some clever "secret" coding can be done using a small mirror and the reflectional symmetry characteristics of many capital letters and some numbers.
Title:  Inversions
Comment:  The inversions gallery of Puzzlemaster Scott Kim, whom Isaac Asimov called "The Escher of the Alphabet". An inversion is a word or name written so that it reads in more than one way. Some inversions read upside down, some flip in a mirror. Some read as two different words, some read as the same word both ways. Web page includes a link to Teacher Resources for Inversions with classroom exercises. Half Words is an intriguing and challenging warm-up exercise suitable for printing. The corresponding teacher notes includes Answers. Don't miss Scott's Origami with its animated vertical reflectional symmetry. His Signature Piece also makes for interesting animations. 
Title:  Ambigrams, Logos, Word Art
Comment:  The generic term for inversions is ambigrams, a word coined by cognitive scientist Douglas Hofstadter. This is the ambigram webpage of John Langdon, known for his Angels & Demons ambigrams. The Ambigram.Matic, the only FREE ambigram generator on the Internet, allows you to create your own ambigram that exhibits rotational symmetry. Ambigram Generator permits the creation of quality ambigrams with different properties. An expensive fee is demanded if you wish to download your graphic. 


ACTIVITY 4 (Symmetry in Flags, First Nations Art, and Quilt Blocks)

Title:  Flags
Comment:  Contains links to images of National Flags, International Maritime Signal Flags, Motor Racing Flags, and the Semaphore Flag Signaling System. Additional flag images are available from the World Flag Database and the Flags of the World web site. The latter includes links to an on-line Colouring Book of Flags with images available in three sizes.
Title:  Binary Numbers and the South Korean Flag
Comment:  What do the markings on the flag of the Republic of South Korea have to do with the binary number system? What do they have to do with the number 7? Can you locate and identify the binary numbers in this Chinese Zodiac Paper Cut (requires Adobe Acrobat Reader)?
Title:  Carving at Skidegate
Comment:  A photo essay of a symmetrical totem pole being carved at Skidegate, Haida Gwaii (Queen Charlotte Islands, British Columbia, Canada). The photos show the carving process from raw log to finished pole.
Title:  Shapes, Lines, Angles & Quilts
Comment:  In this site, quilting is a hands-on application for geometric principles. Requires the copying of traditional quilt block designs from sources such as Quilt Blocks Galore and Quilt Block Collection (requires Adobe Acrobat Reader). For the best commercial resource, consult Mary Ellen Hopkin's The It's Okay If You Sit on My Quilt Book with over 350 color examples of geometric quilt blocks on a gridded background.
Title:  Quilt Maker
Comment:  Design a quilt block by clicking on the squares. Make a quilt by selecting REPEAT, REFLECT or ROTATE. Includes an index of quilt block patterns. Quilters (from ProtoZone) features the same activity in a different configuration. Quilt Reflect challenges students to complete the symmetrical design of a quilt while racing against the clock. All require Macromedia Shockwave Plug-in.
Title:  Quilt Design Program Mini-Reviews
Comment:  Contains a brief review of several quilt design programs. Links permit you to download a demo version of some programs. In the demo version of PCQuilt (Windows or MacIntosh), you will be able to try most of the features of the full version but some of the menu items will be disabled. Quilt-Pro is the Windows-based program of choice.


ACTIVITY 5 (More on Reflectional and Rotational Symmetry)

Title:  Topsy-turvy Matchbox Art
Comment:  Weird and wonderful, this amazing collection of images celebrates the unique art form of topsy-turvy faces. Cards can be viewed upside down or right side up (or is it the other way around?) - either way, a face appears. Cracker Jack's 2-Face Stickers has further examples. This upside down ad for Niblets Corn appeared on the back cover of Life Magazine in 1953.
Title:  The Upside Downs of Gustave Verbeek
Comment:  Gustave Verbeek published a series of upside down comics in The Sunday New York Herald in the beginning of the 1900's. The first part of the comic is read normally, then you can turn it upside down and the story continues! Contains links to five complete comics. In a similar vein, this ad for Niblets Corn appeared on the back cover of Life Magazine in 1953.
Title:  Symmetry
Comment:  A Flash movie by Mike Shepperd featuring two geometric shapes - one with 4th order rotational symmetry and no reflectional symmetry - the other with 2nd order rotational symmetry and 2 lines of reflectional symmetry. Requires Macromedia Flash Player.
Title:  The Diamond 16 Puzzle
Comment:  Rearrange the rows, columns, and quadrants in a 4 x 4 array of 2-colour tiles to solve this online Javascript puzzle and make hundreds of symmetric patterns. For a somewhat more user-friendly version of the puzzle, visit Alexei Vernitski's 16 site. 
Title:  Symmetry in Car Logos
Comment:  Most people associate symmetrical logos with more ethical, socially responsible behavior so many car manufacturers use symmetry of some kind in designing their logos. This Powerpoint presentation explores several examples in an exercise format.
Title:  Car Wheel Covers
Comment:  A good source of objects that have rotational symmetry happens to be hub caps. Canadian Tire Wheel Covers features classic designs. Students can investigate the designs for reflectional and/or rotational symmetry.
Title:  Hex Signs
Comment:  Features details on several symmetrical hex signs with internal links to quality enlargements thereof.
Title:  The Best Brands of the World
Comment:  Features a collection of over 60,000 corporate logos. Challenge students to find familiar logos with reflectional and/or rotational symmetry or to prepare logo exercises. Search by company name. [Downloads are EPS files. Open with a high-end graphics program like Photoshop or Photoshop Elements, specify a large resolution, then adjust to the desired size. Or insert into a Word document (using the Insert / Picture / From File Option) provided the Word EPS filter is installed.] Logotypes has a similar format and usage.
Title:  GoodLogo!com
Comment:  Logo database is searchable by company name and category, but lacks many major corporations. Cick on thumbnails to access enlargements (gif files). Saved graphics can be opened with a simple graphics program or inserted directly into a Word file (using the Insert / Picture / From File Option).
Title:  Family Crest
Comment:  There are over 4,500 Ka-Mon or family crests in Japan. Artistically simpler than their European counterparts, they are a wonderful cultural source for symmetry investigations. This web page provides an excellent introduction to the history and structure of the crests. Japanese Crests has links to highly stylized, symmetrical crests based on butterflies. Kamon features a chart of 259 Japanese family crests.
Title:  Symmetry Webquest
Comment:  A collection of web pages by students of St Kieran's ManlyVale (Sydney, Australia) to help children around the world find out about reflectional symmetry and rotational symmetry. Elegant illustrations include examples from nature, the human body, masks, reflections in water, architecture, geometric shapes, hubcaps, kaleidoscopes, capital letters, advertising logos, and more.
Title:  Symmetry Artist
Comment:  Mathematics and Art come together! Create your own line drawing with reflectional and/or rotational symmetry. Applet permits selection of mirror line orientation, the order of rotation, and the pen width, color and style. Part of Maths Is Fun website. Requires a JAVA-capable browser.


