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Investigating
Patterns |
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Symmetry
and
Tessellations
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ACTIVITY LINKS
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ACTIVITY
1 (What is Mathematics?)
| Title: |
What
is Mathematics? |
| Comment: |
Observations on the the nature of mathematics from Pathways to School
Improvement. Includes: "Mathematics should include experiences that
help students to shift their thinking about mathematics and define mathematics
as a study of patterns and relationships." |
| Title: |
Pythagoras:
Music and Space |
| Comment: |
A brief introduction to Pythagoras' observations on the relationship
between music and number. Graphics show music intervals expressed as mathematical
ratios. If you have a plug-in that can play sound (like
Real
Player), you can pluck the strings to hear the notes. |
| Title: |
How to Make Ukrainian Easter
Eggs
 |
|
| Comment: |
Tells you everything you need to know about the basics of making pysanky
(Ukrainian Easter Eggs). Click on any design and you'll be taken to the
instructions for it. Challenge students to explain why the egg at the right
from Pysanky Designs
(Advanced Designs) is different from the others. [Click
on the egg for the solution.] Symbols
and Colours can be used to investigate the geometric nature of the
designs. There's even a Colouring
Book for kids. |
 |
| Title: |
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. |
ACTIVITY
2 (Introduction to Symmetry)
| Title: |
Photo
Activity for Symmetry |
| Comment: |
The left side of a human face is close to being
the mirror image of the right side - but not exactly. Includes details
on an activity requiring a digital camera and a simple paint program, such
as Paint
Shop Pro or Ultimate Paint.
An evaluation copy of each program is available for download (30-day and
30-use restrictions apply respectively). Visit Symmetrical
Jill Britton for another example. |
| Title: |
SymFace |
| Comment: |
SymFace is a simple and fun way to see
how your face would look if it was perfectly symmetrical. Just follow the
steps using an appropriate digital picture of yourself. [If necessary,
use Ultimate Paint to crop
and/or resize your picture. Visit Ultimate
Paint Lab for details.] The results can be saved and/or printed. Requires
a JAVA-capable browser. |
| Title: |
The Butterfly Web Site |
| Comment: |
The most complete information on butterfly gardening, farming, ecology,
and education. Includes a link to a Photo
Gallery containing butterfly and moth images. Look for images with
a symmetrical configuration. In most cases, a larger picture is available
by clicking on a small picture. |
| Title: |
Manual of Traffic Signs
 |
| Comment: |
Contains graphics of the most commonly used traffic signs in the United
States. You can use the site to challenge students to find symmetrical
signs. |
| Title: |
SymmeToy - Symmetric
Art Program |
| Comment: |
Permits downloading of a trial version of SymmeToy, a Windows
program for creating paint patterns, symmetry roses, tessellating art,
and so forth. |
TOP |
ACTIVITY
3
(Symmetry in the Alphabet)
| Title: |
On Reflection |
| Comment: |
Details how some clever "secret" coding can be done using a small mirror
and the reflectional symmetry characteristics of many capital letters and
some numbers. |
| Title: |
Inversions
 |
| Comment: |
The inversions gallery of Puzzlemaster Scott Kim, whom Isaac Asimov
called "The Escher of the Alphabet". An inversion is a word or name
written so that it reads in more than one way. Some inversions read upside
down, some flip in a mirror. Some read as two different words, some read
as the same word both ways. Web page includes a link to Teacher
Resources for Inversions with classroom exercises. Half
Words is an intriguing and challenging warm-up exercise suitable for
printing. The corresponding teacher notes includes Answers.
Don't miss Scott's
Origami
with its animated vertical reflectional symmetry. His Signature
Piece also makes for interesting animations. |
| Title: |
Ambigrams, Logos, Word
Art
 |
| Comment: |
The generic term for inversions is ambigrams, a word coined by cognitive
scientist Douglas Hofstadter. This is the ambigram webpage of John Langdon,
known for his
Angels
& Demons ambigrams. Ambigram
Generator produces ambigrams with rotational symmetry for any word
or phrase you type in. |
TOP |
ACTIVITY
4 (Symmetry in Flags, First Nations Art, and Quilt Blocks)
| Title: |
Flags
 |
| Comment: |
Contains links to images of National Flags, International Maritime
Signal Flags, Motor Racing Flags, and the Semaphore Flag Signaling System.
