Investigating Patterns
Number Patterns
Fun with Curves
& Topology
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

TOPIC 13 (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. 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 14 (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 15 (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.
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 16 (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 has 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.


TOPIC 17 (Optical Illusions)

Title:  The Exploration's Online Exhibits
Comment:  Online versions of Seeing Exhibits in San Francisco's Exploratorium Museum. Especially recommended are the five interactive illusions in Changing Illusions (requires Macromedia Shockwave Plug-in).
Title:  Welcome to
Comment:  An interactive guide to optical illusions on the internet. Explore live optical effects, illusions, distortions, animations, artwork, stories, PDF projects (including Build an Impossible Triangle), and more. Choose from over 100 colorful exhibits and activities. Most interactive illusions require a JAVA-capable applet. A few (like Rotating Ames Trapezoid) use Macromedia Flash Player. PDF projects require Adobe Acrobat Reader.
Special:  The addition of a stick through a Ames Window heightens the effect as the trapezoid rotates. The video of the live apparatus is worth a visit. An Ames Room (a trapezoidal room that the viewer interprets as rectangular) is a delight!
Title:  Illusionworks
Comment:  The most comprehensive collection of optical and sensory illusions on the web. Consists of interactive demonstrations, scientific explanations, illusion artwork (including Art of M. C. Escher), interactive puzzles, 3D graphics, suggested reading lists, perception links, and much more. Most interactive demonstrations require a JAVA-capable browser. The interactive puzzles and auditory illusion require a Macromedia Shockwave Plug-in.
Title:  Optical Illusions and Visual Phenomena
Comment:  Michael Bach's website provides some of the most interesting optical illusions and visual phenomena including topics of motion, time, luminance, and color. The demonstrations require Quicktime Player and Macromedia Flash Player.
Title:  SandlotScience: The Ponzo Illusion
Comment:  The Ponzo or "railway lines" illusion has applications to linear perspective in art. The activities in One Point Perspective and Two Point Perspective are excellent pedagogical resources. [These include links to a One Point Perspective Student Handout and a Two Point Perspective Student Handout, each featuring step-by-step construction of the perspective form.] You can search CGFA's A Virtual Art Museum for examples in Renaissance Art (like Crivelli's Annunciation, Da Vinci's Last Supper and Raphael's School of Athens). [The Last Supper at Milan explores the hidden structure of Da Vinci's fresco.] The YouTube video Masters of Illusion uses state-of-the-art computer graphics to demonstrate how scientific linear perspective was used by Renaissance artists to create the illusion of three-dimensional space. The Ponzo illusion requires a JAVA-capable browser.
Title:  Vision and Perception
Comment:  Part of Project LITE (Light Inquiry Through Experiments). Depth experiments include an excellent depiction of an Ames' Window (a rotating trapezoid appears as a window swinging in perspective). Requires Macromedia Flash Player Version 10.
Title:  Assembling of an Impossible Triangle
Comment:  Vlad Alexeev's Flash animation of the creation of an impossible triangle is impossible perfection itself. Requires Macromedia Flash Player.
Title:  Ascending and Descending
Comment:  M. C. Escher's famous lithograph Ascending and Descending is based on a drawing by L. S. Penrose. Page includes a delightful animation of an impossible staircase. Click on the graphic above to access a much larger full-sized image. The Image Viewer will allow you to resize the image to fit your screen or zoom in up to 200%. Impossible Staircase, Ascending Descending and Escher-Inspired Animation (all courtesy of YouTube) take the lithograph to another level.
Title:  Virtual Ride
Comment:  Take a virtual ride through three of M. C. Escher's impossible buildings - Belvedere, Waterfall or Ascending and Descending. All require Real Player.
Title:  Andrew Lipson's LEGO® Page
Comment:  Includes links to LEGO® recreations of M. C. Escher's Balcony, Belvedere, Ascending and Descending, Relativity, and Waterfall.
