Investigating Patterns

TOPIC 1 (Topology / Anamorphic Art)

Title:  Rubber Sheet Geometry
Comment:  Five sequential pages providing a brief introduction to topology or "rubber sheet geometry". Includes a simple explanation of genus with an accompanying interactive Exercise on Classification. Dental Dam or Rubber Dam makes an excellent rubber sheet for student investigations. Add a large circle with a suitable marker, then deform it into an ellipse, a square, a triangle, or any other simple closed curve.
Title:  Topology for Tots
Comment:  We grapple with topology from the very beginning of our lives. American mathematician Edward Kasner found it easier to teach topology to kids than to grownups because "kids haven't been brain-washed by geometry". Contains everyday examples.
Title:  Topological Equivalence
Comment:  Two figures are said to be topologically equivalent if one can be transformed into the same shape as the other without connecting or disconnecting any points. Distorted as viewed in a fun-house mirror, Jill Britton's face is topologically equivalent to its rippling counterpart: a single point and its neighbourhood on one correspond to a single point and its neighbourhood on the other. Requires a JAVA-capable browser.
Title:  AlexWarp
Comment:  Click on any part of the photo of Bill Gates, hold the left mouse button down, then drag it to "warp" the photo into a topologically equivalent distortion. Or warp your Favorite Celebrities. You can even warp the Mona Lisa. Explore the warping effect on a rectangular grid in Warp Image. Visit AlexWarp Info for information on putting AlexWarp on your site - as in Warp Jill Britton. You can also run AlexWarp using Java Web Start, which will let you save your creations. All require a JAVA-capable browser.
Title:  QGoo v1.1
Comment:  QGoo is another image morphing applet producing topologically equivalent distortions. Click and drag your mouse on the image using the various settings from the menu. Experiment with other than straight line motions.
QGoo v1.3, the most recent version, includes a pencil tool to add dirt, mustaches, and more. Morph George Bush to work out your frustrations. Page provides instructions on how to create your own QGoo applet. Both versions require a JAVA-capable browser.
Title:  Anamorphic Art: Cylindrical Mirror Anamorphoses
Comment:  Anamorphic art is an art form which distorts an image on a grid and then rebuilds it using a curved mirror. Create your own anamorphic art by printing this Cylindrical Grid. It was used by Jessica Kwasnica to create an Anamorphic Giraffe and by Joey Rollo to create an Anamorphic Elephant. All three files require Adobe Acrobat Reader. A 5 x 8-inch rectangle of flexible Silvered Mylar (2 ml or 5 ml thickness) rolled into a cylinder will make an acceptable mirror.
Title:  Art of Anamorphosis
Comment:  Includes links to What is Anamorphosis?, The Exhibition (with internal links to 13 panels giving an overview), Anamorphosis Gallery, Anamorphosis Software (Anamorph Me!), and Anamorphosis Links. Anamorph Me! can read images in the most common formats and carry out a range of anamorphic transformations on them - including cylindrical mirror (Anamorphic Kitten). Download the software, then decompress with WinZip or freeware ZipCentral. [A 5 x 8-inch rectangle of flexible Silvered Mylar (2 ml or 5 ml thickness) rolled into a cylinder will make an acceptable mirror.] Panels require Adobe Acrobat Reader.
Title:  The Ambassadors
Comment:  Holbein's The Ambassadors (1533) is a famous example of anamorphosis. Do you see the strange object on the floor? Close your left eye, put your face close to the computer screen near the right side of the picture. You will then see a skull! If you can't get it to work, you can cheat and look at a picture of it.