Additional flag images are available from the World
Flag Database and the Flags
of the World web site. The latter includes links to an on-line Colouring
Book of Flags with images available in three sizes. |
| Title: |
Carving at
Skidegate |
| Comment: |
A photo essay of a symmetrical totem pole being carved at Skidegate,
Haida Gwaii (Queen Charlotte Islands, British Columbia, Canada). The photos
show the carving process from raw log to finished pole. |
| Title: |
Tony
Hunt Designs |
| Comment: |
My favourite northwest coast native artist. Tony Hunt clothing by Creative
Embroidery is highly recommended - in particular articles embroidered
with his Symmetrical Sun. |
| Title: |
Quilt Maker |
| Comment: |
Design a quilt block by clicking on the squares. Make a quilt by selecting
REPEAT,
REFLECT
or ROTATE. Includes an index of quilt block patterns. Quilters
(from ProtoZone)
features the same activity in a different configuration. Quilt
Reflect challenges students to complete the symmetrical design of a
quilt while racing against the clock. All require Macromedia
Shockwave
Plug-in. |
| Title: |
QuiltComposer |
| Comment: |
QuiltComposer is shareware that will enable you to design your
own quilts before you make them. Accompanying instructions are simple and
clear. |
| Title: |
Quilt Design Program
Mini-Reviews |
| Comment: |
Contains a brief review of several quilt design programs. Links permit
you to download a demo version of some programs. In the demo version of
PCQuilt
(Windows or MacIntosh), you will be able to try most of the features of
the full version but some of the menu items will be disabled. Quilt-Pro
is the Windows-based program of choice. |
|
TOP |
ACTIVITY
5
(More on Reflectional and Rotational Symmetry)
| Title: |
Matchbox Art
 |
| Comment: |
Weird and wonderful, this amazing collection of images
celebrates the unique art form of topsy-turvy faces. Cards can be viewed
upside down or right side up (or is it the other way around?) - either
way, a face appears. Cracker Jack's 2-Face
Stickers has further examples. This upside down ad for Niblets
Corn appeared on the back cover of Life Magazine in 1953. |
| Title: |
The Upside Downs of Gustave Verbeek
 |
| Comment: |
Gustave Verbeek published a series of upside down comics
in The Sunday New York Herald in the beginning of the 1900's. The
first part of the comic is read normally, then you can turn it upside down
and the story continues! Contains links to five complete comics. In a similar
vein, this ad for Niblets Corn appeared on the
back cover of Life Magazine in 1953. |
| Title: |
Symmetry |
| Comment: |
A Flash movie by Mike Shepperd featuring two geometric
shapes - one with 4th order rotational symmetry and no reflectional symmetry
- the other with 2nd order rotational symmetry and 2 lines of reflectional
symmetry. Requires Macromedia
Flash
Player. |
| Title: |
The Diamond 16 Puzzle |
| Comment: |
Rearrange the rows, columns, and quadrants in a 4 x 4
array of 2-colour tiles to solve this online Javascript puzzle and make
hundreds of symmetric patterns. For a somewhat more user-friendly version
of the puzzle, visit Alexei Vernitski's 16
site. |
| Title: |
The Best Brands of the World |
| Comment: |
Features a collection of over 60,000 corporate logos.
Challenge students to find familiar logos with reflectional and/or rotational
symmetry or to prepare logo exercises. Search by company name. [Downloads
are EPS files. Open with a high-end graphics program like Photoshop or
Photoshop Elements, specify a large resolution, then adjust to the desired
size. Or insert into a Word document (using the Insert / Picture / From
File Option) provided the Word EPS filter is installed.] Logotypes
has a similar format and usage. |
| Title: |
GoodLogo!com |
| Comment: |
Logo database is searchable by company name and category,
but lacks many major corporations. Cick on thumbnails to access enlargements
(gif
files). Saved graphics can be opened with a simple graphics program
or inserted directly into a Word file (using the Insert / Picture / From
File Option). |
| Title: |
Family Crest
 |
| Comment: |
There are over 4,500 Ka-Mon or family crests in
Japan. Artistically simpler than their European counterparts, they are
a wonderful cultural source for symmetry investigations. This web page
provides an excellent introduction to the history and structure of the
crests. Japanese
Crests has links to highly stylized, symmetrical crests based on butterflies.
Kamon
features a chart of 259 Japanese family crests. |
| Title: |
Symmetry Around the
World |
| Comment: |
A collection of web pages by students of St Kieran's ManlyVale
(Sydney, Australia) to help children around the world find out about reflectional
symmetry and rotational symmetry. Elegant illustrations include examples
from nature, the human body, masks, reflections in water, architecture,
geometric shapes, hubcaps, kaleidoscopes, capital letters, advertising
logos, and more. |
| Title: |
Symmetry
Artist |
| Comment: |
Mathematics and Art come together! Create your own line
drawing with reflectional and/or rotational symmetry. Applet permits selection
of mirror line orientation, the order of rotation, and the pen width, color
and style. Part of Maths Is Fun
website. Requires a JAVA-capable browser. |
|
TOP |
ACTIVITY
6 (Symmetrical Strip Patterns)
| Title: |
Frieze Patterns |
| Comment: |
Details the seven combinations of symmetries that can be found in a
frieze pattern. Sequential pages include exercises and corresponding answer
key. |
| Title: |
Frieze
Patterns in Cast Iron |
| Comment: |
Contains examples of all seven frieze pattern types in cast iron decorative
work - in the Sidney Lace of Australia - and the rest of the world.
You can find cast iron patterns everywhere! |
| Title: |
Border Liner |
| Comment: |
Make patterns which go on and on forever, repeating the same pattern
in one direction or the other. BorderLiners
(from ProtoZone)
features a similar interactive activity. Sampler
(also from ProtoZone)
allows you design a cross-stitch sampler that stretches out into seven
border patterns. All require Macromedia Shockwave
Plug-in. |
| Title: |
Great
Math Fun With Paper Dolls |
| Comment: |
With strings of paper dolls, you can decorate a party, make the border
of a bulletin board, create a frame for your favourite picture, and of
course, learn some math in the process. Visit How
To Make Paper Chains to access links to templates for making Christmas
chains, dinosaur chains, Halloween chains and wild animal chains. |
ACTIVITY
7 (Introduction to Polygons)
| Title: |
Connect the Dots |
| Comment: |
Select a numbered dot to begin. Draw a polygon as you click on the
dots sequentially to complete the image. There is an animation on completion.