Title:  Some Ambiguities
Comment:  Changing the point of view offers the possibility of resolving ambiguities sometimes occurring in two-dimensional graphics. Includes an interactive Escher's Triangle allowing you to reveal the basis of the illusion. Requires a JAVA-capable browser.
Title:  Masters of Deception: Escher, Dali & the Artists of Optical Illusion
Comment:  Astonishing creations by masters of the art, such as Escher, Dali and Archimbolo; amazing visual trickery; and an illuminating foreword by the Pulitzer Prize-winning author Douglas R. Hofstadter make this breathtaking collection the definitive book of optical illusions.
Title:  Jacob's Ladder Toy
Comment:  With this 2000-year-old Chinese wooden toy, you can create the illusion that a block is traveling down a ladder of wood and ribbon. Includes instructions for a home-made version. Jacob's Ladder Toy Instructions is suitable for printing. An inexpensive commercial version is available through
Special:  Click on the image above to view a Jacob's Ladder toy in motion. Animation by Edwin Attaway.


TOPIC 18 (Miscellaneous Diversions)

Title:  Color Test
Comment:  This test is simple. There are 8 screens ... and on each screen you are given a word and two buttons to choose from. Select the COLOR of the word ... not the word! Oh, and you only have 4 seconds per screen. Requires Macromedia Flash Player.
Title:  Memory
Comment:  How good is your memory? How fast can you match the pairs of cards? Visit Match'em, Present Match and Memory Game for similar games. Require either Macromedia Shockwave Plug-in or Macromedia Flash Player.
Title:  Simon Says
Comment:  This is the classic game of memory and repetition. Do what Simon (the computer) says. Follow the pattern of tones and lights for as long as you can ... if you can! Requires Macromedia Flash Player.
Title:  Count the F's
Comment:  How many times does the letter F appear in the given sentence? Count them ONLY ONCE; do not go back and count them again. There is no catch!
Title:  Hangman
Comment:  An interactive version of the classic (if a bit morbid) Hangman game. Choose a letter you think is in the word, one letter at a time. If you correctly guess the word, the hangman will do a little dance. If, however, you cannot guess the word and the hangman is completed, the stick figure will be hanged. In HangARoo, try to save the smart mouthed kangaroo from a good old fashioned neck stretching. Requires Macromedia Flash Player. Classic Hangman requires Macromedia Shockwave Plug-in.
Title:  Balloon Pop Word Game
Comment:  Balloon Pop plays just like the classic game of Hangman. Try to guess the mystery word before you pop all seven balloons. Requires Macromedia Shockwave Plug-in.
Title:  Text Twist
Comment:  A challenging word game in which you form words from the letters provided. Find at least one word that uses all the letters to continue playing. Requires Macromedia Flash Player.
Title:  Bookworm
Comment:  Link letters and create words to feed the hungry bookworm. Don't let the fiery red tiles reach the bottom or your library will burn down. Green and gold tiles are worth bonus points. It's vocabulariffic! Requires a JAVA-capable browser.
Title:  Just My Type
Comment:  A Flash game from Scott Kim. Sharpen your eye by matching each close-up snapshot with the letter it came from. Requires Macromedia Flash Player.
Title:  What's In A Name?
Comment:  Only the negative or outside parts of the letters which make a famous person's name are shown in this puzzle. Requires a JAVA-capable browser.
Title:  Wacky Wordies
Comment:  The aim of these word puzzles is to translate an arrangements of letters, numbers and/or symbols into a familiar phrase, saying or cliché. Can you make an educated guess? Visit another Wacky Wordies and More Wacky Wordies for further examples.
Title:  Codes for Scouts and Cubs
Comment:  Codes make great games and they're really very easy to learn. This collection is ordered from "Beginner" codes, then "Substitution" codes, and finally a few tougher puzzles. Try to solve the cryptograms ("coded" sentences) in Cryptograms. Requires a JAVA-capable browser. Includes a link to a page of Statistical Information that may prove helpful.
