Investigating Patterns

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These links are coordinated with the chapters and activities of
"Investigating Patterns: Polyhedra Pastimes"

ACTIVITY 1 (Regular Polygons and Tessellations)

Title:  Symmetry and Tessellations
Comment:  The author's web page containing hundreds of links to pages dedicated to symmetry, tessellations, and the art of M. C. Escher. Coordinated with the chapters and activities of the middle school book Investigating Patterns: Symmetry and Tessellations. [For a pedagogical evaluation, consult the Review in Mathematics Teaching in Middle School (requires Adobe Acrobat Reader).]

ACTIVITY 2 (The Regular Polyhedra)
Title:  A Lesson in Greek
Comment:  What do the Greek prefixes in the names of polygons and polyhedra mean? Learn the simple essential vocabulary. Requires Adobe Acrobat Reader.
Title:  Studying Polyhedra
Comment:  A simple treatment of the regular polyhedra. Includes an applet that allows you to select each polyhedron based on its number of faces, move it with your mouse, control the speed of rotation or stop it altogether, and view the model in various formats.
Title:  Blick-Kote Posterboard
Comment:  Internet source for Blick-Kote Railroad Board, a smooth, bristol-type board, colored on both sides. Individual sheets are 22" ×28" (56-cm × 71-cm). Ideal thickness for polygon panels is 6-ply (about .022" or .56-mm).
Title:  Elastic Geometrics
Comment:  Cut out tabbed polygon panels from construction paper with dies for an Ellison LetterMachine, then assemble the polyhedra with rubber bands. The 3 Die LG Set (16827-LG) consists of an equilateral triangle and a regular hexagon die, a square and a regular pentagon die, and a regular octagon die for the standard/LG machine (1.5-inch sides). The addition of a custom order Elastic Geometrics regular decagon die will allow construction of all 13 Archimedean polyhedra in Activities 20-21.
Title:  Accu-Cut: Platonic Solid Set
Comment:  Features similar tabbed polygon dies for the Accu-Cut Roller Die Cutting Machine. Available in both Large (1-inch sides) and Jumbo (2-inch sides) formats - the latter a preferable size for investigations with regular polyhedra. Individual dies for each of the five tabbed polygons are also available.

ACTIVITY 3 (From Gumdrops to Zometool)
Title:  Geodesic Gumdrops
Comment:  Students learn how to make polyhedra using gumdrops and toothpicks. Find out about the basic principles that hold their structures together. YouTube's Platonic Solids from Gumdrops and Toothpicks is a great resource. Mini-marshmallows can be used in place of gumdrops.
Title:  Straw Shapes
Comment:  Build polygons with bendable (flexible) drinking straws, then join them together with tape to form a regular polyhedron. Lavishly illustrated. Here's another photo of a Straw Icosahedron.
Title:  Straw Polyhedra
Comment:  Detailed lesson plan with graphics describing how to make polyhedra by threading together non-bendable plastic straws with cotton twine. Includes a link to images of a Teachers Workshop for the lesson.
Title:  Bar Straw and Pipe Cleaner Polyhedra
Comment:  Make elegant and expensive polyhedra frames from straws and pipe cleaners. For the best results, use bar straws (available from wholesale food distributors).
Title:  Polydron
Comment:  Polydron is a face-based plastic construction system manufactured in the UK. The durable polygons snap together by of a unique clip hinge, making it easy to create polyhedra. Available in a variety of kits. A Polydron Platonic Solids Set is available from ETA/Cuisenaire. 
Title:  Geofix
Comment:  Geofix is the American name for what is called Geoshapes in Australia, where it's made. Panels are translucent jewel-toned nets. Available in sets of individual polygons (including octagons and decagons) as well as exploration kits.
Title:  Astro-logix
Comment:  An ingenious new construction kit system comprising hubs and tubes, in bright neon colours and glow in the dark plastic. Models shows the many possibilities.
Title:  Roger's Connection
Comment:  Roger's Connection is an award winning ball-and-stick magnetic construction toy and geometric exploration tool for students and teachers. Includes a link to a Gallery of examples.
Title:  Zometool
Comment:  Zometool is a unique color- and shape-coded system that takes ball-and-stick construction to its highest level. Available in a variety of Project Kits. The book Zome Geometry by George Hart and Henry Piccioto is full of fun and beautiful things you can make with the Zome System.
Title:  RND Modelshop
Comment:  All wood polyhedra models by Ron Nelson. Durable classroom visual aid models for "hands on" use. Woven connections are press fit with very few actual glue joints.