Seven different examples with a US Air Force theme. Requires Macromedia
Flash
Player. |
| Title: |
Connect the
Dots |
| Comment: |
Draw a polygon as you click on the dots sequentially to complete the
image. There is an interpretation on completion. All examples involve
Chevron
Toy Cars. Depending on the game you wish to play in this website, you
will need a Macromedia Flash
or Macromedia Shockwave
plug-in. |
| Title: |
Lizards
of M. C. Escher SoftPuzzle
 |
| Comment: |
Through this site you can purchase the Lizards of M. C. Escher SoftPuzzle,
an officially licensed adaptation of Escher's famous reptile. |
| Title: |
Tangrams
 |
| Comment: |
Tangram is a centuries old Chinese tile game consisting of 7 geometric
shapes. Learn its history, make your own tan set, or try solving one of
the shapes. Includes great downloads and interesting links to explore. |
ACTIVITY
8 (Regular Polygons)
| Title: |
Naming
Polygons and Polyhedra |
| Comment: |
What do you call an 11-sided polygon? A 14-sided polygon? A 50-sided
polygon. Doctor Math provides Greek prefixes for the first 50 polygons,
among others. |
| Title: |
Welcome to
the Polygon Playground |
| Comment: |
Come play with polygons! You can drag and arrange hundreds of colorful
polygons to create your own online art. The introductory page includes
a little slide show of ideas to get started, including: patterns, symmetrical
designs, and polygon animals. Requires a JAVA-capable browser. |
| Title: |
Plentiful Pentagons |
| Comment: |
The pentagonal shape is one of the primary geometric forms for organic
shapes (living organisms). Many flowers and starfish have pentagonal shapes. |
| Title: |
Snowflakes |
| Comment: |
All snowflakes have a fundamentally hexagonal shape. Generate symmetrical
snowflakes with this project. Set the slider bar to thin line (left) or
thick line (right), then press draw. Requires Microworld's
Web
Player. Or you can Create
Your Own Snowflakes and then execute a 2D or 3D rotation (requires
Macromedia Flash
Player). SnowFlaker
(from ProtoZone)
challenges you to draw the same snowflake twice (requires Macromedia
Shockwave
Plug-in). |
| Title: |
Paper Snowflakes for Children |
| Comment: |
Everyone can make artist quality paper "snowflakes" by following these
easy patterns! Many patterns were adapted from pictures of real ice crystals.
Make Mother's Day, Valentine's Day, Halloween Snowflakes as well as Christmas
snowflakes! |
| Title: |
Wilson Snowflake Bentley
 |
| Comment: |
Provides a comprehensive resource of everything related to Wilson "Snowflake"
Bentley. Links to Original
Wilson Bentley Images, a collection of 21 of Bentley's snow crystal
photomicrographs. Click on each thumbnail for a closeup view. |
ACTIVITY
9
(Symmetry and Regular Polygons)
| Title: |
BBC
Education - Revision Bite - Symmetry |
| Comment: |
Interactive animated treatment of reflectional and rotational symmetry,
including symmetry in regular polygons and other polygonal shapes. Includes
classroom exercises. |
| Title: |
Foto-TriHexaFlexagon
 |
| Comment: |
Got a colored printer? This simple flexagon program will allow you
to create a pictorial tri-hexa-flexagon from three images. Includes detailed
instructions (uses Ultimate Paint)
and a link to a download of the program file (by
ZöBook
On-Line). |
TOP |
ACTIVITY
10
(Symmetry by Paper Folding and Cutting)
| Title: |
Snowflakes |
| Comment: |
Contains simple instructions for folding and then cutting snowflake-like
designs with two, four, six, and eight repeats. Excellent supporting graphics. |
TOP |
ACTIVITY
11 (Paper & Scissors Pentagon)
| 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. |
TOP |
ACTIVITY
12
(Paper & Scissors Hexagon and Beyond)
| Title: |
Make-a-Flake |
| Comment: |
Allows you to cut polygonal wedges from a virtual piece of folded paper
with virtual scissors (requires Macromedia Flash
Player). Click on Make Your Own Snowflake to begin. Place the
(red dot on the) scissors on an edge of the triangle, then click the mouse
button. Move the scissors within the triangle, then click again. Continue
until the scissors return to an edge. A polygonal wedge will be removed.
Repeat, using Undo/Redo as required. Once you Preview Flake,
you can Cut Some More or Start Over. Snow
Days has a comparable interface. Snowflake
Designer produces a similar result - just click your mouse and pass
it over the circular sector to create each cutout. Or try Snowflake
Builder which allows you to save your snowflake and animate it. |
| Title: |
Kirigami
 |
| Comment: |
Kirigami is the Japanese art of folding and cutting paper. This program
lets you create similar figures on your computer by cutting a polygonal
wedge from a virtual piece of folded paper. You can change the regular
polygon, your wedge, and your paper color. You can then see the figure
that results when the paper is unfolded. Requires a JAVA-capable browser. |
| Title: |
Paper Cutting |
| Comment: |
Become a paper-cutting artist and create beautiful symmetric designs.