ACTIVITY 6 (Symmetrical Strip Patterns)

Title:  Frieze Patterns
Comment:  Details the seven combinations of symmetries that can be found in a frieze pattern. Sequential pages include exercises and corresponding answer key.
Title:  Frieze Patterns in Cast Iron
Comment:  Contains examples of all seven frieze pattern types in cast iron decorative work - in the Sidney Lace of Australia - and the rest of the world. You can find cast iron patterns everywhere!
Title:  Border Liner
Comment:  Make patterns which go on and on forever, repeating the same pattern in one direction or the other. BorderLiners (from ProtoZone) features a similar interactive activity. Sampler (also from ProtoZone) allows you design a cross-stitch sampler that stretches out into seven border patterns. All require Macromedia Shockwave Plug-in. 
Title:  Great Math Fun With Paper Doll Strings
Comment:  With strings of paper dolls, you can decorate a party, make the border of a bulletin board, create a frame for your favourite picture, and of course, learn some math in the process. Visit How To Make Paper Chains to access links to templates for making Christmas chains, dinosaur chains, Halloween chains and wild animal chains.
Title:  Celtic Knot-Design Bookmark
Comment:  Contains instructions for making a symmetrical Celtic Bookmark. Download a high-res version suitable for coloring.
Design custom Celtic artwork with the amazing Celtic Knot Font. Outline style highly recommended.


ACTIVITY 7 (Introduction to Polygons)

Title:  Connect the Dots
Comment:  Select a numbered dot to begin. Draw a polygon as you click on the dots sequentially to complete the image. There is an animation on completion. Seven different examples with a US Air Force theme. Requires Macromedia Flash Player.
Title:  Dot2Dot - Dinosaur
Comment:  Click on each number in order starting from 1, continuing on until the last number is reached and see what you find. Requires a JAVA-capable browser. Connect the Dots with Artt Games offers a choice of level of difficulty. Dot to Dot uses a virtual pencil and joins the points with colored straight line segments. The latter pages require Macromedia Flash Player.
Title:  Lizards of M. C. Escher SoftPuzzles
Comment:  Through this site you can purchase the Lizards of M. C. Escher SoftPuzzle, an officially licensed adaptation of Escher's famous reptile.
Title:  Tangrams
Comment:  Tangram is a centuries old Chinese tile game consisting of 7 geometric shapes. Learn its history, make your own tan set, or try solving one of the shapes. Includes great downloads and interesting links to explore. Rescued from the internet by Web Archive.
Title:  Tangram
Comment:  To play this on-line tangram game, you must flip, rotate and drag the seven pieces to exactly reproduce the given image. When you have mastered the 49 pre-designed patterns, you can make up your own. Enchanted Mind - Tangram Java Puzzle features a similar applet with 18 patterns. Both require a JAVA-capable browser. For a Flash version, visit Interactive Tangram or Tangram Game. In A Movie in Flash Using Tangrams, some 20 tangrams are used to tell a story with on-line questions. All three require Macromedia Flash Player. Windows users can download (includes an editor that lets you create your own collection of puzzles). Decompress with WinZip or freeware ZipCentral. A wooden version is available from Hungry? Try Puzzookies - a commercial tangram cookie - or bake your own!

ACTIVITY 8 (Regular Polygons)
Title:  Naming Polygons and Polyhedra
Comment:  What do you call an 11-sided polygon? A 14-sided polygon? A 50-sided polygon. Doctor Math provides Greek prefixes for the first 50 polygons, among others.
Title:  Welcome to the Polygon Playground
Comment:  Come play with polygons! You can drag and arrange hundreds of colorful polygons to create your own online art. The introductory page includes a little slide show of ideas to get started, including: patterns, symmetrical designs, and polygon animals. Requires a JAVA-capable browser.
Title:  Plentiful Pentagons
Comment:  The pentagonal shape is one of the primary geometric forms for organic shapes (living organisms). Many flowers and starfish have pentagonal shapes.
Title:  Snowflakes
Comment:  All snowflakes have a fundamentally hexagonal shape. Generate symmetrical snowflakes with this project. Set the slider bar to thin line (left) or thick line (right), then press draw.  Requires Microworld's Web Player. Or you can Create Your Own Snowflakes and then execute a 2D or 3D rotation (requires Macromedia Flash Player). SnowFlaker (from ProtoZone) challenges you to draw the same snowflake twice (requires Macromedia Shockwave Plug-in).
Title:  Kids Snow Page
Comment:  Includes links like Snow Science (snow crystals and how to study them) and Snow Art (paper snowflakes and a "self-sticking" snowflake). Learn how to Grow Your Own Borax Crystal Snowflake. All the snow activities are available as an Acrobat Reader document, snowbook.pdf.
Title:  Paper Snowflakes for Children
Comment:  Everyone can make artist quality paper "snowflakes" by following these easy patterns! Many patterns were adapted from pictures of real ice crystals. Make Mother's Day, Valentine's Day, Halloween Snowflakes as well as Christmas snowflakes!
Title:  Wilson Snowflake Bentley
Comment:  Provides a comprehensive resource of everything related to Wilson "Snowflake" Bentley. Links to Original Wilson Bentley Images, a collection of 21 of Bentley's snow crystal photomicrographs. Click on each thumbnail for a closeup view.
Title:  The Giant's Causeway
Comment:  The Giant's Causeway in Northern Ireland is made of 40,000 interlocking basalt columns, most of them hexagonal, resulting from an ancient volcanic eruption.
Title:  Geometric Dissections on the Web
Comment:  A geometric dissection is a cutting of one or more figures into pieces that can be rearranged to form other figures. Features links to animations of dissections (such as Swing-Hinged Dissection of Triangle to Square and Dodecagon to a Hexagon), dissections as puzzles, and more.


