Investigating Patterns
britton.disted.camosun.bc.ca, Sobota 10. leden 2009
Pďż˝evzato z britton.disted.camosun.bc.ca
Investigating Patterns  
Number Patterns Fun with Curves & Topology  
TOPIC LINKS 

TOPIC 1 (Prime Numbers / Magic Squares)
Title: Sieve of Eratosthenes Comment: A natural number is prime if it has exactly two positive divisors, 1 and itself. Eratosthenes of Cyrene (276194 BC) conceived a method of identifying prime numbers by sieving them from the natural numbers. Web page uses the sieve to find all primes less than 50. Includes a link to a Sieve of Eratosthenes Applet which also begins with a size or upper boundary of 50 (maximum 200). Eratosthenes' Sieve contains a similar applet preset to find all primes less than 200. Virtual Manipulative: Sieve of Eratosthenes begins by finding all primes less than 100 (20 rows), but can be reset to display up to and including 100 rows. All three applets require a JAVAcapable browser. Special: Learn all about Eratosthenes, courtesy of YouTube.
Title: Prime Number List Comment: Once you have entered the lower bound and upper bound, this JavaScript applet will display all prime numbers within the selected range. Another Prime Number List will generate prime numbers until you click Stop or until your computer runs out of memory. And if you can bear it ... here's a special Prime Number Generator. All require a JAVAcapable browser.
Title: Prime Factorization Machine Comment: A positive integer (natural number) is either prime or a product of primes. This applet decomposes any positive integer less than 1,000,000 into its prime factors. The bigger the number, the longer it will take. The applet in Factors and Prime Factors will do the same and list ALL the factors. Both applets require a JAVAcapable browser.
Title: GCD & LCM Comment: Details how to find the greatest common divisor (GCD) or greatest common factor (GCF) and the least common multiple (LCM) of two or more integers using prime factorization. GCD/LCM Calculator features an interactive applet (requires a JAVAcapable browser).
Title: Perfect Number Analyzer Comment: This program takes a number and decides whether it is perfect, abundant, or deficient, based on the sum of its proper divisors. Requires a JAVAcapable browser.
Title: Mathematics Enrichment Workshop: The Perfect Number Journey Comment: A positive integer is perfect if it is equal to the sum of all of its proper positive factors (natural divisors other than itself). Investigates perfect numbers, their properties, and their connection to Mersenne primes. Lessons and exercises extend over several pages.
Title: Amicable Numbers Comment: Two positive integers are amicable or friendly if each is equal to the sum of the positive proper factors (natural divisors other than itself) of the other. The first pair {220, 284} was originally found by Pythagoras. Includes as yet unanswered questions. Also available in a PDF Version (requires Adobe Acrobat Reader). You can race the clock finding amicable pairs in Amigos: Amicable Numbers (requires Macromedia Flash Player).
Title: Suzanne Alejandre: Magic Squares Comment: Resources for including a variety of magic squares in the math curriculum, with activities for students and explanations of these interesting puzzles. The LoShu Activity with its internal China cultural links is particularly recommended. Tortoise Patterns features a large divine tortoise graphic and a link to Lo Shu and the Story of Emperor Yu. The Double Six Magic Square (at the Shaanxi History Museum) contains six numbers in length and breadth, the numbers in vertical, horizontal and diagonal lines add up to 111, respectively. Special: Click on Emperor Yu's divine tortoise above to see an animation of the construction of one equivalent variation of the LoShu Magic Square.
Title: 5 x 5 and Other OddNumbered Magic Squares Comment: Details de la Loubère's (diagonal) method for constructing oddnumbered (oddordered) magic squares. Follow by a Magic Squares Applet in which you are asked to create a level 3 (that is, 3rd order or 3 x 3), level 4, level 5, level 6, or level 7 normal magic square by sliding the first 9, 16, 25, 36, or 49 integers into place. Need Help? Watch de la Loubère's Method used squarebysquare on a 3 x 3 grid, a 5 x 5 grid, or a 7 x 7 grid. Applets require a JAVAcapable browser.
Title: Interactive Magic Square 3 x 3 Comment: After viewing the construction of a 3 by 3 magic square, pass your mouse over each of the View Rule texts, then click to see the rule in animation. Use the rules to build your own magic square in 3 x 3 Magic Square. Both require Macromedia Flash Player.
Title: Magic Square Comment: Can you arrange the numbers 1 to 16 in a 4 x 4 square in such a way that the sum of each row, column, and diagonal is the same? It cannot be done by de la Loubère's (diagonal) method for constructing oddordered magic squares. Includes a link to an Answer Key. Both require a JAVAcapable browser. For a version with a jigsaw puzzle twist, visit Magic Square Puzzle (requires Macromedia Flash Player).
Title: Magic Zone: Magic Square Comment: A neat magic trick based on the properties of a 4 x 4 magic square. Requires Macromedia Flash Player.
Title: Melancholia Magic Square Comment: The most famous 4 x 4 magic square is the Melancholia magic square, socalled because Albrecht Dürer used the square in his engraving Melancholie. Learn about its amazing number of magic properties!
Title: Subirachs Magic Square Comment: Structurally this magic square is very similar to the Melancholia magic square, but the numbers in four of the cells have been reduced by 1. The magic sum is 33, the age of Jesus Christ upon his crucifixion. The animated graphic reveals the dozens of regular symmetric combinations of 4 squares which add to 33. Sculptor Josep Subirach included the Magic Square in his Passion Facade for Barcelona's Sagrada Familia cathedral.
Title: Typeography  Magic Square Comment: Pass your mouse over the image to view the magic square grid upside down, and backwards. One of my favourite number patterns! Requires a JAVAcapable browser.
Title: Shin's Magic Square World Comment: Includes links to detailed construction algorithms for the 3 types of magic squares: OddSeries, Multiple of 4 Series, and Other Sizes Series.
Title: Patterns and Magic Squares Comment: Explores the patterns possible when the numbers in certain kinds of magic squares are replaced by specific symmetric symbols. Requires a JAVAcapable browser. For a nonJAVA version, visit Patterns and Magic Squares.
Title: Sudoku Comment: Sudoku roughly translates from Japanese as "solitary number". Despite the name, it is not Asian, but was invented in the 18th century by mathematician Leonhard Euler. It is simply an 81square grid subdivided into nine 3 x 3 grids. Each 3 x 3 grid must end up with the numbers 1 to 9. No column, row or grid can have two of the same number. Unlike magic squares, nothing has to add up to anything else.