ACTIVITY 4 (Regular Polyhedra Nets)
Title:  Unfolding Polyhedra
Comment:  Select a regular polyhedron, then watch it fold and unfold from net to solid. Click and drag to rotate the object. Includes links to more unfoldings.
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:  Polyhedra Earth Maps
Comment:  A similar set of regular polyhedra nets each decorated with a projection of the earth in full color. Dodecahedron Globe from National Geographical Data Center features elevations & nighttime lights. All files require Adobe Acrobat Reader to view and print. See Activity 28 for links to icosahedron maps.
Title:  12-Sided Calendar
Comment:  Download a calendar with desired specifications. Print on card stock, cut along the solid lines, fold along the dotted lines, then use a glue stick to assemble into a dodecahedron. 12 Sided Calendar Cube is a variation thereof with added backgounds. Artabus 2015 Calendar allows you to select an artist for images that decorate your calendar, or even download your own images. All PDF files require Adobe Acrobat Reader.
Title:  Magic Fortune-Teller
Comment:  Is your future so bright you'll have to wear shades? Ask your burning question, roll the magic fortune-teller, and read your forecast. If you don't like the answer, roll again until you do! To prepare the fortune-teller, print the colored net on card stock, cut along the outline, then fold along the heavy inner lines. Use a glue stick to assemble the net into a dodecahedron. Requires Adobe Acrobat Reader.
Title:  Paper PhotoCube
Comment:  Trivista's Paper PhotoCube is a shareware program that allows you to create a printable, foldable, paper photocube from your digital photos. Unzip the compresed file as usual. The installation password is PPC. [Its drag-and-drop feature will cause the program to crash if your operating system is Windows 7. Save the zipped installation file, then install in XP Mode.] This printable Escher PhotoCube was created with the program. [Use Windows 7 Paint or Ultimate Paint to resize to an 8-inch width for the color printout.]
Title:  Cube: Use your digital photographs to make a unique art object!
Comment:  Create a printable, foldable, paper photocube from your digital photos on-line. Upload your photos or select them from Flickr or Facebook - then click on Create. [The uploading of photos requires free membership in] If your photos are not square, they will be cropped by the program [see Squaring Photos]. Save the template to your computer, then print it on heavy card stock. Unfortunately the assembled cube will be painfully small. It is suggested that you use Windows 7 Paint or Ultimate Paint to crop and resize the template appropriately before printing.
Title:  Escher Rotating Photo Cube
Comment:  Show off your computer images with a rotating cube. Visit INTEL 3D Photo Cube to learn how to add a rotating cube to your site. Requires a JAVA-capable browser.
Title:  Captured Worlds
Comment:  Captured Worlds is a commercial set of five pop-out, glue, and fold-up Platonic Solids, rendered in six-point perspective by internationally-known artist Dick Termes. This unique perspective system allows an entire three-dimensional surrounding to be projected onto the polyhedra. Build Your Own Dodecahedron features a print, cut and fold dodecahedon of St. Peter's Cathedral.

ACTIVITY 5 (Euler's Formula)
Title:  Leonhard Euler
Comment:  A brief biography of Leonard Euler (1707-1767), one of the discoverers of the famous formula for polyhedra. Visit Euler Portraits for both color and BW portraits, each suitable for downloading and printing as a transparency.
Title:  Exploring Geometric Solids and Their Properties
Comment:  This Illuminations web page from NCTM features links to five lessons which include interactive applets of regular polyhedra and an exercise on patterns leading to Euler's Formula.
Title:  Euler's Formula
Comment:  Euler's classic formula for polyhedra applied to the regular polyhedra (Platonic solids). The formula is provided in the form:  Faces + Corners  =  Edges + 2 . 

ACTIVITY 6 (Duality and Symmetry)
Title:  Polyhedra
Comment:  Contains links to interactive animations demonstrating duality in the Platonic solids. Select an icon, then click and drag using your mouse to rotate the image.
Special:  View The Platonic Solids video courtesy of YouTube. Includes a computer-animated section illustrating the properties of regular polyhedra, demonstrates why exactly five such solids exist, presents the mathematical heritage of the solids, and explores the intriguing relationships between the solids and their duals.
Title:  Wrapped Polyhedra
Comment:  Contains links to superior interactive graphics created with LiveGraphics3D. The first five demonstrate duality in the Platonic solids. Select an icon, then click and drag using your mouse to rotate the image.