Title:  The Magic Mirror - An Antique Optical Toy
Comment:  Authentic replica of the famed antique toy book complete with a mylar sheet to transform anamorphic images into delightful full color pictures. Another source is The Magic Cylinder Book. The former includes 24 color plates from the original collection at the New York City Museum. [Download the 24 plates as an Acrobat Reader file. A 5 x 8-inch rectangle of flexible Silvered Mylar (5 ml thickness recommended) rolled into a cylinder will make an acceptable mirror.] The latter includes a collection of pictures to view and/or color and an anamorphic art grid (suitable for photocopying) to produce such pictures for yourself.
Title:  Anamorphic Photographs
Comment:  Includes internal links to What Is An Anamorphic Image? and Mirror Anamorphs. The website features several unique visual examples.
Title:  The Anamorphic Art of Kelly M. Houle
Comment:  Artist Kelly Houle's web page includes a link to six of her anamorphic paintings - including Escher 1: Double Reflection and Escher 2: Infinite Reflection. For each, move the cursor over the picture to add the mirror. Kelly's book Gracie's Gallery features activities appropriate for middle school students.
Title:  Morph Magic
Comment:  Out of the mists of history comes a new perspective on a age old pastime. Once the anamorphic jigsaw puzzle has been assembled, the ancient science of the Morph Magic Mirror lets you discover the hidden image. Of particular interest, "The Well" takes you to M. C. Escher's imaginary garden. Step through the gate into this world of the mind and keep an eye out for the master himself.
Title:  The London Underground
Comment:  A map of the London Underground will reveal the layman's need for topological distortions. Such maps show each subway line in a different color, plus the stations on each line. They clearly tell riders what line to take and where to change lines, but are not drawn to scale and do not match geographic reality. This web page includes background information on the underground and its map, suggestions for investigatory activities, and a brief introduction to topology. For a typical comparison of map versus reality, access The London Underground Diagram. A London Tube Map suitable for printing will require Adobe Acrobat Reader.
Title:  Beck's London Underground Map
Comment:  Details the creation and public introduction of the first practical map of the London Underground System. Includes a link to an Enlarged View of a first edition version of the map (1933). A 1909 PUNCH Cartoon reflects the anxieties and spectacle of traveling by "Tube" before Harry Beck completed his schematic map in 1931. For more details on the map design, consult Ken Garland's book Mr Beck's Underground Map. Visit Design Classics: London Underground Map for a historical video, courtesy of YouTube.
Title:  Topological Alphabet Chart
Comment:  The twenty-six letters of our alphabet can be sorted into nine different classes so that all the letters within each class are topologically equivalent and no letters from different classes are topologically equivalent. Print the Alphabet Cards on card stock, then cut them out. Using the chart, sort the letters by placing the corresponding cards against their topological equivalents. Hint: Four of the nine classes have only one letter, three have two letters, one has five letters, and the remaining class has eleven topologically equivalent letters. Click here for the Solution. All files require Adobe Acrobat Reader.
Title:  Mug & Torus Morph
Comment:  "A topologist is a person who doesn't know the difference between a coffee cup and a doughnut." Link is to an animation of a mug (coffee cup) being morphed (transformed) into a torus (doughnut). Here is another Animation. The morphing box above is a 3-D delight!