ACTIVITY 9 (Symmetry and Regular Polygons)

Title:  BBC Education - Revision Bite - Symmetry
Comment:  Interactive animated treatment of reflectional and rotational symmetry, including symmetry in regular polygons and other polygonal shapes. Includes classroom exercises.
Title:  Let's Make a Flexagon
Comment:  Make a tri-hexa-flexagon that produces six different symmetrical hexagonal patterns. Includes links to a YouTube video of the flexagon in action and to a flexagon template (requires Adobe Acrobat Reader).
Title:  Foto-TriHexaFlexagon
Comment:  Got a colored printer? This simple flexagon program by Fernando G. Sörensen of Argentina will allow you to create a pictorial trihexaflexagon from three images. Includes detailed instructions (uses Windows 7 Paint or Ultimate Paint) and a link to a download of the program file.


ACTIVITY 10 (Symmetry by Paper Folding and Cutting)

Title:  Snowflakes
Comment:  Contains simple instructions for folding and then cutting snowflake-like designs with two, four, six, and eight repeats. Excellent supporting graphics.


ACTIVITY 11 (Paper & Scissors Pentagon)

Title:  5-Pointed Star in One Snip
Comment:  Take a thin piece of paper 8-1/2" x 10" (or an exact proportion thereof), fold it as indicated and cut yourself a perfect 5-pointed star. Printable Version requires Adobe Acrobat Reader.
Title:  Some Golden Geometry
Comment:  A sequence of linked pages leading from the Golden Rectangle to the Golden Spiral, the Golden Section, and ultimately to the Golden Triangle. Original webpage by Rashomon.


ACTIVITY 12 (Paper & Scissors Hexagon and Beyond)

Title:  Folding and Cutting a Snowflake
Comment:  Features instructions (require Adobe Acrobat Reader) for folding and cutting a hexagon. The cutouts in the edges of the folded paper create a snowflake appearance. Paper Snowflake Designs has suggestions for artistic cutouts.
Title:  Make-a-Flake
Comment:  Allows you to cut polygonal wedges from a virtual piece of folded paper with virtual scissors (requires Macromedia Flash Player). Click on Make Your Own Snowflake to begin. Place the (red dot on the) scissors on an edge of the triangle, then click the mouse button. Move the scissors within the triangle, then click again. Continue until the scissors return to an edge. A polygonal wedge will be removed. Repeat, using Undo/Redo as required. Once you Preview Flake, you can Cut Some More or Start Over. Snow Days has a comparable interface. Snowflake Designer produces a similar result - just click your mouse and pass it over the circular sector to create each cutout. Or try Snowflake Builder which allows you to save your snowflake and animate it.
Title:  Paper & Scissors Heptagon
Comment:  Extend your paper folding skills beyond the pentagon and hexagon. Features instructions for folding a heptagon. Or try an Octagon, Nonagon, Decagon (based on a pentagon), 11-gon or Dodecagon (based on a hexagon). All require Adobe Acrobat Reader. The instructions were scanned from Paper and Scissors Polygons, the first edition of Paper and Scissors Polygons and More.
Title:  Kirigami
Comment:  Kirigami is the Japanese art of folding and cutting paper. This program lets you create similar figures on your computer by cutting a polygonal wedge from a virtual piece of folded paper. You can change the regular polygon, your wedge, and your paper color. You can then see the figure that results when the paper is unfolded. Requires a JAVA-capable browser.
Title:  Paper Cutting
Comment:  Become a paper-cutting artist and create beautiful symmetric designs. Choose the way you want the paper to fold. Select your color or pattern and start cutting! Click the "Show" button to see your creation. Requires Macromedia Flash Player.
Title:  Snowflake Factory
Comment:  Make your own virtual symmetrical snowflake ... with any number of sides. Select the number of wedges you want to cut through. Select a hole punch. Click on the white wedge and see it go through to all the other wedges! Select a color for your snowflake, and then an animation. Requires Macromedia Flash Player
Title:  Vector Snowflake Application
Comment:  Draw your own snowflakes and save them in PDF format. Clean, cross-platform, open-source, and able to generate closed-polygon vector shape output. The companion SymmetriSketch allows you start with a polygon with any number of sides, to pick and choose different symmetry properties, and then create all kinds of different things that would never be mistaken for frozen water. 
Title:  Origami Help for Beginners
Comment:  Origami is the Japanese art of folding paper. A folded model is both a piece of art and a geometric figure. Learn these simple symbols and directions and you'll be able to make almost any well-diagrammed model in any book.
Title:  Joseph Wu's Origami Page
Comment:  The home page of origami master Joseph Wu. Contains links to the most comprehensive collection of origami materials available on the world wide web. Accessible diagrams include the traditional Origami Crane (a beautiful presentation with very clear instructions presented using photographs).
Title:  Origami Diagrams on the Web
Comment:  Includes classic folding techniques and origami diagrams for models (like a simple Jumping Frog) arranged according to level of difficulty. Jasper's Origami Menagerie features galleries of origami models and links to other origami information on the web.
Title:  Pieces 'n Creases - A Fun Guide to Origami
Comment:  After you visit this web site, you won't able to pick up a square piece of paper without a little folding. Includes links to History of Origami and Fun Activities. Winner of the 1999 Think Quest Junior "Best of Contest".
Title:  Oriland: What Origami Can Be
Comment:  Oriland is a children's world made of origami - half game and half resource. In Oriland Oriversity, you can learn most basic origami points through Animated Origami Symbols (requires Macromedia Flash Player), explore Base Folds that are the basis for folding many models, then take ten Practical Lessons for practicing the folds from the most basic to the complex. More models (with instructions) can be found in the Origami Studio.
Title:  Origami - The Fascination of Folding
Comment:  Fabulous origami web site. Click on any thumbnail for the corresponding instructions. Be sure to explore Origami Tablecards - a collection of 31 tablecards folded from a square.
Title:  Tammy Yee's Origami Page - Easy Paper Folding Crafts for Children
Comment:  Wonderful original artwork combined with origami. If you have access to a color printer, illustrated squares can be printed so the finished project is already decorated.
Title:  Origami for Kids
Comment:  Origami is a fascinating and creative craft for kids, especially if you can find origami models appropriate to their age. Includes links to a variety of models suitable for kids.
Title:  Diagrams, Instructions, and Puzzles
Comment:  PDF instructions for folding 9 origami models from OrigamiUSA. Options include a cup, jumping frog, blow-up bunny and traditional crane. Each requires Adobe Acrobat Reader to view and print.
Title:  Origami Bird
Comment:  Learn to fold a bird from a piece of paper by following animated visual instructions. Designed so you can repeat a step over and over if you need to, or watch the whole bird fold from start to finish. Requires Macromedia Shockwave Plug-in.
Title:  Flapping Bird
Comment:  Learn to fold a flapping bird from a single piece of paper. Excellent graphics. Also available in PDF Format (requires Adobe Acrobat Reader). Animated Origami Diagrams for Flapping Bird and Crane requires Macromedia Flash Player.
Title:  Fabric Origami Workshop
Comment:  Features instructions, photos, and 50 detailed diagrams to make origami boxes and other items by folding fabric instead of paper. Diagrams require Adobe Acrobat Reader.
Title:  Robert Neale's Magic Pinwheel
Comment:  A uniquely wonderful modular design. Despite the fact that the design of the modules is extremely simple, the octagonal wreath can be compressed (in stages) until it becomes a pinwheel. In both these forms, and at each stage in between, the design remains stable. Requires Adobe Acrobat Reader.
Title:  Tea Bag Folding
Comment:  Tea Bag Folding (also known as Miniature Kaleidoscopic Origami) is used to create symmetrical designs and medallions. Includes links to instructions, printable tiles, and commercial sources. Be sure to access the Tea Bag Medallions (Using Tea Bag Wrappers) by Gizella Hagedus.