ACTIVITY 7 (History and Applications)
Title:  3Quarks - GIF Animations - Platonic Solids
Comment:  Brief history of the "Platonic solids" (regular polyhedra). Features an excellent GIF animation of each regular polyhedron (also found in The Platonic Solids).
Title:  Platonic Solids and Plato's Theory of Everything
Comment:  In mathematics Plato's name is attached to the Platonic solids. In the Timaeus there is a mathematical construction of the elements (earth, fire, air, and water), in which the cube, tetrahedron, octahedron, and icosahedron are given as the shapes of the atoms of earth, fire, air, and water. The fifth Platonic solid, the dodecahedron, is Plato's model for the whole universe. Carl Sagan elaborates in Cosmos: Platonic Solids, an excerpt from the famed PBS series Cosmos (courtesy of YouTube).
Title:  Sacred Geometry Set
Comment:  The five Platonic solids are said to be instruments of dialogue between man and God, permitting us to find unity in nature's diversity. These "sacred geometry" sets are carved from Quartz, Amethyst, Hematite or Aventurine and displayed in a special wooden box.
Title:  Johannes Kepler's Polyhedra
Comment:  Includes illustrations from Kepler's 1619 book Harmonice Mundi which graphically shows the Platonic associations of the regular solids with the classical elements and his model of the solar system in which he proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids. Part of Virtual Reality Polyhedra by George Hart. Links require a VRML plug-in like Cortona3D Viewer for your web browser.
Title:  Ernst Haeckel: Die Radiolarien
Comment:  Good approximations of the regular polyhedra are skeletons of radiolaria, minute marine animals. German web page includes all 35 radiolarian plates from Haekel's Report on the Scientific Results of the Voyage of H. M. S. Challenger (1887). The best plates will be found by accessing links from Ernst Haeckel: Kunstformen der Natur.
Title:  Principles of Virus Architecture
Comment:  There are certain common features and general principles of architecture that apply to all viruses. Webpage details their isosahedral nature.
Title:  Math Unit: Crystals
Comment:  Students studying polyhedra enjoy seeing the structures as they occur in the real world. Crystalline structures can be categorized into 7 crystal systems. Students can access links to photographs of beautiful crystals in Systems of Crystals. They can also search Minerals by Name for further examples.
Special:  Paper models for the 32 crystal classes and 7 crystal systems have been reproduced in pdf format (requires Adobe Acrobat Reader). These models can be printed on heavy card stock to construct a physical model.
Title:  Growing Salt Crystals
Comment:  Contains simple instructions showing students how to grow table salt (sodium chloride) crystals in a home environment using ordinary materials. For more advanced experiments, visit Growing Crystals and Crystal Growing Experiments. How To Grow Your Own Crystals features excellent animations. Scientific Explorer Ultimate Crystal Growing Science Kit has 15 different crystal growing activities.
Title:  Polyhedra Dice
Comment:  One practical application of polyhedra is in the design of dice. Includes a photograph of pair of cubical dice from Pompeii, made of bone in the first century AD. Dice is a good source of information. Platonic Dice features links to rotating dice images. Jumbo Plastic Polyhedral Dice and Jumbo Foam Polyhedra Dice are available through Polyhedra Dice Image is suitable for downloading and printing as a transparency.
Title:  Artificial Reefs
Comment:  Artificial Reefs is a modular reef engineering and manufacturing firm dedicated to the development and restoration of our world's fragile oceans and endangered reef ecosystems. The structure of the modules is tetrahedral. Galleries is a archive of images and video taken from many installed Artificial Reefs sites.

ACTIVITY 8 (Tetrahedron from an Envelope)
Title:  Making a Tetrahedron from a Small Envelope
Comment:  Why does the small envelope work to make a tetrahedron, but a business envelope does not? What do the envelope and the tetrahedron have to do with the square root of three?
Title:  Two Piece Pyramid Puzzle
Comment:  Commercial source (The Puzzle & Craft Factory) of the classic puzzle. The aim is to assemble the two wooden pieces into a tetrahedron (4-faced pyramid). Very deceiving, especially if you consider it to be real easy because it only has two pieces.

ACTIVITY 11 (Blow-Up Cube)
Title:  The Origami Cube
Comment:  Contains visual directions for a cube that you can fold from a square sheet of paper. Includes some higher-level mathematics and Internet references.
Title:  Cube (Waterbomb)
Comment:  Contains similar visual instructions for an origami cube or waterbomb. Includes a link to instructions for the preliminary Waterbomb Base. For comprehensive PDF instructions, visit Thinking in 3D: Waterbomb Instructions (requires Adobe Acrobat Reader).
Title:  Paper Ball / Water Bomb
Comment:  Doug Wong's animated graphic instructions for an origami paper ball (cube) or water bomb. A delight! For animated live-action folding, visit Finger Tips Water Bomb. For a video presentation, access How To Make An Origami Water Bomb.
Title:  Origami Balloon
Comment:  Strike a blow for world pieces! The curiously-shattered image of the earth from National Geographical Data Center can be folded into an origami balloon (paper ball / water bomb) that reassembles the map. Download the Origami Balloon Paper. Both require Adobe Acrobat Reader. Assembly uses classic water bomb instructions.
Title:  Photuki for Windows
Comment:  Software to make a water bomb with image faces. Download, then decompress the file Photuki is the primitive program. Photuki2 will allow you to use multiple images and to customize the template output. This printable Escher Photuki Cube was created with Photuki2. Access Potuki Cube Folding Method or Photuki Cube Folding Instructions (my own approach) for further directions.
ACTIVITY 12 (Skeletal Octahedron)
Title:  What is Modular Origami?
Comment:  General information on modular origami, a two stage paperfolding technique which uses multiple sheets of paper. In the first stage each individual sheet of paper is folded into an identical module. In the second stage the modules are assembled into an integrated flat shape or three-dimensional structure. Requires Adobe Acrobat Reader.
Title:  Ornament
Comment:  An example of a simple model that can serve as an introduction to modular (unit) origami. Robert Neale's skeletal octahedron is assembled from six identical origami units (Water Bomb Base). Detailed photographs accompany the instructions.
Title:  Butterfly Ball
Comment:  An extension of the skeletal octahedron (Ornament) to twelve identical units (Water Bomb Base). The butterfly ball is stable when resting on a surface. Toss it in the air and give it a light smack to see it explode into a cloud of butterflies which flutter prettily to earth. Includes a link to detailed instructions as well as to suitable decorated squares. YouTube's Origami Butterfly Ball (Kenneth Kawamura), Origami Butterfly Ball and How to Make a Butterfly Ball have lengthy video instructions.