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TOPIC 2 (Clock or Modular Arithmetic)
Title: Clock (Modular) Arithmetic Pages Comment: Learn about clock or modular arithmetic, which is arithmetic on a circle instead of a number line. Includes links to an explanation of Clock Arithmetic, a Clock Arithmetic Renaming Calculator, a Clock Arithmetic Calculator (+, , x), and an Encoder for secret messages enclosed or decoded using clock arithmetic. All feature interactive applets that require a JAVAcapable browser.
Title: Clock Arithmetic Comment: Convert numbers using various types of clocks. The hand rotates to the calculated time after you have set the clock size and entered the number of hours. Requires a JAVAcapable browser.
Title: Modular Art Comment: One of the ways in which we may use number patterns is in the creation of unique and artistically pleasing designs. Learn how to make designs based on modular arithmetic operations tables. Includes an interactive applet (requires a JAVAcapable browser).
Title: Cayley Quilt Maker Comment: Create colorful posters based on modular arithmetic operations tables by painting your own pattern replacements (tiles). The applet by Scott Lopez includes several presentation options ... including Add or Multiply and Reflection or Rotation. You can select the number of tiles, fill any tile background, and vary the painting pen width and/or color. Requires a JAVAcapable browser. Visit Math Art Posters & Clock Arithmetic for a variety of pedagogical links, including Clock Arithmetic Presentation, Mathart Posters, and several printable Microsoft Word documents. Special:Download CayleyQuilter software for Windows. Save the compressed file cqsetup.zip your hard drive, extract with WinZip or freeware ZipCentral, then doubleclick on the decompressed file CQSetup.exe to install. Copy/paste external graphics with a simple paint program, such as Ultimate Paint. [Use the program to crop and/or resize each graphic to a square. Visit Ultimate Paint Pointers for details.] Print your mod art creation and/or save it as a bit map. Cayley Quilter shows what you can do with the software (requires Adobe Acrobat Reader)
Title: Perpetual Calendar Comment: Learn the historical basis of today's calendar, then investigate a simple way to find the day of the week corresponding to a specific date using modular arithmetic.
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Title: Fascinating Flat Facts about Phi Comment: A compendium of web pages on the golden ratio/golden section/divine proportion (phi). A companion page Some Solid (3D) Geometrical Facts about the Golden Section considers the solid world of 3 dimensions.
Title: The Golden Section Comment: Contains three classic constructions related to the golden section presented in simple selfpaced sequential steps. Requires a JAVAcapable browser.
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.
Title: A Knotty Pentagram Comment: The geometric proportions of a pentagram (fivepointed star) are those of the golden section. Learn how to construct a pentagram by tying a simple overhand knot in a strip of paper.
Title: Flags of the World Comment: Many countries have a flag that contains a fivepointed star or pentagram. How many are there on the USA flag? Why? The flag of Panama (above) has exactly two such stars. Many flags have precisely one. You can search this web site for other flags that contain at least one fivepointed star.
Title: The Hidden Star Puzzle by Sam Loyd Comment: Can you find a perfect five pointed star (pentagram) in the pattern? You will find many more puzzles by the puzzle's author (Sam Loyd) in Topic 18.
Title: Virtual Manipulative: Golden Rectangle Comment: Features an applet which illustrates iterations of the golden section. The plot is created by starting with a golden rectangle, chopping off successive squares, and drawing successively smaller quarter circles. The result is not a true golden (equiangular) spiral, but an adequate approximation thereof. Includes icons for student activities, teacher information, and assistance. Produced by the National Library of Virtual Mathematics for Interactive Mathematics. Requires a JAVAcapable browser.
Title: Golden Rectangle / Triangle / Spiral Series Comment: Click on rectangle or triangle to select a golden figure, then on expand or contract to start the motion. Additional buttons allow you to control the speed and stop the motion. If your screen resolution is less than 800 x 600, you may have to scroll to reach the buttons. Requires a JAVAcapable browser.
Title: Chambered Nautilus Comment: The most famous golden spiral is that exhibited by the chambered nautilus. When done, visit Snail Shell and its entertaining applet. The latter requires a JAVAcapable browser. Netscape users will require version 4.5 or higher.
Title: The Human Face Comment: The human face is based entirely on the golden ratio or Phi. Includes both frontal and side examples. In How "Phi" is My Face? students use pixel measurements to calculate the ratios of different face measurements and to determine how close a ratio is to the golden ratio. Linked Applet requires a JAVAcapable browser. How "Phi" Is My Face? (Template) features instructions on how to create your own Phi applet.
Title: The Golden Proportion, Beauty, and Dental Aesthetics Comment: A fascinating set of pages on the aesthetics of the golden section  particularly dental. Leonardo da Vinci would have coveted the associated Golden Mean Gauge. No matter how much the gauge opens or closes (Animation), the proportion remains the same. Phi Rules for woodworking design are available from Lee Valley.
Title: Golden Section in Art and Architecture Comment: Digital slide show (with several links to animated sequences) detailing the occurrence of the golden section in architecture from the pyramids at Guizeh to the works of LeCorbusier. Also includes its use by artists such as Leonardo da Vinci, Michelangelo, Raphael, Seurat, and Salvador Dali. All graphics are linked to enlargements suitable for downloading.
Title: Leonardo Da Vinci Comment: Explore the divine proportions in Leonardo's painting of the Mona Lisa (La Giocanda) and his study of the proportions of man (The Vetruvian Man). Move your mouse over each image to see the golden rectangles. You can also play a Mona Lisa Movie featuring a similar geometric overlay (requires Macromedia Flash Player). Perhaps you prefer a different Mona Lisa Analysis? Locate your own golden rectangles using this Mona Lisa Applet! Top it all off with Why is the Mona Lisa Smiling? All pages require a JAVAcapable browser.
Title: The Sacrament of the Last Supper Comment: Salvador Dali's The Sacrament of the Last Supper is painted inside a golden rectangle. Golden rectangles were used for positioning the figures. Part of an enormous dodecahedron floats above the table.
Title: Exploring the Golden Rectangle Comment: Activity (requires Adobe Acrobat Reader) in which students construct a golden section and a golden rectangle, then import pictures from the internet and download them into the Geometer's Sketchpad. Using their golden construction, they discover the use of the golden rectangle within famous works of art. Pass your mouse over/off Edouard Manet's The Railway to add/remove a few golden rectangles.