Title:  Ciphers | Crypto Club
Comment:  Since kids enjoy sending and breaking secret messages, cryptography provides a very motivating setting for applying mathematics. Part of a National Science Foundation middle grade cryptography project. Excellent interactive materials that include Things To Do and Tools. Requires Macromedia Flash Player.
Title:  The Dancing Men
Comment:  The Dancing Men is one of the most intriguing Sherlock Holmes stories and involves a simple substitution code of little stick figures. Visit Strange Little Dancing Men Explained at Last to download a dancing men Macintosh TrueType font for your own messages!  [Preview: Dancing Men Alphabet]. Prefer a Windows version? Download Decompress with WinZip or freeware ZipCentral, then add the file <Dancingm.ttf> to your Windows fonts.
Title:  Images of Mathematicians on Postage Stamps
Comment:  High-quality scans of postage stamps from around the world picturing famous mathematicians. Features mathematical events and concepts as well.
Title:  Feynman Online
Comment:  Web site dedicated to Richard P. Feynman (1918-1988), scientist, teacher, raconteur, and musician. For a video of Feynman, detailing his thoughts on education and mathematics, visit The Pleasure of Finding Things Out.
Title:  Hot Math Trails
Comment:  To students, mathematics is usually seen as a sterile subject with little to do with the outside world. The truth is that math is all over and is responsible for many of the things we take for granted. Math trails are designed to acknowledge the many contributions that mathematics has given to our modern world. 
Title:  How to Use SOROBAN (ABACUS)
Comment:  An interactive abacus with instructions for those of you wishing a primitive calculator. Or click and drag the beads in Virtual Abacus. Looking for a more sophisticated calculator? Whether yov're bvilding a road, designing an aqvadvct, or covnting yovr gladiators, Compvter Romanvs (Roman Numeral Calculator) will make the job easier. All require a JAVA-capable browser.
Title:  Popsicle Stick Abacus
Comment:  Make your own popsicle stick abacus. Contains detailed instructions with excellent supporting graphics. Requires Adobe Acrobat Reader to view and print.
Title:  Java Calculator
Comment:  A simple pseudo-electronic calculator that you can add to your own web page. Easy to install. Requires a JAVA-capable browser. This full-featured virtual Scientific Calculator is written in JavaScript. Scientific and Statistical Calculator requires Macromedia Flash Player.
Title:  The Flash Mind Reader
Comment:  Think of any two digit number, add together its digits, then subtract the total from your original number. Look up your result in the table provided and concentrate on the symbol next to it. When you click on the crystal ball, this symbol will "mysteriously" appear! It's great and of course bogus. After you've racked your brain over how it works, take a peek at Flash Mind Reader Demystified. Shaman Psychic and Magic Gopher feature a similar trick. All require Macromedia Flash Player.
Title:  Guess Your Number Game
Comment:  Most of the screens in this game are cleverly designed to divert your attention. Can you figure out how it works mathematically?
Title:  Mind-Reading Trick
Comment:  Follow the instructions carefully. The Magic Swami will attempt to read your mind. [If this trick doesn't work, then you march to the tune of a different drummer.] For a JAVA-version, visit Mind Boggled.
Title:  Swami Omigami
Comment:  Mentally select a card from the six shown and concentrate. The magic swami will magically remove your card. Did you catch on right away? Visit The Wiz, Mysterious Mind Tap or The Amazing Garfield for more versions of the same card trick. Or you can download Majick, an executable (runs on your computer) version. Decompress with WinZip or freeware ZipCentral. All versions require Macromedia Flash Player. For a simpler version, consult The Amazing Amaz-O.
Title:  Think of a Number
Comment:  This totally silly Flash movie gave me a chuckle as I began my search for interesting number tricks. Requires Macromedia Flash Player
Title:  Guess My Number
Comment:  The classic classroom game. Develops skill in halving and estimation. For a spiffy Flash version, visit Guess the Number! (requires Macromedia Flash Player).