ACTIVITY 13 (Pyramids and Prisms)
Title:  NOVA Online /Pyramids / Explore the Pyramids
Comment:  Interactive site tells who build the pyramids and how old they are. Also includes an overview of Ancient Egypt as well as an introduction to hieroglyphs. Photo of Guizeh Pyramids and Large Photo of Guizeh Pyramids are suitable for downloading and printing as a transparency.
Title:  Volume of a Pyramid
Comment:  Two pyramids with congruent bases and altitudes will have the same volume (Flash demo). Uses the yangma (a rectangular-based pyramid whose vertex is vertically above one of the vertices of the base) to find the formula for the volume of a pyramid. Students can fit three yangmas together to form a cube (yangma pattern). Another approach uses Filling 3D Shapes with water to compare volumes (requires a JAVA-capable browser).
Title:  Prisms
Comment:  Rotate any of a selection of prisms on-line. Click on their names to access an enlarged interactive image. A similar directory accesses enlarged interactive examples of Antiprisms and Pyramids.
Title:  American Slide Chart: Pop-Up Calendar
Comment:  The fourteen faces of this rubber band activated pop-up polyhedron calendar (a tetrakaidecahedron) comprise a pair of parallel and congruent regular hexagons and two rows of congruent isosceles trapezoids. The mechanism instantly springs into a 3-dimensional shape when removed from its envelope. Victorine Originals has a pattern for a 2013 collapsible calendar (requires Adobe Acrobat Reader).
Special:  Click on the graphic above to see a lively animation of a pop-up from American Slide Corporation!

ACTIVITY 14 (The Golden Ratio)
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 (3-D) 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 self-paced sequential steps. Requires a JAVA-capable 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 (five-pointed 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:  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:  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.

ACTIVITY 15 (Stellations of Regular Polyhedra)
Title:  Stellating the Dodecahedron
Comment:  Explains the process of stellation on a regular dodecahedron base. Graphics include both the first (standard) and second stellation. The first stellation makes a lovely Christmas decoration and is often seen as such in store windows and commercial displays during the holiday season. It was discovered by astronomer Johannes Kepler. The Kepler Solids has more information and construction details.
Title:  Stellated Polyhedra
Comment:  Click on the icons to manipulate the standard stellations of the octahedron, dodecahedron and icosahedron. [There are no stellations of the cube or octahedron.] Requires a JAVA-capable browser.
Title:  Stellations
Comment:  The applet in this web page is a polyhedron generator and modifier that permits you to truncate and stellate all five Platonic solids. Requires a JAVA-capable browser.
Title:  The Polyhedron Explorer
Comment:  An interactive tool for creating remarkable polyhedral shapes by truncating, indenting, stellating, and adding holes to each of the five Platonic Solids. Requires a JAVA-capable browser.
Title:  Stella Octangula
Comment:  Includes two classroom photographs featuring a stella octangula (stellated octahedron), the octahedron that is its core, and the cube in which it can be inscribed. Each of the faces of the parent octahedron is crowned by a tetrahedron. The visual effect is of two larger intersecting tetrahedra, one rotated 180° with respect to the other.
Special:  Visit YouTube's Stellaoctanglar and the Cube for a terrific video demonstration.
Title:  Christmas Tree Decoration
Comment:  Decorate your Christmas tree with a stella octangula ornament. Includes links to graphic files and assembly instructions. A simpler way to make the stellated polyhedron is to build one of the larger intersecting tetrahedra (red card stock), and then to attach a smaller tetrahedron (white card stock) to each of its four faces. Stella Octangula features suitable printable graphics. PDF files require Adobe Acrobat Reader.
Title:  Stellated Octahedron
Comment:  Better than an origami paper ball (water bomb)! Fold a blow-up stellated octahedron from a single sheet of square paper. Download Escher Reptile Paper for a special Tessellated Effect. Download Colored Escher Reptile Paper so that no two adjacent reptiles will have the same color. [Ensure that you are looking at a section with a "two tone" reptile when you begin Step 12.] All pages require Adobe Acrobat Reader to view and print.

ACTIVITIES 16 & 17 (Origami Star Building Unit)
Title:  Origami: The Sonobe Cube
Comment:  Detailed instructions for both an origami star building unit (a modular or Sonobe unit) and for the assembly of the corresponding cube. Excellent graphics.
Title:  MM's Modular Mania
Comment:  Web site includes Folding Instructions (Own Designs) with links to MM's instructions for several different modular cubes and Sonobes. Click on the images for instructions where available. Folding Instructions has links to modular origami diagrams available on the Internet. Sonobe Assembly Guide has useful tips for larger models (requires Adobe Acrobat Reader).
Title:  A Sonobe Model
Comment:  Detailed instructions by Helena Verrill for a 30 unit sonobe model (stellated icosahedron). Includes a link to instructions for the Basic Unit and a Graphic showing how the flap of one unit fits into the pocket of another. Click on any of the images and you'll get a larger picture of the same thing. Helena's Origami Index Page contain links to other modular units, including a Triangle Unit for building triangular faced origami polyhedra. 
Title:  Origami Polyhedra Instructions
Comment:  Detailed instructions for folding our star building unit, and then using identical units to construct a cube (6 units), a stellated octahedron (12 units), a stellated icosahedron (30 units), and a stellated truncated icosahedron or "Epcot ball" (270 units). For a YouTube video of the stellated octahedron assembly, visit Origami Stellated Octahedron and for the stellated icosahedron assembly, visit Módulo Sonobe. The PDF instructions in Bird Cube and Polyhedra Unit Origami comprise a 6-unit, 12-unit and 30-unit summary. Challenge: Use 90 units to construct a stellated dodecahedron. Download PDF instructions for the same unit. PDF files requires Adobe Acrobat Reader.
Title:  Modular Origami
Comment:  Excellent detailed instructions for making a cube, an octahedron, and an icosahedron from identical sonobe units.
Title:  Knotology
Comment:  Knotology is the process of creating 3-D objects by weaving or plaiting of strips of paper, for example old fashioned ticker tape. Includes polyhedra models based on the Sonobe Basic Unit. Excellent graphics.
Title:  Unit Origami : Multidimensional Transformations
Comment:  Commercial source for the "bible" on unit origami by Tomoko Fuse. A must have for an experienced folder. The photographs in the book are awesome, and frequently inspire the most unlikely kids to fold.
Title:  The Japanese Paper Place
Comment:  The world's largest selection of fine Japanese papers - including solid origami colors in economical 500 sheet packages [ORI2009 (O-4)]. Located in Canada - includes links to American and International retail resellers. US-based Kim's Crane has the largest on-line selection of Origami supplies.