Title: Masterpieces: Start Exploring Coloring Book Comment: Inexpensive art book with 60 famous paintings all ready for coloring or golden section analysis. Terrific resource for both teachers and students.
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 in three parts (PART 1, PART 2, PART 3) courtesy of YouTube.
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Title: Fibonacci Numbers and the Golden Section Comment: Includes links to Fibonacci numbers and the golden section in nature, art, geometry, architecture, music, geometry and even for calculating pi! Fibonacci Numbers and Nature includes a section on Honeybees, Fibonacci Numbers and Family Trees.
Title: Fibonacci Interactive Comment: Features an excellent Java applet by Bruno Van Eeckhout depicting the Fibonacci growth graphical tree (requires a JAVAcapable browser).
Title: Fibonacci Numbers in Nature Comment: Digital slide show (with several links to animated sequences) detailing the occurrence of Fibonacci numbers in flower petals, branching plants, and leaf arrangements (phyllotaxis), as well as the spiral growth patterns in daisy cores, in pine cones and in pineapples. All graphics are linked to enlargements suitable for downloading.
Title: Phyllotaxis Comment: An interactive site for the mathematical study of plant pattern formation. Gallery has links to pictures showing the Fibonacci spirals in various plants. The Pineapple Movie is not to be missed (requires QuickTime Player). Rotate the pineapple with your mouse.
Title: Phi / Fibonacci / Phyllotaxis Comment: Seven sequential articles by the Janus team detailing hypotheses on the occurrence of Fibonacci numbers in nature. Sophisticated, but worth a visit.
Title: Golden Blossoms, Pi Flowers Comment: Details the occurrence of Fibonacci numbers in the spiral growth patterns of florets and seeds in sunflower heads. Fibonacci's Missing Flowers includes several color photos of flowers with a Fibonacci number of petals. Special: A fanciful demo by Bob Ippolito of growth in a Sunflower Head (requires Flash Player).
Title: Fibonacci Bamboozlement Comment: Drag the pieces from the square to the rectangle. Compute and compare areas of the square and the rectangle. Where has the extra square come from? Requires a JAVAcapable browser.
Title: The Fibonacci Association Comment: Official web site of The Fibonacci Association which focuses on Fibonacci numbers and related mathematics, emphasizing new results, research proposals, challenging problems, and new proofs of old ideas.
Title: Fascinating Fibonaccis Comment: Provides links to Fibonacci books and posters by Trudi Hammel Garland. Materials include lots of useful ideas for the classroom. Also available from Amazon.com.
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Title: The Legend of the Chessboard Comment: There are many examples of how poorly our minds are equipped to think exponentially. Oh, we can do it  but we often realize our mistake too late. This story is a classic. For a video version, visit Legend of the Chessboard, courtesy of YouTube.
Title: A Human Counter Comment: Each natural number from 1 up can be formed by adding certain terms of the binary sequence without using any term more than once. This classroom activity can be used to introduce binary numbers. If a sign is required, we write "1". If a sign is not required, we write "0". Requires Adobe Acrobat Reader. Binary Counter includes a nifty interactive animation. Counting in Binary features a YouTube video of a wooden binary counter for decimal numbers from 0 to 63.
Title: Binary Numbers Comment: Binary numbers use the same rules as decimal numbers, that is, the value of any digit (bit) depends on its position in the whole number. Decimal uses base ten; binary uses base two. Special: Click on the graphic above to watch the set of four light bulbs count to 15 in binary.
Title: Decimal and Binary Equivalence Comment: Use the arrows or the slider bar to explore the relationship between decimal and binary numbers from 0 to 255. Access Binary / Decimal Converter Calculator to convert numbers from one system to the other. Both require a JAVAcapable browser. DecimalBinary Conversion is a spiffy Flash alternative (requires Macromedia Flash Player).
Title: How to Count to 1,023 on Your Fingers Comment: If you've ever felt seriously limited by counting on your fingers this is the solution! Count in binary. It give a whole new meaning to the number 4. The linked Hand Counter Applet shows binary and four other ways to count on your fingers including the rather wimpy standard one using base 10. However, if you just can't give up base 10 you can still count to 99 on your fingers with a slight modification. Requires a JAVAcapable browser. For an interactive demo, visit Binary Finger Counting, part of John Selvia's Understanding Binary web site (requires Macromedia Shockwave Plugin).
Title: Binary Fun Comment: The objective of the game is to match a random decimal number shown by the computer, using the 8 binary keys (1  128). If the numbers match, you advance to the next round, and the timer increases as you advance. Requires Macromedia Flash Player.
Title: Multiplying by Doubling Comment: In many sections of Russia, the peasants employed until recently what appears to be a very strange method of multiplication. Learn the method and discover its relation to the binary numbering system.
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 Papercut (requires Adobe Acrobat Reader)?
Title: Digital Images: From Satellites to the Internet Comment: NASAdesigned classroom activity in which students learn about digital images and how satellites send information and pictures to earth using the binary system.
Title: Power Cards Comment: This game is a very simple demonstration of the binary search technique often used for quickly retrieving data from a database. Choose a number from 131. Select all the cards that contain the number by clicking on them, then click on the button for the computer to guess your number. Includes a link to a Print Version and The Trick Explained. Magic Cards presents the cards in a sequential format. Both formats require a JAVAcapable browser. Smartcards is a simple version with links that reveal how the game works as well as to Printable Cards. For a Flash version, visit The Amazing Age Predictor Cards (requires Macromedia Flash Player).
Title: Number Guessing Comment: Think of a number less than 100. The computer will display sets of numbers in succession. For each, press either "Yes" or "No" depending on whether your number is on the screen or not. After a while, the computer will correctly "guess" your number. Includes an explanation using binary numbers. Requires a JAVAcapable browser.
Title: LHS: Tower of Hanoi Facts Comment: In this binary game, the aim is to transfer all the disks from one peg to another peg, moving only ONE disk at a time, and so that a larger disk may not rest on top of a smaller one at any time. Includes a link allowing you to Play Tower of Hanoi online with up to 10 disks. Visit Tower of Hanoi Puzzle Pieces for a printable version of the game that uses up to 6 disks ... or purchase a Tower of Hanoi game from the LHS Museum Store. MazeWorks  Tower of Hanoi allows up to 12 disks and includes an autosolve feature. (The Speed scrollbar determines how fast the computer moves.) Both Towers of Hanoi Puzzle and Tower of Hanoi in Javascript allow up to 10 disks. Click on Help and the computer will make your next move. All applets require a JAVAcapable browser. Tower of Hanoi is a spiffy Flash version with up to 8 disks and a solution feature (requires Macromedia Flash Player). Tower of Hanoi Solution will show you how to solve the game with the minimum number of moves by accessing the binary numbers in sequential order.
