
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.
Requires 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: 
FourthOrder
Magic Squares 
Comment: 
Presents a quick and sure way to create a fourth order normal magic
square, that is, an arrangement of 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,
namely 34. You can also experiment with the applet in GridPuzzle
or Magic
Square. The latter includes a link to an Answer
Key. All three applets requires a JAVAcapable browser. For an interesting
variation, visit Magic Square Puzzle (requires
Macromedia
Flash Player).
Drag the puzzle pieces to the spaces on the board so that each row and
column adds up to 30. 
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: 
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. 
TOP 
TOPIC
2
(Clock or Modular Arithmetic)
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: 
Mod
P Calculator 
Comment: 
This calculator does arithmetic (mod P) where P is an integer that
you set. Integers are limited in size only by the size of the display.
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 flash application
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 Macromedia
Flash
Player. 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 to 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 Ultimate
Paint or Windows 7 Paint to crop and/or resize each graphic to a square.
Visit Ultimate Paint Pointers or Squaring
Photos 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) 
TOPIC
3 (The Golden Ratio)
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: 
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. 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: 
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: 
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: 
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. 
TOP 
TOPIC
4 (Fibonacci Numbers)
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: 
Nature by Numbers 
Comment: 
Created by Cristóbal Vila, this short movie presents a series
of animations illustrating various mathematical principles, beginning with
a breathtaking animation of the Fibonacci sequence. Visit The
Theory Behind This Movie for more information. 
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. Visit YouTube's Mathematics
Amazing for an animation of this classic paradox. 
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: 
Classic book on Fibonacci numbers by Trudi Hammel Garland. Explains
their occurrence in nature, mathematical properties, and historical significance.
Or, for the younger reader, try Trudi's Fibonacci
Fun. 
TOP 
TOPIC
5 (Binary Numbers)
Title: 
Inchworm ... Inchworm 
Comment: 
The classic binary counting song performed by Danny Kaye in the 1952
film "Hans Christian Anderson", courtesy of YouTube. 
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. Another Binary
Counter 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: 
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 (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: 
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.
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.) Towers
of Hanoi Puzzle allows 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: 
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. 
Title: 
The Socratic Method 
Comment: 
Details a teaching experiment to see whether third grade
students could be taught binary arithmetic only by asking them questions. 
Title: 
Binary TShirt 
Comment: 
Stumps them every time! There are only 10 kind of
people in the world: Those who understand binary and those who don't. 
TOP 
TOPIC
6 (Pascal's Triangle)
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: 
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: 
Probability
/ Combinatorics 
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: 
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. Also
available in an online
version (requires Macromedia
Flash
Player). 
TOP 
TOPIC
7 (The Conics)
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: 
You can create Flashlight Conic Sections by projecting the light at
the wall, allowing the wall to be the plane and the light from the flashlight
being the cone. 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: 
Folding
Conic Sections 
Comment: 
Appropriate folding of wax paper circles or rectangles produces envelopes
of creases that will outline another circle, an ellipse, a parabola, or
a hyperbola. 
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: 
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: 
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: 
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: 
MandalaMaker Software 
Comment: 
MandalaMaker is a fullfeatured application which allows you
to create radially symmetrical designs of many kinds, from traditional
Tibetan style mandalas, to striking contemporary art. Highly recommended.
Make
Your Own Mandala! is an online Flash alternative. 
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
Do
Nothing 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: 
Billiards in the Round

Comment: 
Learn all about the mathematical properties of elliptical billiard
tables. For more information, visit The Elliptic
Pool Game. Here is a photo of an elliptical
table in use. 
Special: 
Click on the thumbnail above for an enlarged view of the elliptical
billiard table built by Camosun College mathematics instructor Dan Bergerud. 
Title: 
Mirage: What is a Hologran?

Comment: 
Place any small object 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. Air
Pig explains the phenomenon and includes an excellent video link. The
device is manufactured by OptiGone
International. 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 of the petite pink
pig hologram. 
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. 
TOPIC
8 (Moiré Patterns)
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: 
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. 
TOP 
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: 
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 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. 
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? Visit Wonky
Wheels for a descriptive poster (requires Adobe
Acrobat Reader). 
Special: 
Enlarged
View of one of the Reuleaux triangle windows of the 13thcentury Notre
Dame Cathedral in Bruges, Belgium. 
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. 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 a 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: 
Reuleaux Wheeled Bicycle 
Comment: 
A bicycle patented in China with wheels that are a Reuleaux pentagon
and a Reuleaux triangle. Includes a YouTube video of the bicycle
in motion. 
TOPIC
11
(Cycloids / Spirograph / Famous Curves)
Title: 
Riding on Square Wheels 
Comment: 
A square wheel may be the ultimate flat tire. 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: 
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. The time taken for the bead to travel from any
point on the cycloid to the bottom will always be less than on the corresponding
straight incline. Requires a JAVAcapable browse. For a video version,
visit YouTube's Brachistochrone.
Furthermore, if a couple of marbles (or toy cars) are released from different
points on identical cycloidal ramps, they will arrive at the bottom simultaneously
although one has farther to roll than the other. See YouTube's Brachistochrone
Race. 
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: 
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.] The current crop
of Spirograph toys are not as sophisticated as the original models which
abound on eBay. Replacement pens
(including a multicolor
pen), pins
and baseboards
can be purchased from SoundFeelings. Online spirographs are actually
MORE sophisticated than the original toy. 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,
John Grindall's Flash Spirograph, and Spirograph
Math require Macromedia Flash
Player. 
Title: 
Spirograph 
Comment: 
Wonderful spirograph applet by Liz Vinsel Looney. 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: 
Spirographer

Comment: 
An 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.) 
TOPIC
12 (Fractals)
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: 
Fractals 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: 
Geometric Fractals 
Comment: 
Features 3 classic fractal applets (The Koch Curve, The Dragon Curve,
The Sierpinski Triangle) by the late Jacobo Bulaevsky of Arcytech. For
the original applet webpages, visit Arcytech.
Applets require a JAVAcapable browser. 
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: 
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: 
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.
[Unfortuately the Make a Poster button crashes.] Click on the graphic
above for an enlarged view. Requires a JAVAcapable browser. 
TOP 
PART
2 
All M. C. Escher works (c)
Cordon Art B.V.  Baarn  the Netherlands.
Used by permission. All rights reserved. 