Title:  Amazing Math Trick
Comment:  Follow the correct steps and the result always comes out to 1089. Use the applet to check as many numbers as you wish. Requires a JAVA-capable browser.
Title:  I've Got Your Number!
Comment:  Does Numero Uno, the mind reader, have your number? Mysterious Mind Reader features a similar number trick. The latter requires a JAVA-capable browser.
Title:  7-Up Fido Puzzle
Comment:  Think of a number, make an anagram of it, then subtract the smaller number from the larger. Circle one non-zero digit of the answer, then enter the remaining digits. Your computer will tell you what digit you circled. Requires Macromedia Flash Player. When done, visit Fido Puzzle Explanation.
Title:  Magic Maths
Comment:  Contains links to a wealth of interactive numerical magic tricks (require a JAVA-capable browser). Just one part of a delightful web site from France by Thérèse Eveilleau. Some pages will require Macromedia Flash Player.
Title:  Math Explorer: Mind Reader
Comment:  Here's a cool card trick that's guaranteed to make you look like a mind reader. Learn the algebraic basis of the trick in Analyzing Mind Reader.
Title:  Mulawa Magic
Comment:  Terrific mathmagic tricks from Peter Hewitt. Includes interactive mind reading tricks and card tricks. Requires a JAVA-capable browser.
Title:  Secret Worlds: The Universe Within
Comment:  View the milky way at 10 million light years from the earth. Then move through space towards the earth in successive orders of magnitude until you reach the subatomic universe of electrons and protons. Requires a  JAVA-capable browser. The Powers of Ten concept was advanced by architect Charles Eames. The classic Eames movie Powers of Ten is available on-line courtesy of YouTube.
Title:  Scale Factors (or Why There Are No Giant Spiders!)
Comment:  Giant creatures are the stuff of many science fiction movies ... from giant ants and spiders to 10-storey tall babies. But is it really possible for creatures to be so large? If so, why aren't there any 6 metre tall spiders? Visit YouTube's The Physics of Movie Monsters for an explanation of the Cube Square Law.
Title:  The Monty Hall Problem
Comment:  Suppose you're on a game show, and you're given the choice of three doors. Behind one  door is the grand prize; behind the others, booby prizes. You pick a door and the host, who knows what is behind each door, opens another door, revealing a booby prize. The host then offers you the opportunity to change your selection to the third door. Should you stick with your original choice or switch? Does it make any difference? The answer may be counterintuitive. Includes an applet for your own investigations (requires a JAVA-capable browser). Visit YouTube's Monty Hall Problem: Numb3rs and 21 for a solution..
Title:  36 Humorous Proof Methods
Comment:  I read these with absolute delight and simply had to share them. Anyone who has ever fought with a proof will love this list. Take the time to enjoy!
Title:  A Picture Proof of the Pythagorean Theorem
Comment:  An interactive visual proof (by dissection) of one the most famous theorems of geometry. Virtual Manipulative: Pythagorean Theorem allows manipulation of the constituent pieces. (Click and drag squares and triangles to move them. Click and drag the corner of a square or triangle to rotate it.) Pythagorean Thereom Proof, The Pythagorean Theorem and Animating the Pythagorean Theorem present the same proof in a non-interactive mode. For a YouTube movie of the proof, visit Pythagoras in 60 Seconds. For a Flash movie (by Mike Shepperd), visit Pythagoras' Theorem (requires Macromedia Flash Player). For background on the theorem, visit Pythagorean Theorem History. The YouTube movie Pythagorean Theorem Water Proof features a creative demonstration. Proving the Pythagorean Theorem Graphically and Behold! feature animations of other classic proofs, the former credited to Euclid. Perigal's Dissection and Perigal's Proof showcase another elegent proof of the theorem. The former is a Flash presentation; the latter includes both an animation (courtesty of YouTube) and an interactive version thereof. All applets require a JAVA-capable browser.