Title: Wallingford Toy Works: Tower of Hanoi Comment: Commercial source of a Tower of Hanoi model fitted with 8 moveable disks. Woods: oak & walnut in combination. Another version is available from Nasco (reload page if denied access to information).
Title: Spinout / The Brain / Chinese Rings Puzzle Comment: Provides information on three more binary games. [The Brain is no longer commercially available.] Page includes a link to a virtual JavaScript Spinout Puzzle. [If you have Internet Explorer 4.0 or higher, try the interactive Elephant Spinout. All users can access the 4page Printable Solution. Crazy Elephant Dance and its linked Applet is suitable for all users as well.] Addicts can purchase a real Think Fun Spinout Puzzle. Spinout is a modern variant of the ancient Chinese Ring Puzzle  a favorite since 200 AD. Also known as Cardan's Rings or the Patience Puzzle. All of these games can be solved using sequential movements based on binary numbers. Patience Puzzle Solution shows the steps in solving the Chinese Rings Puzzle with a binary number interpreter for each move. For an interesting project, visit Making a Chinese Ring Puzzle and Solving the Chinese Ring Puzzle. Applets require a JAVAcapable browser.
Title: Nim Comment: Nim is an ancient game of pickup sticks for 2 players. Whoever picks up the last stick loses. This computer version requires Macromedia Flash Player. A winning Nim Strategy involves adding binary numbers. The macabre variation in Nim Skulls is more of a puzzle than a game. Once you solve the puzzle, you can win every time. Requires a JAVAcapable browser.
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Title: Pascal's Triangle Comment: Pascal's triangle is an arithmetical triangle made up of staggered rows of numbers. Read about its history and learn its construction algorithm.
Title: Development of Mathematics in Ancient China Comment: Did you know that Hsiang Chieh Chiu Chang Suan Fa developed Pascal's Triangle about 300 years before Pascal? Visit this web site to learn more.
Title: Interactive Pascal's Triangle Comment: An interactive version of Pascal's Triangle that let's you specify the number of rows. Requires a JAVAcapable browser. Includes a link to a noninteractive version.
Title: Discovering Patterns Comment: Click on each button to see where the Natural Numbers, Triangular Numbers, Tetrahedral Numbers, and Fibonacci Numbers appear in Pascal's Triangle.
Title: Number Patterns in Pascal's Triangle Comment: Includes Natural Numbers, Figurate Numbers (Triangular, Tetrahedral, Pentatope, Hexagonal), Fibonacci Numbers, as well as Powers of 2 and 11.
Title: Pascal's Triangle and Its Patterns Comment: Includes How to Construct Pascal's Triangle, Sums of Rows, Prime Numbers, Hockey Stick, Magic 11's, Fibonacci Sequence, Triangular Numbers, Square Numbers, Points on a Circle, Polygonal Numbers, and Connection to Sierpinski Triangle.
Title: The Twelve Days of Christmas and Pascal's Triangle Comment: A lesson by Judy Ann Brown. Object: Using Pascal's Triangle, find the number of items given each day in the song The 12 Days of Christmas.
Title: Probability / Combinations Comment: How many different ways can you choose two objects from a set of three objects? From a set of five objects? Pascal's Triangle can be used to find combinations.
Title: Binomial Coefficients Comment: Explores the relationship between Pascal's Triangle and the binomial coefficients, culminating in the classic binomial expansion. Requires a knowledge of algebra.
Title: The Pinball Game Comment: Each element in Pascal's Triangle represents the number of different Paths that a pinball can take from the apex of the triangle down to that point. Web page leads from a type of pinball machine to the classic binomial expansion. When done, visit The Normal Distribution for a nifty pinball applet (requires a JAVAcapable browser). Similar applets will be found in Quincunx (formal name for the board by Sir Francis Galton) and Plinko (name used on the TV game show Price is Right). Visit Probability from a TV Game for a report on a demo (with a short video, explanatory links, and a link to Plinko! Build a Board. A Flash version appears in Quincunx and/or Random Walk (requires Macromedia Flash Player). You can even download an offline version. The parody in Executive Decision Maker shows how CEOs really make the big corporate decisions (requires Macromedia Shockwave Plugin.)
Title: Pascal's Triangle Web Unit Comment: Explore this famous triangle through lesson plans that feature questions, answers, discussion, and Student Worksheets. [Click on a picture to go to a printable copy of each of the worksheets.] Of particular interest at the intermediate level is the associated web page Coloring Multiples.
Title: Coloring Multiples in Pascal's Triangle Comment: Allows the user to investigate number patterns in Pascal's Triangle created by placement of multiples. Requires a JAVAcapable browser.
Title: Explore Patterns in Pascal's Triangle Comment: Choose a number to use as a divisor (default value 2). The applet colors the first 128 rows of Pascal's triangle, using black if the corresponding number is evenly divisible by the divisor, and red if it is not. To change the divisor, enter a new number and click the Set Divisor button. The Larger Version of the applet will display the first 256 rows of the triangle. Both versions require a JAVAcapable browser.
Title: Pascal's Triangle Interface Comment: Lets you visualize the entries of Pascal's Triangle with respect to a modulus between 2 and 16. Each distinct value (mod p) is depicted by a unique colored square. Zero values are always depicted by black squares. Enter values for the number of rows (limited to 100), the modulus, and the size of the image, and then submit. Requires a JAVAcapable browser.
Title: Pascal Triangle Applet Comment: Similar to Pascal's Triangle Interface, however the zero (mod p) values are depicted by gray pixels and all nonzero values by black pixels. The modulus may be any integer between 2 and 15000. The number of rows is limited to 650. Click on SHOW and enjoy the image. Requires a JAVAcapable browser.
Title: Pascal's Triangle Comment: Download and decompress this little program by Remco de Korte with WinZip or freeware ZipCentral, then give it a whirl. No installation required. Allows you to reveal/color the multiples of 2 to 24 inclusive in the first 20 rows of Pascal's Triangle and to view a zoom of the pattern continued for the first 128 rows.