ACTIVITY 13 (Angle Relations in Triangles and Quadrilaterals)

Title:  Angle Relations in Triangles and Quadrilaterals
Comment:  How do I know that every triangle has three angles that add up to 180° and that all quadrilaterals have four angles that add up to 360°? Well, try this!
Title:  Triangle Geometry: Triangles
Comment:  Two applets that demonstrate using animation that the sum of the measures of the interior angles of a triangle is 180 degrees. Before you start the second applet, get a piece paper and a pair of scissors so that you can try to perform the demonstration for yourself. Requires a JAVA-capable browser.
Title:  The Sum of the Interior Angles of Any Triangle - By Folding
Comment:  This dynamic worksheet contains a simulation of folding the vertices of a triangle to a point on the base, to illustrate the sum of the interior angles of a triangle. Requires a JAVA-capable browser.
Title:  Triangle
Comment:  A Flash movie (by Mike Shepperd) showing the sum of the interior angles of a triangle. Requires Macromedia Flash Player).
Title:  Paper Folding
Comment:  All of the constructions of Plane Euclidean Geometry that can be performed by straightedge and compass can also be performed by folding and creasing paper. Details how to illustrate that the three interior angles of a triangle is 180 degrees by making just three specific folds to a triangle cut from a sheet of paper.


ACTIVITY 14 (Angle Measures in Regular Polygons)

Title:  Interior Angles of Polygons
Comment:  Details two more methods for finding a formula for the measure of each interior angle of a regular polygon. Both results are equivalent to the formula deduced in the book activity, namely: 180 - 360/n. Web site is part of Cool
Title:  BBC Education - Revision Bite - Polygons
Comment:  Considers regular and irregular polygons, angle properties of polygons, and calculating the interior and exterior angles of regular polygons. Includes classroom exercises.


ACTIVITY 15 (The Regular Tessellations)

Title:  Totally Tessellated
Comment:  A superbly well-done site on tessellations. Winner in the ThinkQuest'98 competition. Features over 100 dynamic pages with quality illustrations, photographs, and links. Main topics: History of Tessellations, Essentials, Mosaics/Tilings, Escher, and Beyond. Background Information has links to polygons and angles, symmetry and transformations, regular polygons, plus links to a simple introduction to tessellations and to the effect of color usage on tessellated designs. Regular Tessellations contains links to real examples.
Comment:  Includes information on all aspects of tessellations, from their history and development to complete galleries of examples by  M. C. Escher, the master of the art. Also included are extensive workshops showing how to design and produce your own artistic tessellations.
Title:  Investigating Tessellations Using Activity Pattern Blocks
Comment:  Features activities for investigating tessellations with pattern blocks as well as student tessellation samples. Includes a link to a Patterns Block Applet produced by the National Library of Virtual Mathematics. You can also experiment online with the shapes in the Pattern Blocks: Exploring Fractions with Shapes Applet by the late Jacobo Bulaevsky. Both applets require a JAVA-capable browser.
Title:  LinoTiler
Comment:  Decorate square tiles on a flat floor in a symmetrical configuration. Then create repeating wallpaper patterns from identical square tiles using wallOVERS (also from ProtoZone). Both interactive activities require Macromedia Shockwave Plug-in.
Title:  Fractiles, Inc.
Comment:  The versatile and elegant geometry of Fractiles allows you to create aesthetically pleasing designs with repeating and non-repeating patterns. And unlike pattern blocks, these MAGNETIC tiles stay put, an especially helpful feature for students with fine motor skill problems.
Title:  Mathartfun
Comment:  Mathartfun is the company that combines art, math, and fun. They make 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.
Title:  Tilings and Designs
Comment:  Commercial source of several sets of shapely tiles that lend themselves to unlimited design making, like mosaic inlays and imaginative scenes and figures.