Choose the way you want the paper to fold. Select your color or pattern
and start cutting! Click the "Show" button to see your creation. Requires
Macromedia Flash
Player. |
| Title: |
Snowflake
Factory |
| Comment: |
Make your own virtual symmetrical snowflake ... with any number of
sides. Select the number of wedges you want to cut through. Select a hole
punch. Click on the white wedge and see it go through to all the other
wedges! Select a color for your snowflake, and then an animation. Requires
Macromedia Flash
Player. |
| Title: |
Vector
Snowflake Application |
| Comment: |
Draw your own snowflakes and save them in PDF format. Clean, cross-platform,
open-source, and able to generate closed-polygon vector shape output. The
companion SymmetriSketch
allows you start with a polygon with any number of sides, to pick and choose
different symmetry properties, and then create all kinds of different things
that would never be mistaken for frozen water. |
| Title: |
Origami Help for Beginners
 |
| Comment: |
Origami is the Japanese art of folding paper. A folded model is both
a piece of art and a geometric figure. Learn these simple symbols and directions
and you'll be able to make almost any well-diagrammed model in any book. |
| Title: |
Joseph Wu's Origami Page |
| Comment: |
The home page of origami master Joseph Wu. Contains links to the most
comprehensive collection of origami materials available on the world wide
web. Accessible diagrams include the traditional Origami
Crane (a beautiful presentation with very clear instructions presented
using photographs). |
| Title: |
Origami - The Fascination
of Folding |
| Comment: |
Fabulous origami web site. Click on any thumbnail for the corresponding
instructions. Be sure to explore Origami Tablecards - a collection
of 31 tablecards folded from a square. |
| Title: |
Origami
for Kids
 |
| Comment: |
Origami is a fascinating and creative craft for kids, especially if
you can find origami models appropriate to their age. Includes links to
a variety of models suitable for kids. |
| Title: |
Origami Bird |
| Comment: |
Learn to fold a bird from a piece of paper by following animated visual
instructions. Designed so you can repeat a step over and over if you need
to, or watch the whole bird fold from start to finish. Requires Macromedia
Shockwave
Plug-in. |
| Title: |
Robert Neale's Magic Pinwheel |
| Comment: |
A uniquely wonderful modular design. Despite the fact that the design
of the modules is extremely simple, the octagonal wreath can be compressed
(in stages) until it becomes a pinwheel. In both these forms, and at each
stage in between, the design remains stable. Requires
Adobe
Acrobat Reader. UFO has
similar directions. |
| Title: |
Tea Bag Folding
 |
| Comment: |
Tea Bag Folding (also known as Miniature Kaleidoscopic Origami) is
used to create symmetrical designs and medallions.
Includes links to instructions, printable tiles, and commercial sources.
Be sure to access the Tea Bag Medallions (Using
Tea Bag Wrappers) by Gizella Hagedus. |
TOP |
ACTIVITY
13 (Angle Relations in Triangles and Quadrilaterals)
| Title: |
Triangle
Geometry: Triangles |
| Comment: |
Two applets that demonstrate using animation that the sum of the measures
of the interior angles of a triangle is 180 degrees. Before you start the
second applet, get a piece paper and a pair of scissors so that you can
try to perform the demonstration for yourself. Requires a JAVA-capable
browser. |
| Title: |
Paper Folding |
| Comment: |
All of the constructions of Plane Euclidean Geometry that can be performed
by straightedge and compass can also be performed by folding and creasing
paper. Details how to illustrate that the three interior angles of a triangle
is 180 degrees by making just three specific folds to a triangle cut from
a sheet of paper. |
TOP |
ACTIVITY
14 (Angle Measures in Regular Polygons)
| Title: |
Interior Angles |
| Comment: |
Details two more methods for finding a formula for the measure of each
interior angle of a regular polygon. Both results are equivalent to the
formula deduced in the book activity, namely: 180 - 360/n. Web site is
part of
Cool Math.com. |
| Title: |
BBC
Education - Revision Bite - Polygons |
| Comment: |
Considers regular and irregular polygons, angle properties of polygons,
and calculating the interior and exterior angles of regular polygons. Includes
classroom exercises. |
TOP |
ACTIVITY
15
(The Regular Tessellations)
| Title: |
Totally
Tessellated |
| Comment: |
A superbly well-done site on tessellations. Winner in the ThinkQuest'98
competition. Features over 100 dynamic pages with quality illustrations,
photographs, and links. Main topics: History of Tessellations, Essentials,
Mosaics/Tilings, Escher, and Beyond.
Background
Information has links to polygons and angles, symmetry and transformations,
regular polygons, plus links to a simple introduction to tessellations
and to the effect of color usage on tessellated designs. Regular
Tessellations contains links to real examples. |
| Title: |
Tessellation.org |
| Comment: |
Includes information on all aspects of tessellations, from their history
and development to complete galleries of examples by M. C. Escher,
the master of the art. Also included are extensive workshops showing how
to design and produce your own artistic tessellations. |
| Title: |
LinoTiler |
| Comment: |
Decorate square tiles on a flat floor in a symmetrical configuration.
Then create repeating wallpaper patterns from identical square tiles using
wallOVERS
(also from ProtoZone).