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Title: Conic Section Models Comment: Applets show the intersections of parallel planes and a double cone, forming hyperbolas, parabolas, and ellipses respectively. Click on an applet, hold the left mouse button down, then drag it to effect the dynamic rotation of the 3D model. Requires a JAVAcapable browser.
Title: Conic Sections Animation Comment: Watch the crosssection of a plane and a double cone. As the plane is rotated, different conic sections emerge. Animation includes the threedimensional image of the cone with the plane, as well as the corresponding twodimensional image of the plane itself. Requires Real Player.
Title: Conic Sections Comment: A Flash animation that shows how an ellipse, parabola and hyperbola can be obtained from a pair of cones. Requires Macromedia Flash Player.
Title: Conics in Clay Comment: The four conic sections can be easily visualized by slicing a doublenapped cone made from clay. A Styrofoam Cup can also be cut in circular, elliptical, parabolic, and hyperbolic cross sections.
Title: Conic Sections Comment: Features an excellent section on Flashlight Conic Sections. Click on each graphic for an enlarged view.
Title: Dissectible Wood Cone Comment: Commercial source of demonstration cone made of contrasting hardwoods. Shows the conic sections of a circle, ellipse, parabola, and hyperbola. Easily assembled/disassembled into five pieces. Reload page if denied access to information.
Title: Dancing Curves Comment: This booklet describes how to build a string cylinder that can be transformed into a double cone by rotating the movable end. Four color slides are included with the booklet. If lines from these slides are projected onto the reflective strings, cross sections of the conic sections are revealed. Outofprint NCTM publication.
Title: The Folded Rectangle Comment: Appropriate folding of a paper rectangle will produce an envelope of creases that will outline a parabola. Interactive picture shows the Geometer's Sketchpad way to model the foldedrectangle construction. Requires a JAVAcapable browser. Part of Daniel Scher's Geometry in Motion Website.
Title: The Folded Circle Comment: Appropriate folding of a paper circle will produce an envelope of creases that will outline either an ellipse or a hyperbola. Interactive pictures show the Geometer's Sketchpad way to model the foldedcircle. Requires a JAVAcapable browser. Part of Daniel Scher's Geometry in Motion Website.
Title: Occurrence of the Conics Comment: Details the occurrence of the ellipse, parabola, and hyperbola in the real world  from planetary orbits to satellite antennas. Supported by extensive graphics.
Title: Mathematical Curves: The Conics Comment: Successive internal links detail realworld examples of circles, ellipses, parabolas and hyperbolas with excellent supporting graphics.
Title: Coolmath: Circles Comment: Four unlinked pages, each page an introduction to a specific conic section. After Circles, visit Ellipses, Parabolas and Hyperbolas. All pages feature superior supporting graphics on applications, many of which are animated.
Title: The Pi Pages Comment: An excellent resource for anyone interested in learning more about Pi. Includes Pi Story, Pi Records, Pi People, Pi Literature, Pi News, and Pi Aesthetics. Click on More to select the language for the recitation of Pi's digits  from English to Mandarin.
Title: Where Does Pi Come From? Comment: Contains a simple explanation of Pi's origins plus several examples of calculating the ratio using actual measurements. Pi Animation has another example with a rolling circle. The former page has a link to Pi to 4000 Decimal Places ... just in case you needed them! For help in remembering Pi's digits, visit Mnemonics for the Number Pi. For a musical recitation of Pi's digits, visit 3.141592653589793...
Title: Approximating Pi Comment: Archimedes (287212 BC) used a fairly simple geometrical approach to estimate pi. See how he did it here. Requires Macromedia Flash Player.
Title: The Area of a Circle Comment: Why is the area of a circle Pi times the square of the radius? This animation gives a geometric justification. Area of a Circle has a more detailed treatment as does The Circle Area Formula. Crocodile's Teeth is a related handson teaching tool. The corresponding Teaching Notes detail its use. Approximating the Area of a Unit Circle with Regular Polygons takes a different approach (requires a JAVAcapable browser).
Title: Native American Geometry Comment: Native American Geometry is a physical, proportional geometry that originates from the circle. Divided into four subsections: Foundations, Anthropology, Designs, and Education. Foundations investigates constructions using only compass and straight edge. Designs investigates the construction secrets of symbols such as the YinYang and the CBS eye.
Title: Construction of an Islamic Pattern Comment: In the Islamic culture, the circle is the unit of measure. By following a few simple steps, you can construct the starhexagon pattern, a popular Islamic allover pattern, using only compass and straight edge. Requires Adobe Acrobat Reader.
Title: Trinity Knot Comment: A compass can be used to construct a Celtic Trinity knot that is perfectly symmetrical and easy to create. Here is a Graphic Summary. Source of text and graphics: Anon Celtic Art.
Title: What is a Mandala? Comment: Mandala is the Sanskrit word for circle. A mandala is a pattern, an integrated structure organized around a unifying center. Details the occurrence of mandalas in science, religion and art. Includes information on Education.
Title: Crop Circles Comment: Contains thumbnail links to aerial photographs of crop circles which have appeared in various fields throughout the world. Although theories abound as to their origin and significance, crop circles remain a beautiful, engaging, and ongoing mystery. Lucy Pringle's Crop Circle Photographs has links to higher resolution photos.
Title: Ellipse Game Comment: Includes an animation showing how its foci can be used to draw an ellipse. In the ellipse game, your aim is to find a focus. You will be given an ellipse of random dimensions and you will have to try to click on one of the foci. Requires a JAVAcapable browser.
Title: Ellipses with Pins and String Comment: In this ellipse applet, you can adjust the position of the foci (the pins) by clicking on one of them and dragging it left or right. Requires a JAVAcapable browser. Here's a simple Animated noninteractive graphic of the process and a Static version.
Title: Oval Mat Cutters Comment: Learn the basis of an oval mat cutter and an elliptical compass. [The mechanism, based on the Trammel of Archimedes, is also used in a DoNothing Machine (vacuum grinder)  a toy for keeping executives busy.] Includes an animation using the relevant mathematics which involves parametric equations. Visit How it Works or Ellipse Device for a similar animation. [When you are done with the latter, use the right arrow to proceed to Ellipse Foci and Ellipse Merge.] To interact with the device, explore the applet in The Ellipsograph. If you feel challenged, try using what you have learned to draw both a circle and an ellipse with an Analog Gadget. Applets require a JAVAcapable browser.