ACTIVITY 16 (From Crackers to Honeycomb to Soap Bubbles and Beyond)

Title:  Bees, Microscopy & Mathematics
Comment:  The comb of the honeybee is organized into regular hexagonal cells, showing a repeating, symmetrical mathematical pattern. Such a pattern could be equally well achieved by cells based on the equilateral triangle or square. So why the hexagon?
Title:  Mathematics of the Honeycomb
Comment:  The elegance of the honeycomb was not fully appreciated until modern mathematical methods were applied. Excellent film courtesy of YouTube.
Title:  Bees-Online
Comment:  An educational web site about honey bees and beekeeping. Download the only free e-book on the Internet about honey bees. Bumblebee Crafts for Kids has free printable templates and instructions for insects crafts suitable for preschool, kindergarten and grade school children. The hands-on game in BeeBopper asks you to bop bees right out of their hive and leave as few as you can. The latter requires Macromedia Shockwave Plug-in.
Title:  Polar Smash
Comment:  Another game based on a tessellation of regular hexagons - designed as a Christmas promo. Either one or two players - jump up and down on the ice to send your opponent swimming. Requires Macromedia Shockwave Plug-in
Title:  Carbon Nanostructures
Comment:  Graphite is formed from carbon atoms arranged in a honeycomb pattern. Discover why this form of carbon is used in pencils. Learn about the strongest fibers known (tubes of graphite or carbon nanotubes) and their applications.
Title:  Soap Bubbles
Comment:  This site from the Exploratorium in San Francisco has information on the shape of bubbles and investigates what happens when bubble meets bubble. Contains formulae for long-life bubble solutions, links to other sites on bubbles (like Create Giant Bubbles), as well as a "bubbliography".
Title:  Interlocking Pavers & Molds
Comment:  Features tessellated, interlocking concrete pavers and molds. Like Escher's reptile, the tessellating shapes are modified hexagons.
Special:  Click on the photo above to view the tessellation of salamanders in greater detail.
Title:  Shmuzzle Puzzle
Comment:  A standard jigsaw puzzle is comprised of pieces of many different shapes that fit together only one way. A Shmuzzle Puzzle is comprised of Escher-like pieces of only one shape that fit together many ways - but only one right way! Be sure to try CyberShmuzzling (requires Macromedia Flash Player).


ACTIVITY 17 (A Regular Tessellation Prism Kaleidoscope)

Title:  Escher: Lizard/Fish/Bat
Comment:  A large color rendition of M. C. Escher's classic lizard/fish/bat tessellation. Graphic is suitable for printing as a vivid color transparency.
Title:  You Can Create a Kaleidoscope
Comment:  Features simple instructions on how to assemble a basic kaleidoscope mirror system. Finger Tips Kaleidoscope features similar directions. Make an inexpensive Kaleidoscope from a Potato Chip Can (requires Adobe Acrobat Reader). Silver Mirricard makes an excellent mirror. Heavyweight (.02" thick) clear plastic sheets are available from AccuCut.
Title:  Duck Into Kaleidoscope
Comment:  This simple devise will create hundreds of images of whatever you place inside it. Make Multiple Images of Yourself! The basic kaleidoscope is a triangle, but mirror tiles can deformed into other shapes and angles as well.
Title:  Kaleidoscope Collector - How Kaleidoscopes Work
Comment:  Features graphics showing the basis of and the dramatically different images produced by the three mirror, two mirror, twin mirror, and four mirror scope.
Title:  Kaleido-Headache
Comment:  Mind-boggling kaleidoscope show created by Ann-S-Thesia (Ann Stretton). Unfortunately the individual animations are no longer available. But this is simply phenomenal!
Title:  Symmetrizer
Comment:  Press play, then move your mouse around to see this on-line kaleidoscope in action. Allows change of repeating character and superposition of grid.How? features an animated explanation. Requires Macromedia Flash Player.
Title:  Kaleidoscope Painter
Comment:  Paint your own kaleidoscope canvas! Just drag the mouse slowly around the canvas and let the computer show your artistic talents. Requires a JAVA-capable browser.
Title:  Kaleidodraw
Comment:  This interactive kaleidoscope from the Long Island Children's Museum allows you to select the symmetry of a circular canvas, change pen points, and change color. KaleidoDraw (from ProtoZone) features a similar interactive activity. Both require Macromedia Shockwave Plug-in.
Title:  Build Yer Own
Comment:  Click and drag the shapes provided into the circle. When the blue light passes over them, they show up like a kaleidoscope image. Another Virtual Kaleidoscope features a similar interactive activity. Both require Macromedia Flash Player.
Title:  KaleidoMania! Interactive Symmetry
Comment:  KaleidoMania! is a unique, new tool for dynamically creating symmetric designs. The CD-ROM software and activities will make your students eager to explore the mathematics of symmetry.


ACTIVITY 18 (Tessellating with Irregular Polygons)

Title:  Triangle
Comment:  Every triangle tiles the plane (tessellates) as this Flash movie by Mike Shepperd will demonstrate. So does every Quadrilateral. Both demos require Macromedia Flash Player.
Title:  Perplexing Pentagons
Comment:  The regular pentagon does not tile the plane, however there are tessellating pentagons. Fourteen types have been discovered. Are all the types now known? The problem remains unsolved.
Title:  Pentagons That Tile The Plane
Comment:  The problem here is to find a single tile shape with five sides which can tile an infinite floor without leaving holes. Discusses various solutions to the problem.
Title:  Tessellating Alphabet
Comment:  Each letter of this tessellating alphabet fits together with copies of itself to tile the plane. Click a letter to see how it tiles. Some letters tile in more than one way.


ACTIVITY 20 (The Semiregular Tessellations)

Title:  Totally Tessellated: Semiregular Tessellations
Comment:  You can view color renditions and real examples of the semiregular tessellations in this page of the Totally Tessellated web site.Templates allows you to print renditions of each tessellation suitable for coloring activities.
Title:  Virtual Manipulatives: Tessellations
Comment:  Features an interactive applet which uses five regular polygons to build tessellations. Includes icons for student activities, teacher information, and assistance. Produced by the National Library of Virtual Mathematics. Requires a JAVA-capable browser.


ACTIVITY 22 (Demiregular Tessellations)

Title:  Totally Tessellated: Demiregular Tessellations
Comment:  You can view color examples of more demiregular tessellations in this page of the Totally Tessellated web site. Templates allows you to print renditions of each tessellation suitable for coloring activities.