Both interactive activities require Macromedia
Shockwave
Plug-in. |
| Title: |
Fractiles, Inc. |
| Comment: |
The versatile and elegant geometry of Fractiles allows you to create
aesthetically pleasing designs with repeating and non-repeating patterns.
And unlike pattern blocks, these MAGNETIC tiles stay put, an especially
helpful feature for students with fine motor skill problems. |
| Title: |
Tessellations |
| Comment: |
Tessellations is the company that combines art, math, and fun.
They make Puzzellations, Colorful Connectables and Perpetual Patterns puzzles,
plus a variety of other products inspired in part by the Dutch graphic
artist M. C. Escher, but also by mathematicians such as Roger Penrose and
Benoit Mandelbrot. |
| Title: |
Tilings and Designs |
| Comment: |
Commercial source of several sets of shapely tiles that lend themselves
to unlimited design making, like mosaic inlays and imaginative scenes and
figures. |
TOP |
ACTIVITY
16 (From Crackers to Honeycomb to Soap Bubbles and Beyond)
| Title: |
Bees,
Microscopy & Mathematics
 |
| Comment: |
The comb of the honeybee is organized into regular hexagonal cells,
showing a repeating, symmetrical mathematical pattern. Such a pattern could
be equally well achieved by cells based on the equilateral triangle or
square. So why the hexagon? |
| Title: |
Bees-Online |
| Comment: |
An educational web site about honey bees and beekeeping. You can download
Fun
Bee Software and
Bee
Games, such beekeeping jig saw puzzles, beekeeping crosswords, the
only free e-book on the Internet about honey bees, and a honey bee screensaver.
The hands-on game in BeeBopper
asks you to bop bees right out of their hive and leave as few as you can.
The latter requires Macromedia Shockwave
Plug-in. |
| Title: |
Polar Smash |
| Comment: |
Another game based on a tessellation of regular hexagons - designed
as a Christmas promo. Either one or two players - jump up and down on the
ice to send your opponent swimming. Requires Macromedia
Shockwave
Plug-in |
| Title: |
Carbon
Nanostructures |
| Comment: |
Graphite is formed from carbon atoms arranged in a honeycomb pattern.
Discover why this form of carbon is used in pencils. Learn about the strongest
fibers known (tubes of graphite or carbon nanotubes) and their applications. |
| Title: |
Soap Bubbles
 |
| Comment: |
This site from the Exploratorium in San Francisco has information on
the shape of bubbles and investigates what happens when bubble meets bubble.
Contains formulae for long-life bubble solutions, links to other sites
on bubbles (like Create
Giant Bubbles), as well as a "bubbliography". |
| Title: |
Interlocking Pavers
& Molds
 |
| Comment: |
Features tessellated, interlocking concrete pavers and molds. Like
Escher's reptile, the tessellating shapes are modified hexagons. |
| Special: |
Click on the photo above to view the tessellation of salamanders in
greater detail. |
| Title: |
Shmuzzle Puzzle
 |
| Comment: |
A standard jigsaw puzzle is comprised of pieces of many different shapes
that fit together only one way. A Shmuzzle Puzzle is comprised of
Escher-like pieces of only one shape that fit together many ways - but
only one right way! Be sure to try CyberShmuzzling
(requires Macromedia
Flash
Player). |
TOP |
ACTIVITY
17 (A Regular Tessellation Prism Kaleidoscope)
| Title: |
Escher: Lizard/Fish/Bat
 |
| Comment: |
A large color rendition of Escher's lizard/fish/bat tessellation. Suitable
for a color transparency. |
| Title: |
Kaleido-Headache
 |
| Comment: |
Mind-boggling kaleidoscope show created by Ann-S-Thesia (Ann Stretton).
Simply phenomenal! |
| Title: |
Symmetrizer |
| Comment: |
Press play, then move your mouse around to see this on-line kaleidoscope
in action. Allows change of repeating character and superposition of grid.How?
features an animated explanation. Requires Macromedia
Flash
Player. |
| Title: |
Kaleidoscope
Painter |
| Comment: |
Paint your own kaleidoscope canvas! Just drag the mouse slowly around
the canvas and let the computer show your artistic talents. Requires a
JAVA-capable browser. The author's FLAKaleidoscope
(requires Macromedia
Flash
Player) allows you watch colorful and hypnotic patterns emerge from
a selection of simple drawings. |
| Title: |
Kaleidodraw |
| Comment: |
This interactive kaleidoscope from the Long Island Children's Museum
allows you to select the symmetry of a circular canvas, change pen points,
and change color. KaleidoDraw
(from ProtoZone)
features a similar interactive activity. Both require Macromedia Shockwave
Plug-in. |
| Title: |
Build Yer Own |
| Comment: |
Click and drag the shapes provided into the circle. When the blue light
passes over them, they show up like a kaleidoscope image. Another
Virtual Kaleidoscope features a similar interactive activity. Both
require Macromedia
Flash
Player. |
| Title: |
KaleidoMania! Interactive
Symmetry |
| Comment: |
KaleidoMania! is a unique, new tool for dynamically creating
symmetric designs. The CD-ROM software and activities will make your students
eager to explore the mathematics of symmetry. |
TOP |
ACTIVITY
18 (Tessellating with Irregular Polygons)
| Title: |
Triangle |
| Comment: |
Every triangle tiles the plane (tessellates) as this Flash movie
by Mike Shepperd will demonstrate. So does every Quadrilateral.