Title: Reflective Properties of Ellipses Comment: If the interior of ellipse is silvered to produce a mirror, rays originating at one focus are reflected to the other focus. Interactive applet demonstrates the property. Try changing the distance between the foci. Requires a JAVAcapable browser.
Title: Planet Paths: Studying Planetary Orbital Paths Comment: NASAdesigned classroom activity in which students learnthat planets travel in elliptical orbits around the sun and that planetary motion obeys laws defined by Kepler. Includes links to a Playground Ellipse Activity and to interactive web sites demonstrating orbital motion (including Planet Watch simulator shareware). Nature of the Universe has similar material with excellent graphics  including an animated Pin and String Ellipse.
Title: Billiards in the Round Comment: Learn all about the mathematical properties of elliptical billiard tables. The same author's Tricky Tables contains similar information. For more information, visit The Elliptic Pool Game. A photo of a table in use will be found at ElliptiBilliards. Purchase your own Elliptic Billiard Table from Math Love. Highly recommended! Special: Click on the thumbnail above for an enlarged view of the elliptical billiard table built by Camosun College mathematics instructor Dan Bergerud.
Title: The Taut String Comment: A parabola can be constructed with a right triangle, Tsquare, tack and a piece of string the same length as the vertical arm of the triangle (Graphic). Study the path in the interactive model. Can you prove it is parabolic? [Unfortunately the model draws only half of a the curve. Visit Generating a Parabola for an animation showing the entire curve.] Requires a JAVAcapable browser. Part of Daniel Scher's Geometry in Motion Website.
Title: Mirage Comment: Place any small object (like the petite pink pig provided by the manufacturer) in this incredible device and you will see a perfect 3D version of the object floating above the reflective circle. The mirage is produced by two parabolic mirrors facing one another. Understanding the Pig explains the phenomenon and details other experiments with the parabolic mirrors. Mirage (Pink Pig in a Poke) has display suggestions. Retail sellers include Sandlot Science and Eyetricks.com. Visit Real Image to learn how to make a $2 version from a silvered plastic Christmas tree ornament. Special: Click on the thumbnail above for an enlarged view.
Title: Parabolic Golf Comment: Click anywhere within the putting green and watch where the ball lands. The curved area behind the hole is a parabolic reflector. As long as the path is parallel to the sides, the ball will go to the focus of the parabola (the hole) every time!
Title: Fling the Cow Comment: A very popular sport in most of North Eastern New Guinea is known as cow flinging. The object of the game is to fling a cow onto a target. (No cows are injured during this sport. Studies show the cows actually enjoy their parabolic flight.) Click on the catapult to fling the cybercow (for points). The longer you hold the button, the farther it will fly. Requires Macromedia Flash Player. Download an offline Windows version. Decompress with WinZip or freeware ZipCentral to its own folder, then click on the program file. No installation required. Add your own comments to the farmer or delete him altogether.
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Title: Moire Patterns Comment: Moiré patterns are created whenever one semitransparent object with a repetitive pattern is placed over another. Includes a link to a Moiré Pattern Graph (a pattern of concentric circles suitable for printing on two sheets of transparency film). Watch moiré in motion at the Spatial Beats web page (requires Macromedia Shockwave Plugin.)
Title: Moire1 Comment: The basic pattern in this applet consists of lines radiating out from a common center. One copy of the pattern is fixed, and the other drifts about, creating a changing moiré pattern. You can start and restart the applet by shiftclicking on the pattern. You can also clickanddrag to control the motion of the pattern yourself. Requires a JAVAcapable browser.
Title: Pinwheel Moire Comment: Particularly striking Moire pattern from Sandlot Science. The curves in Circle Pattern III are hyperbolas. Many patterns in the web site require a JAVAcapable browser. More Moires requires Macromedia Flash Player.
Title: Moire Patterns Comment: This applet generates a virtually unlimited variety of moiré patterns. Colorful and amusing to watch. See also: Moire Circles Animation. Both require a JAVAcapable browser.
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TOPIC 9 (Line Designs & Curve Stitching)
Title: A Rhythmic Approach to Geometry Comment: Generate the illusion of classic curves (like parabolas, curves of pursuit, spirals, cardioids and other roses) out of straight lines using interactive files created with the Geometer's Sketchpad. Requires a JAVAcapable browser.
Title: Making Maths: Curve Stitching Comment: If you think that sewing isn't for you, think again. These curve stitching patterns look fantastic and once you've got the hang of it, they take next to no time to do. All you need is a little bit of coordinate knowhow!
Title: Curve Stitching Comment: Curve stitching utilizes basic geometric forms, making curves and circles out of straight lines. Focus is on the curve stitching of angles. A companion web page, not linked to this page, considers Curved Stitching Based on Circles.
Title: CurveStitch Designs Comment: Simple introduction to curve stitching. Includes a link to CurveStitch Isometric Cube and A Family of New Designs. The latter begins with a square and some parabolas, then a little generalization goes a long way.
Title: Math Cats: String Art Comment: Print out patterns for making your own string art pictures. Includes some ideas and patterns to get you started. Check out the instructions for creating an Icosihenagon (21sided polygon) design in the same web site.
Title: String Art 1 Comment: Create line designs interactively! Generates two overlapping designs based on the angles formed by concurrent intersecting lines. You can choose the angle size (length), the number of angles (parts), and the color used in each layer. All generated "curves" are parabolas. The same author's Mystic Rose features a variable number of points evenly spaced around a circle in which every point is joined to every other point. Both require Microworld's Web Player. The applet in another Mystic Rose promotes the investigation of "jumping" (requires a JAVAcapable browser).
Title: Line Designs Comment: This book contains patterns for over 100 designs using a straightedge, compass, needle, and thread. It includes reproducible worksheets, activity directions, and teacher reference guides.
Title: Line Designs for the Computer Comment: Utilizes 33 program from the first edition of the book Curve Stitching by Jon Millington to create line designs. The programs were written for the historic Spectrum computer. Includes a JAVA Emulator to view the programs online (requires a JAVAcapable browser). Primitive technology  but the results are fascinating! The book is highly recommended  a curve stitching bible.
Title: Envelopes of Lines and Circles Comment: Uses the power of The Geometer's Sketchpad to examine the curves formed by sets of lines or circles that move along some defined paths. An Instructor's Evaluation Edition of Sketchpad can be downloaded from Key Curriculum Press. Movie illustration requires QuickTime Player.