ACTIVITIES 18 (Tetrahedron Kite)
Title:  Tetrahedral Principle in Kite Structure
Comment:  Article by Alexander Graham Bell detailing his work with tetrahedral kites. Reprint from National Geographic Magazine which Bell founded. Alexander Graham Bell's Flights of Fancy features several rare photographs. The movie Tetrahedral Kite in Flight requires Real Player. Tetrahedral Kite was inspired by Alexander Graham Bell's historical models ... and his discovery on the telephone.
Title:  Tetrahedral Kite Information
Comment:  Anthony Thyssen's plans and notes for tetrahedral kites. Includes links to Bell Tetrahedral Kite (extensive Bell photos ... including the delightful Tetrahedral Kiss) and Tetrahedral Kite Using Straws. Be sure to visit the 100 Cell Tetra, a project by Diane Hislop's fourth grade class in Bakersfield, California - complete with links to photo enlargements.
Title:  Tetrahedron Kites
Comment:  Photos of students constructing tetrahedron kites from string, straws, and tissue paper. Includes a link to Build Your Own Tetrahedron Kite which includes materials, tissue template, assembly instructions, and directions on flying. Excellent Graphics.
Title:  Making a Tetrahedral Kite
Comment:  Excellent instructions (with a wealth of supporting photographs) detailing the construction of a tetrahedral kite from thread, straws, tissue paper, and tape. YouTube's Tetrahedron Kite features a 16-cell kite in flight.
Title:  Build Your Own Sierpinski Tetrahedra
Comment:  Photographic essay detailing the construction of a 16-cell Sierpinski tetrahedron - a 3-D fractal with the structure of a tetrahedron kite. Requires Triangular Grid Paper.
Title:  Tetralite Kites
Comment:  Commercial source of construction plans for ultra-lightweight, collapsible, multi-cell tetrahedral kites. Page includes several options including Pictures, Diagrams and Links.

ACTIVITY 19 (Polyhedra Bubbles)
Title:  Bubbles
Comment:  What is so fascinating about bubbles? Why does a bubble form a sphere at all? Why not a cube, tetrahedron, or other geometrical figure? Prepared by Ron Hipschman of the San Francisco Exploratorium.
Title:  Soap Bubble Shapes
Comment:  Using pipe cleaners and drinking straws, you can make 3D geometric frames: cubes, tetrahedrons, or shapes of your own design. When you dip these frames in a soap solution, the soap films that form on the frames are fascinating and colorful.
Title:  Kubic Bubbles
Comment:  North American source for Kubic Bubbles, a soap film experiment kit featuring a tetrahedron, cube, octahedron and triangular prism..
Special:  Click on the thumbnail above for an enlarged view of the polyhedra bubbles in their frames.
Title:  Zometool Lesson Plans
Comment:  Many models you can build with Zometool will create fantastic bubble forms when dipped in a soap bubble solution. PDF lesson plan (beginning on page 50) includes instructions and a bubble recipe. The YouTube video Science Labs: Cube Bubble and More! is a good resource. Bubbles II: Minimal Surfaces (beginning on page 96) contains more bubble projects. Lesson plans require Adobe Acrobat Reader.