Both demos require Macromedia
Flash
Player. |
| Title: |
Perplexing Pentagons |
| Comment: |
The regular pentagon does not tile the plane, however there are tessellating
pentagons. Fourteen types have been discovered. Are all the types now known?
The problem remains unsolved. |
| Title: |
Pentagons
That Tile The Plane |
| Comment: |
The problem here is to find a single tile shape with five sides which
can tile an infinite floor without leaving holes. Discusses various solutions
to the problem. |
| Title: |
Tessellating
Alphabet |
| Comment: |
Each letter of this tessellating alphabet fits together with copies
of itself to tile the plane. Click a letter to see how it tiles. Some letters
tile in more than one way. |
TOP |
ACTIVITY
20 (The Semiregular Tessellations)
| Title: |
Totally
Tessellated: Semiregular Tessellations |
| Comment: |
You can view color renditions and real examples of the semiregular
tessellations in this page of the Totally Tessellated web site.
Templates
allows you to print renditions of each tessellation suitable for coloring
activities. |
ACTIVITY
22 (Demiregular Tessellations)
ACTIVITY
23 (Islamic Tessellations)
| Title: |
Historical
and Geographical Connections |
| Comment: |
Contains historical and geographical links for tessellations and tilings
in the Moorish architecture of Spain, Chinese and Japanese Designs (with
JAVA applets by Dr. William Chow), Islamic Art and Architecture, and Persian
Architecture. |
| Title: |
Magic Carpet |
| Comment: |
Weave symmetries into an instant rug by drawing in a rectangular panel.
Interactive software (from ProtoZone)
requires Macromedia
Shockwave
Plug-in. |
| Title: |
Some
Arabic Tiles |
| Comment: |
Features colorful tessellating Arabic tiles with links to an outline
version of each. Download the tiles and use the paint bucket tool to make
your own color schemes. |
| Title: |
Geometrical Patterns |
| Comment: |
Download seamless decorated tessellating tiles for your web page background
or simply view the full background on a whole page. You can even change
the background color. Options include Moorish (Islamic), Japanese, Celtic,
and Medieval. |
| Title: |
Taprats |
| Comment: |
Taprats is an applet for generating symmetric Islamic-style
star patterns. Includes a detailed on-line user's manual and a library
of built-in tilings that can be used to construct many famous Islamic designs.
Requires a JAVA-capable browser. |
| Title: |
Islamic
Art and Geometric Design |
| Comment: |
Presenting beautiful Islamic works from the collection of The Metropolitan
Museum of Art along with a series of pattern-making activities involving
straightedge and compass, these materials are designed for use in the classroom
and may be readily adapted by teachers to create exciting lessons in art,
culture, math, and geometry. |
TOP |
ACTIVITY
24 (The Art of M. C. Escher)
| Title: |
Escher
in het Paleis (Escher in the Palace) |
| Comment: |
A permanent Escher exhibition in the center of The Hague. Famous prints
are showcased, complemented by changing displays of selected personal documents.
The exhibition includes a Virtual Reality display which enables
visitors to embark on a voyage of discovery through and within Escher's
work. |
| Title: |
Escher
on Display |
| Comment: |
Escher's work can be seen in several other DemoUrls in the Netherlands.
They are listed here in chronological order, with accompanying photographs.
A large print of Escher's Metamorphosis III
used to be displayed in the Kerkplein post office in The Hague. It has
been moved
to Lounge 4 of Amsterdam's Schiphol airport (beyond passport control). |
| Special: |
Escher in The Hague (Jill Britton's
Odyssey) |
| Title: |
NGA
- M. C. Escher - Life and Work
 |
| Comment: |
A tour of four virtual rooms at the National Gallery of Art in Washington
DC devoted to the art of M. C. Escher. For each graphic, a larger picture
with accompanying commentary is available by clicking on a small picture.
The gallery has 426
Works by Escher with over 80 images available on-line. |
| Title: |
M. C. Escher |
| Comment: |
A Flash experience brought to you by Renee Perry. Requires Macromedia
Flash
Player. Includes four sections: bio, interpretation, visual analysis,
and a gallery of the artist's work. The interpretation section features
a QuickTime movie
of Escher's Reptiles. Nothing short of terrific! |
| Title: |
Biography
of M. C. Escher |
| Comment: |
Comprehensive biography condensed from the book M. C. Escher - His
Life and Complete Graphic Work (Bruno Ernst © 1981). Includes
additional material from original essays by Escher and supporting graphics. |
| Title: |
The Mathematical
Art of M. C. Escher
 |
| Comment: |
An excellent introduction to the mathematics in Escher's art. Divided
into areas: tessellations, polyhedra, the shape of space, the logic of
space, and self-reference. |
| Title: |
The Shrine of Escher |
| Comment: |
Contains links to an Escher biography and an annotated art gallery
with details on eight Escher prints - including
Moebius Strip II,
Belvedere,
Waterfall,
and Day and Night. |
| Title: |
3Quarks
- Applets - Metamorphose |
| Comment: |
Presents Escher's Metamorphosis II as a panorama picture. Use
the scroller to move the picture. To move it with a mouse, click on the
panorama. Includes a link to Background
information. Requires a JAVA-capable browser. |
| Title: |
Snakes |
| Comment: |
Contains links to both a short computer animated movie and stills based
on Escher's last graphic work Snakes. Movie
requires
QuickTime Player. |
| Title: |
The
Mathematics in the Art of M. C. Escher |
| Comment: |
Public lecture by the late H. S. M. Coxeter that touches on various
mathematical aspects of M. C. Escher's art, its centre-piece an examination
of Escher's circular woodcuts. That's a very nervous Jill Britton at
the very end (46:16) asking the question "Do you regard Escher as a mathematician?"