TOPIC 2 (Jordan Curves / Mazes / Networks / Map Coloring)

Title:  Jordan Curve Theorem
Comment:  A simple closed curve in a plane separates the plane into two regions of which it is the common boundary. An obvious theorem ... but extremely important in topology. Includes an analysis of the classic Three Utilities Problem (Gas/Water/Electricity) and the "crossings rule" for simple closed curve mazes. Features a link to the amazing Fishy Maze (requires Adobe Acrobat Reader).
Title:  Extraordinary Mazes
Comment:  Download free printable mazes, learn to draw mazes, explore the history of mazes, and more. The photo is of the maze at Hampton Court, the oldest hedge maze in Britain. Click on it for an enlarged view.
Title:  Math aMazes
Comment:  A history of mazes from The Story of the Minotaur to How to Solve a Maze Using a Packet of Peanuts and a Bag of Crisps. Includes a link to a printable version. The article is adapted from one originally published as part of the Posters in the London Underground series. Click on any of the images in the latter page for an enlarged version and, where available, explanatory notes and further reading.
Title:  The Secret of the Maze
Comment:  Details the hand-on-wall rule for solving a maze with only one entrance and exit. [In effect, put your hand on the wall at the entrance and keep it on the wall until you exit the maze.] Includes a link to a right-hand and left-hand solution.
Title:  Hampton Court Maze
Comment:  Find the path from the entrance to the Hampton Court Maze to its center. Includes a link to the Solution and a Print & Play version for individual use or classroom distribution. 
Title:  Mazed Things
Comment:  Unique mazes by Isaac Thayer based on animal, holiday or miscellaneous topic themes. All mazes are suitable for printing and classroom distribution.
Title:  Javascript MazeMaker
Comment:  Maneuver the red dot through the arbitrary maze in as few moves as possible. Includes a Show Solution button.
Title:  Leonard Euler and the Bridges of Königsberg
Comment:  The problem of the Seven Bridges inspired the great Swiss mathematician Leonard Euler to create graph or network theory, which led to the development of topology. Euler's Solution will lead to the classic rule involving the degree of a vertex.
Special:  Click on the graphic above to view an enlargement of Königsberg and its bridges as it was in Euler's day.
Title:  The Seven Bridges
Comment:  Four areas of land are linked to each other by seven bridges. Is it possible to cross over all these bridges in a continuous route without crossing over the same bridge more than once? Try it on-line! Requires Macromedia Shockwave Plug-in. Experiment with different numbers of areas (islands) and bridges in Konigsberg Plus (requires Macromedia Flash Player).
Title:  Bridge Crossing Challenge
Comment:  Printable activity challenging students to solve problems similar to the Bridges of Königsberg problem. Excellent graphics. Requires Adobe Acrobat Reader.
Title:  New York City Bridges & Tunnels Problem
Comment:  Printable activity requires students to draw a network which represents the four land masses and thirteen brides/tunnels comprising the greater New York City area. Is the network traversable? Requires Adobe Acrobat Reader.
Title:  Trace Puzzles
Comment:  Can you draw each of the given figures in one continuous route without retracing any line and without lifting your pencil off the paper? Interactive version requires a JAVA-capable browser. (Use the forward/reverse buttons to change the puzzle.)
Title:  Coloring Maps
Comment:  Simple printable one-page handout on map coloring by Terry McCabe. Supporting graphics are in color. Requires Adobe Acrobat Reader.
Title:  The Four Color Problem
Comment:  Investigate map coloring interactively and on-line. Create a "map of countries" of any number, shape, and size, or let the computer create a map for you. How many colors are required to color the map? See if you can create a map that requires two colors, or three colors, or four colors. If you create one that "requires" five colors, you will upset mathematicians worldwide. Requires Macromedia Shockwave Plug-in
Title:  Map Game: Change the Lightbulb
Comment:  This on-line game (requires Macromedia Shockwave Plug-in) invites you to color a  map of the 48 continental US states with 6 (beginner), 5 (intermediate) or 4 (advanced) colors. For a less "jazzy" computer investigation, follow the instructions in "Paint" Map Coloring. To investigate the problem with real crayons (or color numbers), print Outline USA Map (requires Adobe Acrobat Reader).
Title:  USA Map Jigsaw Puzzle
Comment:  The geography game in USA FLASH Puzzle uses five colors to color a map of the 48 continental US states, but were five colors necessary? This interactive jigsaw puzzle uses the minimum number of colors. [The drag-and-drop feature requires Internet Explorer or Netscape 4. It is not supported by Netscape 6 or higher.]
Special:  Click on the map above for an enlarged view.
Title:  The Four Color Map Problem
Comment:  Download the software for this game based on the four color map problem. Ten pages of six maps each are provided to color, and are presented in increasing levels of difficulty. Part of Colorful Mathematics which aims to present advanced mathematical concepts to K-12 students in a game-oriented approach.


TOPIC 3 (Math-e-Magic / Mobius Strip)