Title:  The Archimedean Solids: A Lesson in Greek
Comment:  Finding those tongue-twisting names of the Archimedean solids confusing? Learn their basis to simplify the nomenclature. Requires Adobe Acrobat Reader.
Title:  Archimedean Solids (Pappus)
Comment:  Contains a translation of the manuscript by Pappus which gives the first known mention of the thirteen Archimedean solids, which Pappus lists and attributes to Archimedes. 
Title:  The Platonic and Archimedean Solids (Science U)
Comment:  Contains links to both JAVA and VRML models of the 5 Platonic and 13 Archimedean solids. VRML models requires a plug-in like Cortona3D Viewer for your web browser
Title:  Archimedean Polyhedra
Comment:  Investigates semiregular polyhedra, their generation by progressive truncation, and their relationships to semiregular tessellations. Excellent colored graphics.
Title:  Polyhedra
Comment:  Web site by Albert P. Carpenter covering polyhedra based upon or derived from the five Platonic and thirteen Archimedean solids. Excellent graphics.
Title:  Truncatering
Comment:  The word truncate is specifically the act of symmetrically slicing away pieces. Truncations are the main method of changing a regular polyhedron into a new semiregular polyhedron.
Title:  Close Packing
Comment:  Among the Platonic solids, only the cube tessellates (fills) space ... a property it shares with the truncated octahedron, an Archimedean solid. Discusses other space-filling polyhedra and their use in commercial packaging.
Title:  A Non-Traditional Soccer Ball
Comment:  Traditionally soccer balls are truncated iscosahedra (5.6.6). This soccer ball is a rhombicosidodecahedron ( Click here for an enlarged view.
Title:  Polyhedra Models
Comment:  Contains links to templates for making paper models for each of the 5 Platonic solids and the 13 Archimedean semiregular polyhedra. Requires Adobe Acrobat Reader.
Title:  Paper Models of Polyhedra
Comment:  Contains links to more than 35 paper models of polyhedra available for free. Select the polyhedron, then print the corresponding net on card stock. Includes a link to detailed general instructions.
Title:  Polyhedra Plated with Paper Strips
Comment:  Construct polyhedra by plaiting folded paper strips. Every strip is a sequence of specific quadrilaterals, and every folding-line is an edge or a diagonal of a quadrilateral. Includes downloadable patterns.
Title:  The Many Faces of Polyhedrons
Comment:  Download Windows software which allows you to interactively manipulate virtual regular and semiregular polyhedrons, as well as a few others. With it you can rotate polyhedrons by dragging with a mouse, display them as wireframes or with colored faces, and transform them into each other in various ways (including duality conversions and truncation). Absolutely terrific!
Title:  Poly
Comment:  Shareware program for exploring and constructing polyhedra. With Poly, you can manipulate polyhedral solids on the computer in a variety of ways. Flattened versions (nets) of polyhedra may be printed and then cut out, folded, and taped, to produce 3D models. Includes a link to a set of eight Die-Cast Metal Models.
Title:  Shapescape
Comment:  An easy to use Windows program that automates the process of designing and constructing polyhedral models. Pictures and colors can be applied to each polyhedron face. Result can be flattened so that a pattern can be printed to make a 3D model. Program can also produce animated GIF files of rotating polyhedra.
Title:  Stella: Polyhedron Navigator
Comment:  Commercial software programs for exploring and constructing polyhedra. Great Stella can create all the uniform polyhedra (which includes the Platonic, Archimedean, and many more), their duals, and all stellations thereof. Small Stella is a cut down version ideal for schools just starting out. Both include nets to build the models. The printout option is not included in the demo versions (available on-line).
Title:  Green Onyx Paperweight
Comment:  Commercial source of an onyx paperweight in the shape of a cuboctahedron. Math teachers can use it to to inspire their students to learn geometry by calculating its surface area and volume.
Title:  Octabug
Comment:  The jitterbug motion of an octabug, a combination of rotation and contraction, smoothly transforms a closed octahedron into an open cuboctahedron and vice versa.
Title:  Hoberman Transforming Sphere
Comment:  The Hoberman Sphere is an expanding/contracting icosidodecahedron that combines geometry, engineering and aesthetics. A big hit with kids of all ages!
Title:  Polyhedra Poster
Comment:  Polyhedron Poster by Pedagoguery Software where you'll find the 5 Platonic solids, the 15 Archimedean solids, 10 Prisms and anti-prisms, and all 92 Johnson solids!

Title:  Archimedean Solids
Comment:  Contains a diagram and symbol for each Archimedean solid, and the kind and number of regular polygons in its makeup. Students should be able to deduce the number of faces (F), corners (C), and edges (E) in the solid, then verify the result using Euler's formula (F + C = E + 2).

ACTIVITY 24 & 25 (The Polyhedra of M. C. Escher)
Title:  M. C. Escher: The Official Website
Comment:  Home page of the M. C. Escher Foundation and Cordon Art B.V. Includes information about the use of M.C. Escher's work, a short biography, news, bibliography, links and some fun stuff like a Virtual Ride through some of his works.
Title:  The Polyhedra of M. C. Escher
Comment:  Part of Virtual Reality Polyhedra by George Hart. Includes a list of Escher's works that feature strong polyhedra content, as well as links to virtual reality versions of the polyhedra cited. Requires a VRML plug-in like Cortona3D Viewer for your web browser
Title:  The Mathematical Art of M. C. Escher: Polyhedra
Comment:  Commentary on three of Escher's works featuring strong polyhedra content, including Stars and Order and Chaos. Click on any graphic for an enlarged view.
Special:  Click on the thumbnail above to access a enlarged version of Stars.
Title:  Escher Cuboctahedron
Comment:  Colored net of a cuboctahedron decorated with tessellating fish by M. C. Escher. Print on the heaviest weight of card stock your printer will allow, score along all straight line segments between adjacent polygons, cut out the net, then assemble into a cubocahedron using transparent tape to form additional edges as required.