... and requesting Coxeter's autograph in the process. Requires Real
Player. |
| Title: |
Adventures
in Perception
 |
| Comment: |
Offers an on-line resource to Adventures in Perception, a film
which details M. C. Escher's life and work, features an interview with
the artist, and shows him carving a woodblock for and printing his last
work Snakes. Appropriate for middle school students and above. The
site allows access to a short free clip from the film (requires
Real
Player). A VHS video version is available for purchase. |
| Special: |
View the entire film in three parts (PART
1, PART
2, PART
3) courtesy of YouTube. |
| Title: |
The
Life and Works of M. C. Escher |
| Comment: |
Featuring a rare interview with Escher himself, this new video takes
an in-depth look at the world-renowned graphic artist and his ever-popular
work. Traces his career path and personal life, revealing the people and
places that inspired him to achieve greatness. Far too cerebral for the
middle school student, but an essential resource for the mathematics educator.
Approximately 60 minutes. YouTube's Escher
Interview features a brief excerpt from the film. |
| Title: |
The Fantastic
World of M. C. Escher |
| Comment: |
Through first-person accounts by his friends and mathematicians, computer
animated recreations of his work, and a look at his sources of inspiration
in Italy and Spain, The Fantastic World of M.C. Escher is an engaging
exploration that will enrich the understanding and appreciation of Escher
and his art. Approximately 50 minutes. |
| Title: |
Escher,
Van Gogh and Seurat: Art at Play |
| Comment: |
Canadian animation wizardry brings the work of Escher, Van Gogh and
Seurat) to life in this playful series of three video films for all ages.
M.C.
Escher: Sky and Water 1 is a 3-minute exercise for your brain, capturing
the essence of motion in the Dutch master's print. Inspiring, informative,
and fun! |
| Special: |
View the entire Sky
and Water 1 courtesy of YouTube. |
| Title: |
Escher Interactive |
| Comment: |
Escher Interactive is an educational and entertaining CD-ROM
for Windows. Exploiting animation, video, sound commentary and interactive
manipulation of images, it features nine modules, each revealing a separate
facet of Escher's work. |
Title:
|
M.
C. Escher's Legacy
 |
| Comment: |
The proceedings of the International Escher Congress held in Rome,
Italy in July 1998 (the centenary of Escher's birth). This book contains
40 articles, richly illustrated with original art works in addition
to well-known and little-known works by Escher. A CD-ROM complements the
articles, containing color illustrations of work by contemporary artists,
movies, animations, and other demonstrations. New softcover edition. |
| Title: |
Visions
of Symmetry |
| Comment: |
New edition of Doris Schattschneider's definitive book on Escher's
tessellations. Features an illustrated epilogue that reveals new information
about Escher's inspiration and shows how his ideas of symmetry have influenced
mathematicians, computer scientists, and contemporary artists. |
| Title: |
World of Escher |
| Comment: |
On-line store offering high quality Escher products (T-shirts, computer
accessories, ties, posters, puzzles, books). T-shirts
has several options. Art
Gallery features popular Escher graphics. Contest
has galleries of quality Escher-like art (in-store contest entries and/or
winners). |
ACTIVITIES
25 & 26 (Tessellating Template)
| Title: |
Square Template and Dot Paper Masters |
| Comment: |
Make a tessellating template by sliding cutouts between opposite sides
of a cardboard square, then draw the tessellation by tracing around the
template with its residual corners aligned to the markings on dot paper.
Contains printing masters for a sheet of nine 2-inch cardboard squares
and a letter-size sheet of 2-inch dot paper (requires
Adobe
Acrobat Reader). Visit Escher in the Classroom
for more details. |
TOP |
ACTIVITY
27 (A Pop-Up Sponge Jigsaw Puzzle)
| Title: |
Pop-Up Sponge |
| Comment: |
Pop-Up (compressed sponge) starts out wafer thin and rigid. Cut out
a shape, submerge it in water, then watch it expand! Used in the tessellating
jigsaw puzzle activity of Investigating Patterns:
Symmetry and Tessellations. Perfect for tessellating sponge printing
as well. The creators of the Ellison® LetterMachine (TM) offer pop-up
sponge in 8 1/2" x 12" sheets and includes 6 sheets of one color (blue,
lemon, natural, mint, raspberry, white). Also available in Assorted
12 Pack (2 of each color) and Assorted
48 Pack (8 of each color). |
ACTIVITY
28 (Mirroring the Craft of M. C. Escher)
| Title: |
Acryl-Pull
Handle
 |
| Comment: |
Make your own transparent plastic stamp mounts. Attach large (½"
x 3½") clear plexiglass handles to 4-inch clear plexiglass squares
(1/8" to 3/16" thickness). Click here
for more information on fabrication (requires
Adobe
Acrobat Reader). |
| Title: |
Escher in the Classroom |
| Comment: |
A condensation of Jill Britton's article for the proceedings of the
International Escher Congress held in Rome, Italy in July 1998 (the centenary
of M. C. Escher's birth). The book of the proceedings (M.C. Escher's
Legacy: A Centennial Celebration) is available from Amazon.com. |
| Title: |
What
Is a Print? |
| Comment: |
Includes links to interactive demonstrations on the principal printmaking
techniques - woodcut, etching, lithography, and screenprint. Escher was
a master of both the woodcut and the lithograph. Requires Macromedia
Flash
Player. |
| Title: |
Patterning |
| Comment: |
Contains exercises to give children a new understanding about how commonly
used patterns and designs based on nets are produced. Includes use of vegetable
stamps. Handwork
contains links to related pedagogical web pages on crafts. |
TOP |
ACTIVITY
29 (Exploring Tessellating Art)
| Title: |
The
Mathematics Behind the Art of M. C. Escher |
| Comment: |
A student project for Mathematics in Art and Architecture, a
course offered by Helmer Aslaksen at the National University of Singapore.