Title:  Pythagorean Triple Calculator
Comment:  Three integers a, b, and c that satisfy a2 + b2 = c2 are called Pythagorean Triples. There are infinitely many such numbers and there also exists a formula that will generate triples, although not all of them. The formula was known to Euclid and was used by Diophantus to obtain Pythagorean triples with special properties. This calculator will find triples using the formula (requires a JAVA capable browser).
Title:  Irrational Numbers Can "In-Spiral" You
Comment:  Introducing students to the Pythagorean Theorem presents a natural context for investigating what irrational numbers are and how they differ from rational numbers. This artistic project allows students to visualize, discuss and create a product that displays irrational and rational numbers.
Title:  Pick's Formula
Comment:  You probably know how to find the area of polygonal regions such as squares, rectangles, parallelograms and triangles. But how would you find the area of irregular polygonal regions? The pdf document (requires Adobe Acrobat Reader) includes a link to a geoboard applet (requires a JAVA capable browser).
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!
Title:  Puzzle Parlor
Comment:  Twelve interactive online games from the puzzle makers at Kadon Enterprises designed to give students practice with problem solving skills using mathematical and spatial sense intelligence.
Title:  Information on Pentomino Puzzles
Comment:  A pentomino consists of five equal-sized squares attached edge-to-edge to form some shape. The problem is to fit the 12 possible pentomino pieces into various shapes, often rectangles. Links include a really nifty 8 by 8 java Pentominos Puzzle Solver. Pentominoes features a similar applet. Applets require a JAVA-capable browser.
Title:  Alice in Wonderland - An Interactive Adventure
Comment:  Enjoy activities with Alice and her Wonderland friends. Interactive puzzles require a JAVA-capable browser. Alice's Adventures in Wonderland and Through the Looking-Glass and What Alice Found There were written by Lewis Carroll, (aka Charles Dodgson), mathematics professor at Oxford University. Alice has links to each book (complete with the classic illustrations by John Tenniel) as Adobe Acrobat Reader files. The book The Annotated Alice summarizes the mathematics disguised within the Alice books. See also The Hidden Math Behind Alice in Wonderland.
Special:  Lewis Carroll's Jabberwocky is often considered to be the greatest nonsense poem written in the English language. Illustration by John Tenniel and narration by Griffin Rogers.
Title:  The Williams Family
Comment:  One of the 500 puzzles (an interactive logic puzzle) in the Questacon Maths Centre - an outreach program which travels around Australia making mathematics fun for everyone. Requires Macromedia Shockwave Plug-in.
Title:  Paper Pool Game
Comment:  Adjust the length and width of the table with the sliders. Explore tables of various sizes. How many hits occur before the ball reaches a pocket? And how is the number of hits related to the size of the table? Expand Instructions and Exploration for details. Requires a JAVA-capable browser.
Title:  Water Puzzle
Comment:  The problem is how to measure out exactly 4 litres of water with only a 5-litre jug and a 3-litre jug? It was featured in the film Die Hard with a Vengeance. Pour-to-Score asks you to find from 1 quart to 8 quarts with only an 8-quart container and a 3-quart container. All three require Macromedia Flash Player.
Title:  Nine Dots Puzzle
Comment:  Classic puzzle requiring the connecting of an square array of nine dots using just four straight contiguous lines. Requires a JAVA-capable browser.
Title:  Toothpick Puzzle
Comment:  Form five squares by moving three of twelve toothpicks. Interactive applet uses JavaScript. Give up? Here's a Solution. Figi Toothpick Puzzle requires Macromedia Flash Player.
Title:  Matchstick Puzzle
Comment:  By moving two matchsticks and a button, try to make the image of the fish point in a different direction. Requires a JAVA-capable browser. For more matchstick puzzles, visit Ze's Lil Match Puzzles (requires Macromedia Flash Player).
Title:  The Frog Puzzle
Comment:  Click on the lily pads to make the frogs move across the pond by hopping or sliding (requires Macromedia Flash Player). Can you work out how many moves you would need to shift any number of frogs? Includes a link to a Solution. Frog Leap Test is yet another Flash version used as a 2nd grade computer test in China.