Title: Herrschners Online: South Maid® Crochet Cotton Comment: Online source of 100% mercerized South Maid® Crochet Cotton in a wide range of colors. Perfect medium for curve stitching exercises.
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TOPIC 10 (Curves of Constant Width)
Title: Reuleaux Triangle Comment: Because a circle has the same width in all directions, it can be rotated between two parallel lines without altering the distance between the lines. Is the circle the only curve with constant width? Special: Enlarged View of one of the Reuleaux triangle windows of the 13thcentury Notre Dame Cathedral in Bruges, Belgium.
Title: Rolling Reuleaux Triangle Comment: Move your mouse slowly between the two bars to roll the Reuleaux triangle. When done, move the Rolling Rounded Reuleaux Triangle. Both require a JAVAcapable browser.
Title: Shapes of Constant Width Comment: There are shapes (curves) of constant width other than the circle. A Reuleaux triangle is the simplest example of a such a shape. [The applet can be in one of three modes. Click inside it to change modes.] The companion applet in a Star Construction of Shapes of Constant Width shows how to construct other, less regular, shapes of constant with by starting with star polygons. Both require a JAVAcapable browser. Special: An excerpt from the classic film Curves of Constant Width, courtesy of YouTube.
Title: Reuleaux Triangle Comment: Contains links to files that allow you to visualize the rolling of a Reuleaux triangle between an appropriate pair of parallel lines and inside a square with sides of the same width using The Geometer's Sketchpad. In each, the path of the centroid is shown. [An Instructor's Evaluation Edition of Sketchpad can be downloaded from Key Curriculum Press.] If your browser has JAVA capability, you do not need Sketchpad. Just follow the link to the applet Rolling Reuleaux Triangle.
Title: Rolling with Reuleaux Comment: Considers the construction of curves of constant width from any polygon with an odd number of sides, as well as their applications to coins and to rotary drills that bore square holes. Like a circle, a Reuleaux triangle fits snugly inside a square having sides equal to the curve's width no matter which way the triangle is turned. As it rotates, the curved figure traces a Path that eventually covers just about every part of the square (except for a little rounding at the corners). Note the locus of the center of the triangle.
Title: Reuleaux Pentagon in Hexagon Comment: Perhaps a Reuleaux triangle rotating in a square is too ... square for you. So here's the Reuleaux pentagon rotating even more happily within a hexagon.
Title: Drilling Square Holes Comment: A bit with the shape of a curve of constant width can be used to drill a square hole. Square & Hexagonal Drill has more information. Visit Square Hole Drilling for a video, courtesy of YouTube.
Title: Wankel Engine Comment: A Reuleaux triangle is used for the rotor in Wankel engines (Photo of Wankel Engine in Deutsches Museum). Wankelmotor features a similar animation. A simpler version will be found at Rotary Engine Applet. All require a JAVAcapable browser. How Rotary Engines Work features a Flash animation (requires Macromedia Flash Player).
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TOPIC 11 (Cycloids / Spirograph / Famous Curves)
Title: Riding on Square Wheels Comment: Maybe you can't fit a square peg in a round hole, but that doesn't mean you can't ride a bike with square wheels.
Title: Edwin's Animated Images Comment: Features links to delightful animations by Edwin Attaway. Options include a Cycloid, Curtate Cycloid, Prolate Cycloid, and Cardioid.
Title: Roulettes Comment: Animated gifs of roulettes from Eric Weisstein's World of Mathematics. Include the Trochoids (Cycloid, Curtate Cycloid, Prolate Cycloid), various Hypocycloids (Deltoid, Astroid), and various Epicycloids (Cardioid, Nephroid, and 5cusped Ranunculoid).
Title: Applet: Cycloids  Maths Online Gallery Comment: Cycloids emerge as the paths traced out by the motion of points on a wheel (disk) which rolls on a straight line. To access the applet, click on the red button on its own window. By means of two scroll bars, the wheel can be moved and the position of the marked point relative to the center of the wheel can be adjusted. Requires a JAVAcapable browser.
Title: Cycloids Comment: Another applet for drawing cycloids. Press the Start button to start the small circle rolling. Click any time in the drawing area to attach the point under the cursor to the circle. [The point will be assigned a random color so its motion can be traced.] Requires a JAVAcapable browser.
Title: Cycloid as Brachistochrone Comment: An inverted cycloid is the brachistochrone, that is the curve between two points in a vertical plane, along which a bead needs the shortest time to travel. Features a race between a bead on an inverted cycloidal ramp and one on a linear ramp. Requires a JAVAcapable browser.
Title: The Cycloid Family Comment: Compact visual presentation of the family of cycloids  cycloid, curtate cycloid, and prolate cycloid. Cardioid features a similar animation. All require Macromedia Flash Player.
Title: Cycloids, Hypocycloids, Epicycloids Comment: Looks at the power of The Geometer's Sketchpad to investigate cycloids, hypocycloids, and epicycloids. An Instructor's Evaluation Edition of Sketchpad can be downloaded from Key Curriculum Press. Movie illustrations require Apple's QuickTime Player.
Title: SpiroGraph Comment: Remember those little plastic wheels that spun around and made those fascinating patterns? [Here's a commercial from the 1960's showing Spirograph in action.] This applet allows you to generate all the spirographs your heart desires interactively. You can also Download the applet for offline viewing. Want more? Visit Anu Garg's Spirograph or Brett Allen's easytouse Java Spirograph. All require a JAVAcapable browser. Nathan Shields' Spirotica and John Grindall's Flash Spirograph require Macromedia Flash Player.
Title: Spirograph Comment: Allows you more choices in ring and wheel size as well as pen color than in the original 1960's toy. The author's Real Spirograph limits you to the original rings, wheels, and pen colors. It also mimics the negative effects you could experience with the real toy  like the rings shifting or sliding, or the the pen coming out of the hole and drawing (mistakenly) across the picture. It even mimics the fact that the pens could run out of ink very quickly. Both require a JAVAcapable browser.
Title: Epicycles for Windows Comment: Download a working copy of Epicycles for Windows, an excellent Spirograph program. Allows you to pull the gear around the ring with the mouse  just like you used to do with Spirograph  or to draw the patterns automatically. Includes print option, but printouts will include a "trial copy" watermark. An inexpensive registered copy is available online.
Title: Spirographer Comment: Another inexpensive shareware Spirograph program (Windows and Mac). Decompress the program file with WinZip or freeware ZipCentral. Very simple to use. A User's Manual is available online. During the 30day evaluation period, some of functions are restricted. A registration key (US $20) is available through Kagi.