Flash Player 6 is required to view the excellent animations. |
| Title: |
Doggy
Mosaic |
| Comment: |
Try to pave an area with the figures of the dogs shown without any
gaps or overlaps. Move the dogs by dragging them. Rotate the dogs by double
clicking on them. Requires Internet Explorer 4.0 or higher, with JavaScript
enabled.
Netscape users beware! |
| Title: |
The
Seventeen Wallpaper Tilings |
| Comment: |
Some of the most fascinating tilings are the so-called wallpaper tilings.
There are exactly seventeen of them. Click on the various icons to see
an animation of the corresponding tiling being built. Each is labeled with
the name used by the International Union of Crystallography since 1952. |
| Title: |
Seventeen
Kinds of Wallpaper Patterns
 |
| Comment: |
Features examples of the seventeen kinds of wallpaper patterns in Japanese
culture. Includes Japanese nomenclature for the names of the patterns. |
| Title: |
Escher Patterns |
| Comment: |
Features tessellations by Japanese artist Yoshiaki Araki - including
his classic tessellating elephant Maozou. Click on the icons to
view the tessellations. |
| Title: |
Escher
Tilings |
| Comment: |
A collection of Escher-like tilings by Belgian artist Jos Leys. Many
ultilize hyperbolic ("circle limit") geometry. Click on each graphic for
an enlarged view. |
| Title: |
Tessellations &
Punography |
| Comment: |
Features artistic and thematic tessellations with accompanying verse.
Enjoy "Quantum Leap", a tessellanimation movie with Ozzie the kangaroo,
as the extensive graphics load. |
| Title: |
David Bailey's World of Tessellations |
| Comment: |
Features links to the artistic tessellations of David Bailey, each
generally accompanied by explanatory text, whereby the various "intricacies"
behind the tessellation are discussed. |
| Title: |
Escher
Tile Gallery
 |
| Comment: |
Contains 17 Escher tiles each of which forms a repeating pattern of
interlocking figures when placed side to side and vertex to vertex. Terrific! |
| Title: |
Tessellated Polyhedra
 |
| Comment: |
Build a durable model of each of the five regular polyhedra (Platonic
solids) from a card stock net fitted with tabs. Each has been decorated
with a tessellation of animals in full color. |
| Title: |
Artlandia
SymmetryWorks |
| Comment: |
Artlandia SymmetryWorks is an Adobe
Illustrator plug-in that lets you create the highest-quality professional
ornaments, designs, and art, and interactively explore symmetry and wallpaper
tilings. |
| Title: |
M. C. Escher Font
Set
 |
| Comment: |
This font set is based on the lettering styles and artwork of M. C.
Escher with both a formal lettering and a handwritten-style type. Rounding
out the set is a collection of extras drawn from and inspired by Escher's
work. Available for Win or Mac. |
TOP |
ACTIVITY
30 (Explorations with TesselMania!)
| Title: |
TesselMania! Escher Simulations |
| Comment: |
Permits downloading of a compressed file containing 20 TesselMania!
Escher
simulations (for use with the Windows version of TesselMania! only). |
| Title: |
Tess |
| Comment: |
With Tess, you can quickly create attractive symmetric planar
illustrations. While you draw, Tess will automatically maintain
the symmetry group you have chosen; 11 rosette, all 7 frieze, and all 17
wallpaper groups are included. Visit Downloads
to access a fully-functional evaluation version of the Mac/Win software. |
| Title: |
Tessellate |
| Comment: |
Allows you to generate a new tessellating shape starting from a rectangle,
triangle or hexagon. Select Tessellate to see the complete pattern.
You can choose several different colors to enhance the pattern, and can
observe the different effects that colors have on tessellations. Requires
a JAVA-capable browser. |
| Title: |
Tessellation Demo |
| Comment: |
Jim McNeill's animated "how to" website for creating translation tessellations.
Brief - but excellent! Includes a link to a book, video and posters. |
TOP |
| More Books and Web Pages by
Jill Britton |
|
 |
 |
 |
 |
Fun
with Patterns
 |
| links |
|
|
|
links
|
All M. C. Escher works (c)
Cordon Art B.V. - Baarn - the Netherlands.
Used by permission. All rights reserved. |