Title:  Move the Trains
Comment:  Do you remember those little smart puzzles where two trains switch on a railroad? Try this interactive version by Ishihama Yoshiaki, with everything as it should be: locomotives, coaches, and sidetracks. Requires a JAVA-capable browser.
Title:  Get Connected
Comment:  Another Flash puzzle from Scott Kim. Connect everyone to the web shifting pieces until all wires match. There are only seven folks to be connected, but don't be mislead. Requires Macromedia Flash Player.
Title:  The Fox and the Duck Puzzle
Comment:  An interesting interactive variation of the classic The Wolf, the Goat, and the Cabbage problem. Be sure to read the hints/rules before you begin. Crossing the River with a Wolf, a Goat, and a Cabbage is a feline version of the classic puzzle from MathCats. All three require a JAVA-capable browser. Here's a flash variation involving Missionaries and Cannibals.
Title:  The Frustrated Farmer
Comment:  An animated version of the Fox and Duck Puzzle, complete with sound effects. Requires Macromedia Shockwave Plug-in. The Frustrated Farmer: Manipulatives has pictures of the farmer, fox, chicken, and bag of corn suitable for printing.
Title:  Sam Loyd Official Site
Comment:  Sam Loyd (1841-1911) was America's greatest puzzle expert and invented thousands of ingenious and tremendously popular puzzles. Visit The Pony Puzzle for an interactive Sam Loyd Puzzle (requires Microworld's Web Player).
Title:  Get Off the Earth Puzzle
Comment:  This is my favorite of Sam Loyd's puzzles. For a hint, visit The Vanishing Leprechaun - a rectangular puzzle with a similar solution. Visit Nine Trolls for a similar interactive puzzle and Who Turned to Doggie Doo? for an animated variant (require a JAVA-capable browser). You can even download a Printable Version (decompress with WinZip or freeware ZipCentral).
To celebrate 110 years since Sam Loyd  received a patent for his Get Off the Earth puzzle, a Commemorative Edition has been created using Loyd's original artwork. This edition is an exact working replica of the one released in 1896. The link contains details as well as purchase information.
Title:  Trick Mules Puzzle
Comment:  Position the three pieces so that the two jockeys are each properly sitting on top of a mule. Requires Macromedia Shockwave Plug-in. Although created by Sam Loyd, millions of copies were distributed as P. T. Barnum's Trick Donkeys. The Three Hares contains a concise history of the classical Chinese puzzle. Cracker Jack Cowboy Puzzle and Cracker Jack Parrot Puzzle are similar in design. Saddle the Horses features an animated solution by (requires Macromedia Flash Player) and a link to a set of Printable Cards.
Special:  Click on the image above for a printable Trick Mules Puzzle puzzle in full color (requires Adobe Acrobat Reader).
Title:  The Fifteen Puzzle
Comment:  The most famous of the puzzles attributed to Sam Loyd, however Loyd had nothing whatsoever to do with either the puzzle or its popularity. [See The 15 Puzzle (requires Adobe Acrobat Reader).] The 15 pieces are to be arranged in ascending order, with the hole in the bottom right corner. The graphics are made to resemble an actual physical puzzle, with all the little grooves and ledges that hold it together. You get to slide the pieces manually instead of simply clicking on them to add a bit of realism to the game. Requires a JAVA-capable browser.
Title:  Rush Hour 12
Comment:  Rush Hour is a variation on the traditional sliding-block puzzle. In the game, your car (usually red) is stuck in traffic! You must slide the blocking cars and trucks out of the way to escape the gridlock. Contains links to 12 on-line versions of the game. Each will require either a JAVA-capable browser or Macromedia Flash Player.
Title:  MazeWorks
Comment:  Contains links to ten interactive diversions by David Herzog and the late Bob Kirkland - including Peg Solitaire and Sliders (8 Puzzle and 15 Puzzle). Select the links option to access even more puzzles. All require a JAVA-capable browser.