Title: Famous Curves Index Comment: Click on the name of a curve to see its history and some of its associated curves. If your browser supports JAVA, you can experiment interactively with each curve and its associated curves. All of the applets can be accessed directly via the Famous Curves Applet Index.
Title: A Visual Dictionary of Special Plane Curves Comment: Covers the history, description, formulas, and properties of about 30 curves. The work is heavily enhanced with illustrations, Quick Time movies, Geometer's Sketchpads, and Mathematica notebooks.
Title: Wise Turtle Stories Comment: Three entertaining stories about curves by wise turtle from Logo country. The first story is about Spirals, the second about Wheels, and the third is about Fractals. This site was developed as an entry for the ThinkQuest'98 competition.
Title: Pendulums: Patterns from Sand Comment: Back in the 1970's, I used to create intricate geometric designs with a toy called a PendulArt. It consisted of a pen that remained stationary and a tray that swung beneath the pen, acting as a pendulum. As the tray moved, the pen would trace out a complicated, diminishing pattern  known as a Lissajous Figure. Learn how to make Lissajous patterns by dropping sand from a swinging pendulum. Sand Pendulums has a similar experiment designed for teams (requires Adobe Acrobat Reader). Lissajous Sand Pendulum features a movie of the process (requires QuickTime Player). The device is available commercially as a Sand Pendulum. Make similar patterns online by moving the horizontal and vertical sliders in Questacon's Lissajous Patterns (requires Macromedia Shockwave Plugin.)
Title: Tacoma Bridge Disaster Comment: My favorite example of a real world curve. Movie clip includes commentary. Learn more about the infamous Galloping Gertie by visiting The Tacoma Narrows Bridge Disaster. Includes a link to a contemporary newsreel. All clips require QuickTime Player. The lengthy colorized Tacoma Narrows Bridge will take a while to download (requires Real Player).
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Title: What IS a Fractal? Comment: A simple Flash cartoon giving a brief illustration of Natural, Mathematical and Artistic fractals. Includes fractal links. Requires Macromedia Flash Player.
Title: 17 Captivating Examples of Fractals Found in Nature Comment: Fractals are purely a wonder  too irregular for Euclidean geometry; iterative and recursive and seemingly infinite. They turn up in food and germs, plants and animals, mountains and water and sky.
Title: Fractals Comment: Brief introduction to simple fractals (Koch snowflake and the Sierpinski triangle). Part of Mathematrix  a web site devoted to exploring mathematical recreations. Special: Koch Curve Zoom
Title: A Fractals Lesson  Recognitions Comment: This fractals site is for kids, to help them understand what the weird pictures are all about  that it's math  and that it's fun! Includes links to Why Study Fractals?, Making Fractals (The Sierpinski Triangle and the related Sierpinski Meets Pascal, The Jurassic Park Fractal, and The Koch Snowflake), and Fractal Properties (including SelfSimilarity). Requires a JAVAcapable browser. Every lesson has a print version for classroom use.
Title: Fractals  An Introduction Comment: Learn simple methods to construct fractals. Linked to six successive web pages featuring several JAVA applets (The Koch Curve Fractal, The Sierpinski Triangle Fractal, and The Dragon Curve or Jurassic Park Fractal) for handson construction and interaction with simple fractals. Requires a JAVAcapable browser.
Title: The Snowflake Curve Comment: Clear explanation of Koch's snowflake. Most effective if followed by the interactive Koch's Snowflake applet (requires a JAVAcapable browser). Koch's Snowflake Fractal presents the iterative process as a Flash movie (requires Macromedia Flash Player).
Title: Sierpinski's Triangle Comment: Allows the user to step through the process of building the Sierpinski Triangle (also known as the Sierpinski Gasket or Sierpinski Sieve). Requires a JAVAcapable browser. Click on What? for a background explanation. Another interactive option is More Sierpinski Triangle (requires Microworld's Web Player). Sierpinski Triangle presents the iterative process as a Flash movie (requires Macromedia Flash Player). Top it off with the interactive Sierpinski Tetrahedron. The latter page requires a JAVAcapable browser.
Title: Sierpinski's Carpet Comment: This activity should be tried after the previous Sierpinski's Triangle for comparison purposes. Requires a JAVAcapable browser. Click on What? for a background explanation.
Title: The Chaos Game Comment: Perhaps the most amazing thing about fractals is that totally random processes can lead to totally deterministic results. The results of the Chaos Game will always trace out the Sierpinski Triangle. Experience a computer simulation of the process in The Fractal Game or the interactive Sierpinski Triangle. Both require Macromedia Shockwave Plugin. A terrific interactive turtle version can be accessed in The Sierpinski Triangle (requires Microworld's Web Player).
Title: Pascal's Triangle: Sierpinski Triangle Comment: Part of the Math Forum's Pascal's Triangle Web Unit. Details the fascinating connection between the Sierpinski Triangle and Pascal's Triangle.
Title: Dragon Curve Comment: Choose which Dragon curve (large fractal or small fractal) you would like to see rendered in a Flash movie. Requires Macromedia Flash Player.
Title: Fractal Grower Comment: JAVA software and documentation designed to introduce the curious person to fractals. Simple, stepbystep processes of paper folding can give rise to infinite varieties of fractals  including the classic Dragon curve. Requires a JAVAcapable browser. Netscape users will require version 4.5 or higher.
Title: PopUp Fractal Comment: One of the simplest fractal cards to make. However, it shows a lot of the properties of a fractal, including selfsimilarity and infinite detail. Fractal Cut, Fractal Cut PopUp Cards and Fractal Paperfolding have similar instructions. Fractal Cards offers other alternatives. Several files require Adobe Acrobat Reader.
Title: Fractal Tool Comment: This applet allows you to play with and create fractals. View preset iterations of various shapes and/or choose to create your own iterations. Requires a JAVAcapable browser.
Title: Fractal Color Scheme Chooser Comment: Create digital fractal art with a color scheme that you choose yourself. Click on the Make a Poster button to generate your fractal creation as a 720x480 GIF image. Requires a JAVAcapable browser.
Title:  Big Al's Fractals: Images Index Page 
Comment:  An excellent collection of 120 fractal images freely available for noncommercial use. Click on any of the small pictures (like the one above) to view a full screen version. Options include Seamless Tiled Backdrops for your web page. 
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PART 2 
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