Title:  Get Knotted! Animated Knots for Scouts
Comment:  Includes links to animations of 15 useful knots, with helpful comments on each. From the 42nd Brighton Scout Group, East Sussex, UK. Learn to Tie These Knots features 9 standard knots, with links to animations of each, courtesy of Boy Scout Troop 9, Billings, Montana. Knots Gallery has 16 animated links. Tying the Knot has links to 30 popular knots.
Title:  String Figures
Comment:  String figures are made around the world; hundreds of patterns have been recorded. Includes a link to animated instructions for Jacob's Ladder. Visit WWW Collection of Favorite String Figures for more links, which include a Kid's Guide to Easy String Figures. Figures are described, illustrated, and most have streaming video clips showing how to make them.
Title:  Judah Penetration Trick
Comment:  My favourite topological magic trick! A shoelace is wrapped securely around a pencil and a paper soda straw. When the ends of the shoelace are pulled, it appears to penetrate the pencil and cut the straw in half. The original trick was created by Stewart Judah, a Cincinnati magician. In place of the (alas long-gone) paper soda straw, use a strip of heavy paper. Requires Adobe Acrobat Reader.
Title:  Hanging Boots Puzzle
Comment:  Another favourite! Cut out and assemble the puzzle pieces. Then challenge a friend who does not know how the puzzle pieces were put together to remove the boots without tearing the paper or forcing the boots through the hole. Requires Adobe Acrobat Reader.
Title:  Through the Postcard
Comment:  Can you make a hole in a simple  postcard so that a person of ordinary stature will be able to pass through it? Click on Secret for the solution and the link to a Print & Play version of the postcard for practice.
Title:  String and Ring Puzzle
Comment:  This ancient puzzle is easy to make and uses inexpensive materials. Includes a detailed solution. Available commercially under a variety of names, such as Two Bead Puzzle and Yoke Puzzle. Here's one actually shaped like an Ox Yoke!
Title:  Buttonhole Puzzle
Comment:  The challenge in this puzzle by Sam Loyd is to attach a pencil to and remove it from a buttonhole. It seems impossible, but it can be done - merely an application of topological theory!
Title:  The Handcuffs Puzzle
Comment:  This is a classic topological puzzle that has been around for at least 250 years. It is very challenging, but it does give students a chance to get students up and moving. It can also make a good party game (for adults too). Includes an animated solution.
Title:  Livewire Puzzles
Comment:  Home-based Canadian business specializing in the production and sale of wire disentanglement puzzles. Includes a link to Do-It-Yourself Puzzles (require Adobe Acrobat Reader to view and print).
Title:  Tavern Puzzle Collection
Comment:  Tavern Puzzles® are reproductions of a type of puzzle traditionally forged by blacksmiths to amuse their friends at country taverns and inns. Each puzzle is mechanical in nature; removal of the object piece does not rely on force or trickery. My personal favourites include Old Shackles and Iron Heart (YouTube Iron Heart Solution).
Title:  Moebius
Comment:  A Moebius strip is a loop of paper with a half twist in it. Includes information on how to make a Moebius strip and what to do with a Moebius strip. The YouTube video Mobius Strip includes four experiments.
Title:  The Möbius Strip
Comment:  Another simple introduction to the Möbius strip. Considers their use as conveyor belts, continuous-loop recording tapes, and electronic resistors.
Title:  Möbius Mysteries
Comment:  Details the paradox of the double Möbius strips. Includes background, presentation details and links to two detailed student worksheets.
Title:  Recycling Topology
Comment:  It's hard to miss the triangle of three bent arrows that signifies recycling. Was it originally meant to be a Mobius strip, perhaps to symbolize the never-ending nature of recycling?
Title:  Mobius Strip
Comment:  A short looping animation by Vlad Holst of the endless cycle of reincarnation. The mobius strip is taken as symbol of eternity. Requires Quicktime Player.
Title:  Moebius Strip II
Comment:  "An endless ring-shaped band usually has two distinct surfaces, one inside and one outside. Yet on this strip nine red ants crawl after each other and travel the front side as well as the reverse side. Therefore the strip has only one surface." -- M. C. Escher on Moebius Strip II. Use Moebius Strip II for high-res printing (color transparency).
Special:  Access a Mobius Strip II Animated Movie by Mike Wilson (courtesy of YouTube).
Title:  What's a Klein Bottle?
Comment:  In 1882, Felix Klein imagined sewing two Möbius Loops together to create a single sided bottle with no boundary. Its inside is its outside. It contains itself. Acme's Classical Klein Bottles (albeit pseudo versions) can be ordered on-line. Klein Bottle has links to an excellent Artistic Rendering and Movies of the Klein Bottle.
Title:  Homemade Topological Shapes
Comment:  Do you have way too much time on your hands? Then try knitting a Mobius Strip or a Klein Bottle. Instructions for a Klein Bottle Pi Hat require Adobe Acrobat Reader. How to Knit a Mobius Strip and Crocheted Mobius Scarf Instructions feature simple patterns. No time? Order a Klein Bottle Hat. You can even purchase a matching Mobius scarf.