ACTIVITY 26 (Kaleidocycles)
Title:  Rings of Tetrahedra (Kaleidocycles)
Comment:  A kaleidocycle is a ring of linked tetrahedra. It is possible to form a chain with an even number of  tetrahedra. What makes it special is that you can continually twist it inwards or outwards while it shows different sides of each tetrahedron.
Special:  Click on the graphic above to see the cycling motion of a kaleidocycle consisting of six congruent tetrahedra.
Title:  Other Kaleidocycles
Comment:  Contains LiveGraphics3D applets that allow you to move kaleidocycles with your mouse. Use the internal links to access six other interactive kaleidocycles. All require a JAVA-capable browser.
Title:  Animated Kaleidocycles
Comment:  Features an applet by Marcus Engel to create and animate kaleidocycles. You can even add an M. C. Escher texture. Requires Internet Explorer and Sun Java software.
Title:  Origami Activities
Comment:  Make your own kaleidocycle with earthquake-education themes. The detailed instructions combine the rapid production of tetrahedra from envelopes with various geological themes printed in color on hexagons to produce customized kaleidocycles covering the topics of earthquakes, plate tectonics, and the geology of New Mexico.
Title:  A Moving Sculpture Made from Paper
Comment:  As you turn this little paper sculpture inside-out, it changes colors. First yellow, then blue, then red, then green, and then yellow again. You can keep turning it inside-out, cycling the colors, as long as you like. Includes a link to a Printout and detailed instructions. The movie in Hexagonal Kaleidocycle shows an alternate way of assembling and flexing this fascinating paper toy.
Special:  View an Animated Movie of a kaleidocycle flexing.
Title:  M. C. Escher Kaleidocycles
Comment:  This one-of-a-kind publication by Doris Schattschneider and Wallace Walker includes 17 die-cut prescored kaleidocycle models, all printed with Escher tessellations, and a lavishly illustrated book of instructions. Visit Kaleidocycle to see an Escher kaleidocycle cycling.
Special:  Download a B/W or a Color kaleidocycle net based on Escher's Lizard/Fish/Bat tessellation. The files will require Adobe Acrobat Reader to view and to print. Click on the thumbnail above for an enlarged view of the assembled model. Visit M. C. Escher Kaleidocycle on YouTube to see it cycling. 
Title:  Make Your Very Own Photo Kaleidocycle
Comment:  Make your own full color kaleidocycle from your own photos (requires 4 photos). You can choose from either a hexagonal (6 sides) or octagonal (8 sides) shape. Includes detailed instructions. Requires Firefox or Google Chrome as a browser; unfortunately your own images may not upload in Internet Explorer.

ACTIVITY 27 (Folding and Unfolding Cubes)
Title:  Magic Cube / Magic Cube Puzzle
Comment:  A "magic cube" is a hinged configuration of 8 single cubes. When you manipulate a magic cube, three new surfaces appear. Contains simple instructions for constructing a magic cube from wooden blocks. Magic Folding Cube features similar instructions with excellent graphics and supportive Flash animations (require Macromedia Flash Player). Picture Block includes an animated gif. Picture Cube Puzzle and Eight Cube Picture Puzzle feature printable instructions (require Adobe Acrobat Reader). Suitable 1-inch wooden cubes are available in bulk from EAI Education. Use 2-inch wide packaging tape for the hinges.
Title:  Folding and Unfolding Cubes
Comment:  Pattern by Ellen Yi-Luen Do for a magic cube which folds and unfolds in a continuously changing pattern of pictures. Print on card stock using a landscape orientation.
Title:  Poster Point: Magic Cubes
Comment:  Commercial magic cubes are usually picture cubes, endlessly folding and unfolding to reveal a series of nine different images (six outside and three inside). Poster Point offers nine different cubes, several involving famous artists. Mini Smiley Face Cube Puzzles can be purchased by the dozen.
Title:  MagiCube
Comment:  Home page of the promotional 3D Cube that Unfolds revealing 9 custom graphics on one interactive premium. Magic Cubes Catalog Index contains links to the cubical products - many with associated Flash demos. All demos require Macromedia Flash Player.
Title:  The Puzzling Rhombic Dodecahedron
Comment:  A particular non-regular tetrahedron and its closely related polyhedra, the cube, the rhombic dodecahedron, and the first stellation of the rhombic dodecahedron, share dimensions and space filling properties. Explore the connections between these polyhedra and how to use their common dimensions to create a net or pattern to build three intriguing puzzles, which exploit their space filling properties.
Title:  Rhombic Dodecahedron
Comment:  The rhombic dodecahedron is a polyhedron with 12 faces each of which is a rhombus. It can be obtained by dividing a cube into 6 congruent pyramids with apex at the center of the cube, then flipping every pyramid outside the cube (Animation). Al Mendle has turned the shape into an elegant 2014 Calendar (requires Adobe Acrobat Reader to view and print).
Title:  Escher's Solid
Comment:  The usual stellation of the rhombic dodecahedron was a favorite polyhedron of M. C. Escher and is often referred to as Escher's Solid. It shows up in at least two of his works: decorating the top of one of the towers in Waterfall and in a Study for Stars [instead of the compound of three octahedra which appears in final version - Stars]. It was also the subject of a wooden take-apart puzzle he designed. In Diagonal Burr Puzzle, six identical cardboard pieces assemble into the space-filling polyhedron. Available commercially in solid hard wood as Star Clusters.
Title:  Yoshimoto Cube
Comment:  Cube / Stellated Rhombic Dodecahedron puzzle invented by Naoki Yoshimoto in 1971. Gold and silver Yoshimoto Cube can be purchased on-line from the Museum of Modern Art. This Animation of the puzzle requires Macromedia Flash Player.
Special:  Click on the graphic above to view a puzzle animation. Or visit YouYube's Yoshimoto Cube for a live action video.