Bonus:  Download a copy of any of the programs on the Mazeworks site. All files are in Zip format (decompress with WinZip or freeware ZipCentral).
Title:  Escher Sliding Puzzle
Comment:  An interactive 15-puzzle featuring a scrambled Escher graphic. You can also try the Jill Britton Sliding Puzzle. Access to download the sliding puzzle applet. Decompress with WinZip or freeware ZipCentral. All require a JAVA-capable browser.
Title:  Escher Jigsaw Puzzle
Comment:  An interactive jigsaw puzzle featuring an Escher tessellation. Selected pieces can be rotated by clicking the right mouse button. For a less challenging activity (no rotation), visit Jill Britton Jigsaw Puzzle. Access to download the jigsaw puzzle applet with instructions and examples. Decompress with WinZip or freeware ZipCentral. All require a JAVA-capable browser.
Title:  Escher Tic-Tac-Toe
Comment:  Try your luck at an interactive game of good old tic-tac-toe! This one uses M. C. Escher icons. Automatic scoring and alternating "first moves" are built in. Visit JavaScript Source to learn how to make your own tic-tac-toe game. Tic Tac Toe features the classic game (with added sound effects). Both require a JAVA-capable browser. For a Flash version, visit TIC-TAC-TOE (requires Macromedia Flash Player).
Title:  Hang Gliding Puzzle
Comment:  This puzzle is very ingenious. No mathematics contained here ... but you've probably had enough of that! Requires Macromedia Flash Player.
Title:  Mr. Potato
Comment:  Just For Fun! An interactive game by Steve White. Works just like the old Mr. Potato Head game. You can also embellish Mrs. Pepper, Mr. Lemon, and Miss Onion. In Jill Britton Potato Head, guess who has added her head to the collection of vegetables? Access to download the applet with instructions and examples. Decompress with WinZip or freeware ZipCentral. All require a JAVA-capable browser. Visit Mr. Potatohead for a Shockwave version of the classic game (requires Macromedia Shockwave Plug-in). For a Flash vegetable dressing game, visit Mr.VegHead (requires Macromedia Flash Player). When you have finished adding embellishments, you can animate your creation.
Title:  Q*Bert
Comment:  Based on the arcade classic Q*Bert by Jeff Lee. In the game you must jump on all the squares and change the color while avoiding the falling balls ... and other pitfalls. Very addictive! Requires Macromedia Flash Player.
Title:  SheepGame
Comment:  Another favorite game. It's very simple! All the sheep run away from your mouse cursor. So use that to steer them into the pen. Download SheepGame to play on your computer when you are not on the WWW. Don't forget to try SheepGame's sister game DogGame. You are a dog with a nose for bones, and must dig up as many as you can in one minute. All games require Macromedia Flash Player.
Special:  Move your cursor/mouse over the sheep above to watch them jump. Click here for an Enlarged View. Requires Macromedia Flash Player.
Title:  Snowcraft
Comment:  This is cute ... and addictive! Two teams of kids (one of which you control) in an all-out snowball fight. Requires Macromedia Shockwave Plug-in.
Title:  Smack the Penguin
Comment:  Another addictive game in which a yeti wielding a club attempts to hit a penguin. Click once to get the penguin to jump. Then, as the penguin is falling, click and release to swing the club. If the yeti hits the penguin, it will fly across the icy field. The objective is to get the penguin to fly as far as possible. Download an off-line version. Decompress with WinZip or freeware ZipCentral. Both require Macromedia Flash Player.
Title:  Orisinal : Morning Sunshine
Comment:  Home to over 40 games done in Macromedia Flash (require Flash Player). My favorite is Bum Bum Koala which features a koala using his hind end to bump snails off his trees.
Title:  Touch Me Tender
Comment:  This web page really made me laugh! Move your cursor/mouse over the frog slowly, touching it everywhere (but don't click). Requires a JAVA-capable browser. Visit Fly Eatin' to learn what it's like to snatch flies gracefully with a click of your mouse. Requires Macromedia Flash Player.



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