ACTIVITY 23 (Islamic Tessellations)

Title:  Historical and Geographical Connections
Comment:  Contains historical and geographical links for tessellations and tilings in the Moorish architecture of Spain, Chinese and Japanese Designs (with JAVA applets by Dr. William Chow), Islamic Art and Architecture, and Persian Architecture.
Title:  Symmetric Patterns at the Alhambra
Comment:  Features symmetric patterns at the Alhambra with accompanying math problems. Includes a link to the Four Types of Symmetry in the Plane with connections to a number of tessellation pages.
Title:  Symmetry & Pattern: The Art of Oriental Carpets
Comment:  A joint effort between The Textile Museum and The Math Forum. Link to Rug Gallery includes images of carpets with a variety of geometrical patterns and symmetrical designs. Educational Resources contains suggestions for the classroom.
Title:  Magic Carpet
Comment:  Weave symmetries into an instant rug by drawing in a rectangular panel. Interactive software (from ProtoZone) requires Macromedia Shockwave Plug-in.
Title:  Some Arabic Tiles
Comment:  Features colorful tessellating Arabic tiles with links to an outline version of each. Download the tiles and use the paint bucket tool to make your own color schemes.
Title:  Geometrical Patterns
Comment:  Download seamless decorated tessellating tiles for your web page background or simply view the full background on a whole page. You can even change the background color. Options include Moorish (Islamic), Japanese, Celtic, and Medieval.
Title:  Islamic Art and Geometric Design
Comment:  Presenting beautiful Islamic works from the collection of The Metropolitan Museum of Art along with a series of pattern-making activities involving straightedge and compass, these materials are designed for use in the classroom and may be readily adapted by teachers to create exciting lessons in art, culture, math, and geometry. Requires Acrobat Reader. The full package on CD is available from


ACTIVITY 24 (The Art of M. C. Escher)

Title:  M. C. Escher: The Official Website
Comment:  Home page of the M.C. Escher Foundation and Cordon Art B.V., worldwide and exclusive representative of the works and writings of the artist. Includes a link to a shop of Escher merchandise. Take a Virtual Ride through three of his most famous works - Belvedere, Waterfall or Ascending and Descending (require Real Player). Read his biography and quotes, or download a FREE Escher Interactive Puzzle (self extracting Winzip file).
Title:  Escher in het Paleis (Escher in the Palace)
Comment:  A permanent Escher exhibition in the center of The Hague. Famous prints are showcased, complemented by changing displays of selected personal documents. The exhibition includes a Virtual Reality display which enables visitors to embark on a voyage of discovery through and within Escher's work.
Title:  Escher on Display
Comment:  Escher's work can be seen in several other locations in the Netherlands. They are listed here in chronological order, with accompanying photographs. A large print of Escher's Metamorphosis III used to be displayed in the Kerkplein post office in The Hague. It has been moved to Lounge 4 of Amsterdam's Schiphol airport (beyond passport control). Webpage is part of Math & The Art of M. C. Escher.
Special:  Escher in The Hague (Jill Britton's Odyssey)
Title:  M. C. Escher Mindscapes
Comment:  A wide variety of Escher resources from the National Gallery of Canada. Content is divided into five sections: Escher's Life, Escher's Themes, Escher's Techniques, Escher's Gallery (40 works from the NGC collection), and References. Includes an excerpt from the 1969 film (Adventures in Perception) featuring M.C. Escher at work, two excerpts from a video interview with the artist, plus video and sound interviews with his eldest son George. Video clips require QuickTime Player.
Title:  M. C. Escher
Comment:  A Flash experience brought to you by Renee Perry. Requires Macromedia Flash Player. Includes four sections: bio, interpretation, visual analysis, and a gallery of the artist's work. The interpretation section features a QuickTime movie of Escher's Reptiles. Nothing short of terrific!
Title:  Biography of M. C. Escher
Comment:  Comprehensive biography condensed from the book M. C. Escher - His Life and Complete Graphic Work (Bruno Ernst © 1981). Includes additional material from original essays by Escher and supporting graphics.
Title:  The Mathematical Art of M. C. Escher
Comment:  An excellent introduction to the mathematics in Escher's art. Divided into areas: tessellations, polyhedra, the shape of space, the logic of space, and self-reference.
Title:  The Shrine of Escher
Comment:  Contains links to an Escher biography and an annotated art gallery with details on eight Escher prints - including Moebius Strip II, Belvedere, Waterfall, and Day and Night.
Title:  Jill Britton's Escher Gallery
Comment:  Features links to high resolution Escher Graphics - including Drawing Hands, Moebius Strip II, Reptiles, Day and Night, and (my personal favourite) Sky and Water.
Title:  3Quarks - Applets - Metamorphose
Comment:  Presents Escher's Metamorphosis II as a panorama picture. Use the scroller to move the picture. To move it with a mouse, click on the panorama. Includes a link to Background information. Requires a JAVA-capable browser.
Title:  M. C. Escher: Metamorphose III Panorama
Comment:  Presents Escher's Metamorphosis III as a continuous band. Use your mouse to pan the image. Requires Macromedia Shockwave Plug-in. Visit Metamorphose III for a graphic version suitable for downloading.
Title:  Snakes
Comment:  Contains links to both a short computer animated movie and stills based on Escher's last graphic work Snakes. Movie requires QuickTime Player
Title:  The Mathematics in the Art of M. C. Escher
Comment:  Public lecture by the late H. S. M. Coxeter that touches on various mathematical aspects of M. C. Escher's art, its centre-piece an examination of Escher's circular woodcuts. That's a very nervous Jill Britton at the very end (46:16) asking the question "Do you regard Escher as a mathematician?" ... and requesting Coxeter's autograph in the process. Requires Real Player.
Title:  Coxeter, Martin Gardner & M. C. Escher
Comment:  Prolific mathematics and science writer Martin Gardner, known for popularizing recreational mathematics, died on May 22, 2010. He was 95. This YouTube video features the Escher excerpt from David Suzuki's 1996 "The Nature of Things" program on Gardner. View the entire video courtesy of Vimeo.
Title:  Inspirations
Comment:  Wonderful short film by Cristóbal Vila inspired by Escher's works and a free vision of what Escher's workplace could have been like. Inspirations from Maths includes brief explanatory notes about those elements in the film which have a highly mathematical nature.
Title:  Adventures in Perception
Comment:  Offers an on-line resource to Adventures in Perception, a film which details M. C. Escher's life and work, features an interview with the artist, and shows him carving a woodblock for and printing his last work Snakes. Appropriate for middle school students and above. The site allows access to a short free clip from the film (requires Real Player). A VHS video version is available for purchase.
Special:  View the entire film courtesy of YouTube.
Title:  Escher Documentary
Comment:  Featuring a rare interview with Escher himself, this new video takes an in-depth look at the world-renowned graphic artist and his ever-popular work. Traces his career path and personal life, revealing the people and places that inspired him to achieve greatness. Far too cerebral for the middle school student, but an essential resource for the mathematics educator. Approximately 60 minutes.
Title:  The Fantastic World of M. C. Escher
Comment:  Through first-person accounts by his friends and mathematicians, computer animated recreations of his work, and a look at his sources of inspiration in Italy and Spain, The Fantastic World of M.C. Escher is an engaging exploration that will enrich the understanding and appreciation of Escher and his art. Approximately 50 minutes.
Title:  Sky and Water 1
Comment:  Canadian animation wizardry brings the work of Escher to life in this playful 3-minute exercise for your brain, capturing the essence of motion in the Dutch master's print. Excerpted from the NFB film Escher, Van Gogh and Seurat: Art at Play.
Title:  Escher Interactive
Comment:  Escher Interactive is an educational and entertaining CD-ROM for Windows. Exploiting animation, video, sound commentary and interactive manipulation of images, it features nine modules, each revealing a separate facet of Escher's work.