ACTIVITY 28 (Icosahedron Globes)
Title:  Mercator Projection
Comment:  A map of the classic Mercator Projection, suitable for downloading and printing as a transparency. Use for comparison with the Fuller Projection (follows).
Title:  Fuller Projection
Comment:  The Fuller Projection, or Dymaxion Map, solves the age-old problem of displaying spherical data on a flat surface using a low-distortion transformation. It is rendered by juxtaposing a grid of triangles on the globe and transferring the data to corresponding triangles on an unfolded icosahedron. Includes a link to a Dymaxion Projection Animation. Downloading of the unfolding Dymaxion Map as a 1.4 meg ZIPped 640x480 QuickTime movie is highly recommended. Decompress with WinZip or freeware ZipCentral.
Title:  Buckminister Fuller Institute
Comment:  Web site devoted to the illustrious architect. Large Dymaxion Map is suitable for downloading and printing as a transparency. Or view the Map In Animation moving from a globe into a flat surface. Purchase Dymaxion maps and large fold up globes from the Institute's Store. If you browser is JavaScript enabled, try the on-line Bucky Map Puzzle. (To move a puzzle piece, Netscape users should click on it once before dragging it around.)
Title:  Icosahedron Globe
Comment:  Print the 20 tabbed triangles in the four sequential graphic files (Globe1, Globe2, Globe3,Globe4) on card stock using a color printer, then score and join the 30 numbered pairs of tabs to create an icosahedron decorated with the continents of the earth. Click on the thumbbail above for an enlarged view.
Title:  Planetary Icosahedrons
Comment:  Icosahedron models of the sun, planets, and their moons. Select the graphic of your choice, then the size appropriate for your printer. Print, cut, fold, and tape or glue. The net for the Earth Icosahedron has been resized for convenient printing on legal-size cardstock in portrait format.
Title:  Surface of the Earth Icosahedron Globe
Comment:  Information on a cut-out-and-fold-up icosahedron globe in full color available for purchase from the World Data Center for Marine Geology & Geophysics.

ACTIVITY 29 (Geodesics)
Title:  Geodesic Domes: History
Comment:  Concise history of geometric domes with a number of appropriate links, including the National Geographic based Alexander Graham Bell and the Octet Truss.
Title:  Spaceship Earth
Comment:  Color photograph of the familiar Epcot dome suitable for downloading and printing as a transparency. The structure is a true sphere or geosphere. It weighs over 16 million pounds, is 165 feet in diameter, 180 feet in height, and encloses some 2,200,000 cubic feet of space inside. On the outside the surface consists of 954 triangular panels.
Title:  Expo 67 - United States Pavilion
Comment:  The largest dome in the world is the former USA pavilion at Expo 67 in Montreal. It is an 18-frequency dome and measures 250 feet in diameter. Photos include an aerial view of the dome suitable for downloading and printing as a transparency. (Sparks from welding repair work set the dome's Acrylic Skin on Fire (Video) in 1976, destroying it.)
Title:  Dome Geometry
Comment:  Features an applet allowing you to interact with an exhibit on dome frequency. Part of Synergetics on the Web by Kirby Urner. Requires a JAVA-capable browser.
Title:  Dome Formulas
Comment:  Includes chord or strut factors for various frequencies of geodesic domes, where the strut factor is the dome radius divided by the strut length.
Title:  An Inexpensive Technique for Building Models of Geodesic Domes 
Comment:  Numbers and drawings are nice, but models are better. Includes suggestions for geodesics constructed with plastic straws and pipe cleaners.
Title:  Geodesic Club House
Comment:  Plans for a newspaper dome that you can fit inside. Includes a link to a Printer Version. Don't miss the picture of the grade 2 class at College Park School inside their dome.
Title:  Astro-Logix Geodesic Sphere
Comment:  To build a geodestic sphere, you will require 12 5-way hubs, 30 6-way hubs, 60 77-mm tubes (pink), and 60 88-mm tubes (green or orange). Includes link to Instructions (requires Adobe Acrobat Reader).

ACTIVITY 30 (The Buckyball)
Title:  A Brief History of C60
Comment:  C60 is a molecule that consists of 60 carbon atoms, arranged as 12 pentagons and 20 hexagons. The shape is the same as that of a soccer ball.
Title:  Learning about Buckyballs
Comment:  Students wondering why we study about polyhedra can find one reason by looking through the information in the links to Buckyballs. Includes a link to a Buckyball Applet.
Title:  Soccer-Ball-Shaped Molecules Detected in Space
Comment:  Astronomers using NASA's Spitzer Space Telescope have discovered buckyballs in space for the first time. Video courtesy of YouTube.
Title:  Darling Models: Fullerene C60
Comment:  Source of a 14-inch diameter demonstration model of Fullerene (Buckyball) available in 4 colors. Also available in a Glow Version.
Title:  NOVA Online: Race to Catch a Buckyball
Comment:  This episode of NOVA documents the chance discovery of an entirely new form of carbon: soccer-ball-shaped miraculous molecules called Buckyballs.
Special:  View the entire film courtesy of YouTube.


More Books and Web Pages by Jill Britton
Fun with Patterns

All M. C. Escher works (c) Cordon Art B.V. - Baarn - the Netherlands.
Used by permission. All rights reserved.

NCTM Illuminations

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