TOPIC 4 (Flexagons)

Title:  Predict Your Future!
Comment:  This folded flexagon first appeared in Japan during the early 1600s. The modern version is often used by school children to predict the future of such important life questions as How many children will I have?and Who will I marry? Origami Fortune Teller and Instructions for Fortune Teller have similar instructions. Cootie Catcher is an interactive version (requires Macromedia Shockwave Plug-in). Virtual Fingertip Fortune Teller requires Macromedia Flash Player. The companion Fingertip Fortune Teller can be printed and assembled. Point Fortune Teller has printable templates and instructions (requires Adobe Acrobat Reader) as does The Misfortune Teller. Instructional Folding Video has excellent instructions (requires Quicktime Player).
Title:  Tetra-tetra-flexagon
Comment:  A tetra-tetra-flexagon is made from a folded paper rectangle that is 4 squares long and 3 squares wide. Features detailed instructions. Want something more complicated? Try making a cyclic Hexa-tetra-flexagon from a square piece of paper. The latter will require Adobe Acrobat Reader. Visit YouTube for a detailed video on the cyclic version.
Title:  Flexifier
Comment:  A simple online tetra-tetra-flexagon generator. Choose four square on-line photos, then click on generate. [A good source of square on-line photos is Square Flower Photographs. Place your mouse over the desired photos in turn, press the right mouse button, then select Properties to access and copy the corresponding photo URL. Paste each URL in turn into Flexifier.] Print the result in color, cut out the two large rectangles, and glue them back to back. Follow the directions in Tetra-tetra-flexagon to assemble.
New:  FotoFlexifier, a simpler revision of Flexifier by Gerhard Drinkman. Cut out the one large rectangle, fold it in half horizontally, then glue the two halves together. Requires Firefox or Google Chrome as a browser; unfortunately it fails in Internet Explorer.
Title:  Jacob's Ladder Book
Comment:  Instructions for making a tetra-tetra-flexagon book. The author calls it a Jacob's Ladder Book due to the almost magical way the pages open...and open...and open again. Requires Adobe Acrobat Reader.
Title:  Flexagon
Comment:  Includes a history, instructions for making a hexa-hexa-flexagon, and directions for flexing the flexagon. Jürgen Köller's Flexagons has even more information and includes an excellent set of flexagon links.
Title:  Hexaflexagons
Comment:  Includes Background, How to Make a Hexahexaflexagon, How to Flex a Hexaflexagon, and Applications. Adapted from Martin Gardner's Book Mathematical Puzzles and Diversions. Another Hexaflexagons includes both trihexaflexagons and hexahexaflexagons. Visit 6-Color Hexahexaflexagon for a YouTube flexing video.
Title:  Hexaflexagons
Comment:  Martin Gardner's classic Scientific American article on flexgons. Requires Adobe Acrobat Reader. Visit Martin Gardner and Flexagons for a supportive YouTube video.
Title:  3D Simulation of a Hexahexaflexagon
Comment:  Shows a hexahexaflexagon cycling through all its 6 sides. It flexes at the same corner for as long as it can, then it moves to the next door corner. Click near the flexagon to start or stop it flexing. Requires a JAVA-capable browser. Be sure to visit the Flexagons home page for links to free printable templates & instructions, and a detailed page of flexagon theory.
Title:  Hexaflexagons
Comment:  Features Jim Gallant's son Max (age 7 at the time) making several hexa-hexa-flexagons at a time. Includes a link to a small movie of the flexing motion (requires Quicktime Player).
Special:  Click on the image above for a direct link to the flexagon movie.
Title:  Flexagons
Comment:  Includes links to printable models of a Trihexaflexagon, Tetrahexaflexagon, Pentahexaflexagon, and Hexahexaflexagon. Part of Mathematrix- a web site devoted to exploring mathematical recreations.
Title:  Kaleidoscope Tri-Hexa-Flexagon
Comment:  Make a tri-hexa-flexagon that produces six different kaleidoscope-like patterns. Print on a sheet of legal (8.5 x 14 in) paper, then print the Second Page on the reverse. Includes instructions and registration marks to assist perfect alignment. Requires Adobe Acrobat Reader.
Title:  Let's Make a Flexagon
Comment:  Instructions for another tri-hexa-flexagon that will produce six different 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.
Title:  Flex Mex
Comment:  YouTube video showing how to make a delicious Texa Mexa Hexa Flexagon complete with guacamole.
Title:  Hexaflexagon Toolkit
Comment:  Download this freeware and make yourself a pictorial hexahexaflexagon! The template features six Xmas pictures, but the program allows you to use your own images. Flexagon Creator is an on-line version of the Toolkit Program. It will run in Google Chrome, but NOT in Internet Explorer. The pop-up window will contain the strip for folding as well as the figures for the instructions.


Books and Web Pages by Jill Britton
Fun with Patterns
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All M. C. Escher works (c) Cordon Art B.V. - Baarn - the Netherlands.
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