M. C. Escher's Legacy
Comment:  The proceedings of the International Escher Congress held in Rome, Italy in July 1998 (the centenary of Escher's birth). This book contains 40 articles, richly illustrated with original art  works in addition to well-known and little-known works by Escher. A CD-ROM complements the articles, containing color illustrations of work by contemporary artists, movies, animations, and other demonstrations. New softcover edition.
Title:  Visions of Symmetry
Comment:  New edition of Doris Schattschneider's definitive book on Escher's tessellations. Features an illustrated epilogue that reveals new information about Escher's inspiration and shows how his ideas of symmetry have influenced mathematicians, computer scientists, and contemporary artists.
Title:  World of Escher
Comment:  On-line store offering high quality Escher products (T-shirts, computer accessories, ties, posters, puzzles, books, stationery).
Title:  Instructional Images
Comment:  On-line store offering high quality visual tools for math, science, and social studies educators. Contains links to Escher and World Mathematics posters.


ACTIVITIES 25 & 26 (Tessellating Template)

Title:  Square Template and Dot Paper Masters
Comment:  Make a tessellating template by sliding cutouts between opposite sides of a cardboard square, then draw the tessellation by tracing around the template with its residual corners aligned to the markings on dot paper. Contains printing masters for a sheet of nine 2-inch cardboard squares, a letter-size sheet of 2-inch dot paper, and a letter-size sheet of stretched 2-inch dot paper for tessellation sponge printing on coated paper (requires Adobe Acrobat Reader). Visit Escher in the Classroom for more details.


ACTIVITY 27 (A Pop-Up Sponge Jigsaw Puzzle)

Title:  Pop-Up Sponge
Comment:  Pop-Up (compressed sponge) starts out wafer thin and rigid. Cut out a shape, submerge it in water, then watch it expand! Used in the tessellating jigsaw puzzle activity of Investigating Patterns: Symmetry and Tessellations. Perfect for tessellating sponge printing as well. The creators of the AccuCut LetterMachine offer pop-up sponge in 9" x 12" sheets and includes either 6 sheets or 12 sheets of one color (beige, white, yellow, blue). Also available in Assorted 48 Pack (12 of each color). ICSponge is a supplier of high quality compressed sponge in 6 different colors.
Title:  Lumocolor Medium Black Permanent Marker Pen
Comment:  On-line source of Staedtler permanent medium black Lumocolor markers (S3179). Used in the tessellating jigsaw puzzle activity of Investigating Patterns: Symmetry and Tessellations.


ACTIVITY 28 (Mirroring the Craft of M. C. Escher)

Title:  Craft Foam Stamps
Comment:  YouTube video detailing how to create a craft foam stamp from a sticky back (self-adhesive) craft foam like Funky Foam or Darice Foamies. For a tessellating stamp, attach the tessellating shape to a transparent acrylic block rather than the wood block used in the video. Or fabricate your own transparent stamp mounts as detailed below. Add interior interpreting features with a ballpoint pen. Print with large stamp pads on your own dot paper ... matching the residual corners of the tessellating shape with the dots on the paper.
Title:  Acryl-Pull Handle
Comment:  Make your own transparent plastic stamp mounts. Attach large (½" x 3½") clear plexiglass handles to 3½ to 4-inch clear acrylic squares (1/8" to 3/16" thickness). Convenient 4-inch acrylic blocks are available from See the YouTube video "How to Glue Acrylic" for information on fabrication. Suggestion: Fix a 3½" square of Clear Grafix Cling Vinyl to the plate of a mount before positioning the self-adhesive tessellating craft foam region. Both the see-through material and foam can be easily removed when all is done. Used in the tessellating stamp activity of Investigating Patterns: Symmetry and Tessellations.
Title:  Carter's Felt Ink Pad Size #3 (4¼ x 7½)
Comment:  On-line source of the extra large stamp pads used in the tessellating stamp activity of Investigating Patterns: Symmetry and Tessellations. Ink with Roll-On Stamp Pad Ink.
Title:  Escher in the Classroom
Comment:  A condensation of Jill Britton's article for the proceedings of the International Escher Congress held in Rome, Italy in July 1998 (the centenary of M. C. Escher's birth). The book of the proceedings (M.C. Escher's Legacy: A Centennial Celebration) is available from
Title:  What Is a Print?
Comment:  Includes links to interactive demonstrations on the principal printmaking techniques - woodcut, etching, lithography, and screenprint. Escher was a master of both the woodcut and the lithograph. Requires Macromedia Flash Player.
Title:  Patterning
Comment:  Contains exercises to give children a new understanding about how commonly used patterns and designs based on nets are produced. Includes use of vegetable stamps.


ACTIVITY 29 (Exploring Tessellating Art)

Title:  Staedtler Mars All-Purpose Translucent Vellum
Comment:  On-line source of Staedtler 100% rag vellum (translucent bond) in 8-1/2" x 11" sheets. Used in the Escher-based activities of Investigating Patterns: Symmetry and Tessellations. Sheets feed easily into a laser printer.
Title:  The Mathematics Behind the Art of M. C. Escher
Comment:  A student project for Mathematics in Art and Architecture, a course offered by Helmer Aslaksen at the National University of Singapore. Flash Player 6 is required to view the excellent animations.
Title:  Doggy Mosaic
Comment:  Try to pave an area with the figures of the dogs shown without any gaps or overlaps. Move the dogs by dragging them. Rotate the dogs by double clicking on them. Requires Internet Explorer 4.0 or higher, with JavaScript enabled.
Title:  The Seventeen Wallpaper Tilings
Comment:  Some of the most fascinating tilings are the so-called wallpaper tilings. There are exactly seventeen of them. Click on the various icons to see an animation of the corresponding tiling being built. Each is labeled with the name used by the International Union of Crystallography since 1952. A poster of the 17 Symmetry Groups, is available from Instructional Images.
Title:  Seventeen Kinds of Wallpaper Patterns
Comment:  Features examples of the seventeen kinds of wallpaper patterns in Japanese culture. Includes Japanese nomenclature for the names of the patterns.
Title:  Tessellation Database
Comment:  View the artwork of a myriad of talented tessellation artists. Permits the searching of tessellations by subject matter. The Virtual Museum will require a VRML plug-in like Cortona3D Viewer for your web browser.
Title:  Escher Patterns
Comment:  Features tessellations by Japanese artist Yoshiaki Araki - including his classic tessellating elephant Maozou. Click on the icons to view the tessellations.
Title:  Tessellation World of Makoto Nakamura
Comment:  Includes a link to tessellation Animation and interactive Jigsaw Puzzles. Imaginative and highly recommended. Requires a JAVA-capable browser.
Title:  Ozzigami: Tessellations & Other Punography
Comment:  Features artistic and thematic tessellations with accompanying verse. Enjoy "Quantum Leap", a tessellanimation movie with Ozzie the kangaroo, as the extensive graphics load.
Title:  Escher Tilings
Comment:  A collection of Escher-like tilings by Belgian artist Jos Leys. Many ultilize hyperbolic ("circle limit") geometry. Click on each graphic for an enlarged view.
Title:  David Bailey's World of Tessellations
Comment:  Features links to the artistic tessellations of David Bailey, each generally accompanied by explanatory text, whereby the various "intricacies" behind the tessellation are discussed.
Title:  Particles of Infinity - Walk to the Garden of Escher
Comment:  Escher-like artwork by French artist and author Alain Nicholas. Companion to his book "Parcelles d'infini: Promenade au jardin d'Escher". Includes links to three galleries of tessellations.
Title:  Hop's Escher Tessellation Tiles
Comment:  Imaginative Escher-like artwork by Hollister (Hop) David. Hop's Gallery includes tessellations, geometric paintings, and various geometric designs.
David Hop's Magic Mirror Image Coloring Book is a unique collection of 30 tessellations of realistic figures ranging from bug-eyed frogs and flying pigs. 
Title:  Escher Tile Gallery
Comment:  Contains 17 Escher tiles each of which forms a repeating pattern of interlocking figures when placed side to side and vertex to vertex. Terrific!
Title:  Tessellated Polyhedra
Comment:  Build a durable model of each of the five regular polyhedra (Platonic solids) from a card stock net fitted with tabs. Each has been decorated with a tessellation of animals in full color.
Title:  Artlandia SymmetryWorks
Comment:  Artlandia SymmetryWorks is an Adobe Illustrator plug-in that lets you create the highest-quality professional ornaments, designs, and art, and interactively explore symmetry and wallpaper tilings.
Title:  M. C. Escher Font Set
Comment:  This font set is based on the lettering styles and optical artwork of M. C. Escher with both a formal lettering and a more casual handwritten-style type. Rounding out the set is a collection of extras drawn from and inspired by Escher's work.


ACTIVITY 30 (Explorations with TesselMania!)

Title:  General Instructions for TesselMania!
Comment:  At the request of its author, Kevin Lee, we are discontinuing our distribution of Tesselmania! software.  Instead, we recommend that you download the preview version (either Mac or Windows) of Kevin's new tessellation program TesselManiac! via its website It is a limited time preview and will no longer run after January 15, 2014. will be selling an educational version of TesselManiac! for $30 (with discounted lab packs) at the beginning of the new year.
Special:  Watch TesselMania's Tile Magic generate the tessellating bunny of our background using vertex rotation on a parent square.
Title:  Tess
Comment:  With Tess, you can quickly create attractive symmetric planar illustrations. While you draw, Tess will automatically maintain the symmetry group you have chosen; 11 rosette, all 7 frieze, and all 17 wallpaper groups are included. Visit Downloads to access a fully-functional evaluation version of the Mac/Win software.
Title:  Tessellate
Comment:  Allows you to generate a new tessellating shape starting from a rectangle, triangle or hexagon. Select Tessellate to see the complete pattern. You can choose several different colors to enhance the pattern, and can observe the different effects that colors have on tessellations. Requires a JAVA-capable browser.
Title:  Creating a Freeform Tessellation with Microsoft Paint
Comment:  Contains instructions for creating a freeform tessellation involving translation (slide) using the Paint software found on all Windows microcomputers. [Rescued from the Internet by Internet Archive: Wayback Machine.] The YouYube video Tessellations Using Microsoft Paint features similar instructions.
Title:  Tessellation Tutorial
Comment:  Jim McNeill's animated "how to" website for creating translation tessellations. Brief - but excellent! See also Jim's YouTube video Tessellations: How to Create Them.


More Books and Web Pages by Jill Britton
Fun with Patterns

All M. C. Escher works (c) Cordon Art B.V. - Baarn - the Netherlands.
Used by permission. All rights reserved.

NCTM Illuminations

Jill Britton Home Page
Copyright Jill Britton