 PolygonFlux  John Miller
An iPad app for exploring the path taken by a point bouncing around inside a regular polygon of 3 to 14 sides. Set the initial angle and starting location, the number of bounces, and the number of sides for the polygon; collect an album of resulting "fluxagons"
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 Polymorf  Rick Engel
A math manipulative, construction set toy, model building kit, and 3D puzzle all in one, used for modeling polyhedra and the structure of molecules and crystals. Build lifelike geometric models of creatures, vehicles, and structures. For all ages.
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 Polyominoes  Slavik Jablan
A plane topological disk (i.e. a plane region without "holes") consisting of n edgetoedge adjacent squares is called a polyomino. This site offers a shape description: in every border square cell of a polyomino we could introduce twosided mirrors perpendicular
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 Positive and negative Euler characteristic (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
A symmetry pattern on the sphere always gives rise to a quotient orbifold with positive Euler characteristic. In fact, if the order of symmetry is k, then the Euler characteristic of the quotient orbifold is 2/k, since the Euler characteristic of the
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 Printables: Student Handouts  The Learning Network
PDF format printable handouts. Browse by grade or subject; check out the featured handouts; or search for a topic of your choice. Other resources also available through the Lesson Planning Center links on the site.
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 Problem Pictures  Richard Phillips
A CDROM of 184 attractive, unusual, or puzzling color photographs accompanied by activities for mathematics teaching. Images may defy explanation, motivate investigation, or inspire the viewer to answer a question. Other pictures may be the starting
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 QuasiTiler (Gallery of Interactive Geometry)  Eugenio Durand; The Geometry Center
Generate Penrose tilings or design your own nonperiodic tilings of the plane with this interactive application. In the process, you can select and visualize plane crosssections of a lattice in anywhere from 3 up to 13 dimensions. Geometry of Quasicrystals
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 Quilt Block Patterns  Hester ButlerEhle
Links to a variety of quilt block patterns (fourpatch, fivepatch, ninepatch, and more). Several are suitable for replication using pattern blocks.
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 Quilt Design Program MiniReviews  Robert G. Holland, The Virtual Quilt Company
A brief review of several quilt design programs. Links permit you to download a demo version of some programs. QuiltPro is the Windowsbased program of choice.
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 Randomness and Geometric Probability (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material on randomness and geometric probability: Buffon's needle; random domino tilings; random polygons, etc.
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 R. Buckminster Fuller's Octet Truss  Hop David
A discussion of Buckminster Fuller's invention based on the octahedron/tetrahedron tessellation.
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 Recreational Mathematics Topics  Steven Dutch; University of Wisconsin  Green Bay
Contents: Symmetry, Crystals, Polyhedra and Tilings: an extensive, illustrated collection including reflectional and rotational symmetry; repeating patterns in two and three dimensions; building crystal forms with unit cells; regular, Archimedean, radial,
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 Rectangle Pattern Challenges  Cynthia Lanius
Examine different stges of rectangle patterns, and describe what you must do to get from one to the next. Observe the designs looking for patterns. Use the symmetry of the design to help you count. Organize your information into a table. On square grid
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 Rectilinear Geometry (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material on geometric problems defined on squares and rectangles, higher dimensional boxes, cubes, and hypercubes, right triangles and other rightangled figures, and "taxicab geometry" in which distances are measured
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 ReInventing FourthGrade Mathematics  Jack J. Lopez; Kodak
A series of four related activities through which students acquired a firm, active knowledge of some of the basic terminology of Euclidean geometry, learning the vocabulary for describing various geometric shapes and transformations. Students photographed
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 Room 8 Geometry Book  Ms. Woods' First Grade
Shape drawings by firstgraders: Prelude; Shapes  Triangles, Circles, Ovals, Squares, Rectangles, Diamonds, Pentagons, Hexagons, Octagons, Other Shapes; Forms  Cubes, Rectangular Prisms, Pyramids, Spheres, Cylinders, Cones; Concepts  Corners, Angles,
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 Sacred Science: Essays on Mathematics, Physics and Spiritual Philosophy  Thomas J. McFarlane
Essays in philosophy and religion intended to reveal the harmony in mathematical, physical and spiritual knowledge: Quantum Mechanics and Reality; Symmetry in Science and Religion; Science: Physical and Spiritual; Kant and Mathematical Knowledge; The
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 Science Museum Math Exhibit  The Geometry Center
A museum exhibit on triangle tilings that contains sophisticated concepts such as tilings of the sphere and the relation between tilings and the Platonic and Archimedean solids: the sum of the angles of a planar triangle is always 180 degrees. Repeated
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 Science Museum of Minnesota
Natural history and technology. See the world's largest Camptosaurus dinosaur skeleton, touch a tornado, play the congas, do some serious Internet surfing, or test the waters of the Mississippi River. The museum is Minnesota's most popular indoor cultural
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 sciencetotouch  Jürgen RichterGebert
Projects for interactively visualizing science, from one of the coauthors of the Cinderella geometry software package: iOrnament, a drawing and explanation app for the 17 crystallographic wallpaper groups; ixquadrat, a museum exhibit of symmetry, Escher,
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 Scott Kim, Puzzle Master
A puzzle and graphic design creator, Scott Kim offers links to his online puzzles, a gallery of inversions (or ambigrams, words or names written so they read in more than one waysome read upside down, some flip in a mirror, some read as two different
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 Semiregular Tessellations  Totally Tessellated
Color renditions and real examples of the semiregular tessellations. Templates allow you to print renditions of each tessellation suitable for coloring activities.
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 Setting the Stage with a Story  Suzanne Alejandre
Article about the development, and implementation, of online lessons created by Varnelle Moore at a Math Forum summer institute. Originally published in the June, 2013, issue of the CMC ComMuniCator, journal of the California Mathematics Council.
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 Seventeen Kinds of Wallpaper Patterns  Mathematics Museum (Japan)
Examples of the 17 wallpaper groups in Japanese culture. Also available in Japanese.
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 Shmuzzle Puzzle  Sam Savage
Inspired by M.C. Escher's famous tessellation drawings, these salamanders interlock in multiple ways. Physical and online puzzles, a contest, history of the shmuzzle, resources for teachers, and other shmuzzle products.
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 A Short Introduction to Symmetry and Groups  Jenny Olive
A beautiful and satisfying area of maths that also has many important applications in areas such as crystallography and particle physics. Olive gives some examples of how symmetry can be described by groups, for which no previous maths is required, and
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 The Silicon Mirror and Kaleidoscope  Elliot Leonard
The Silicon Mirror is an interactive tool for creating tessellations. It
operates on the same principles as a kaleidoscope. The Silicon Kaleidoscope is a screen saver that uses the Silicon Mirror to create kaleidoscopic images that flow in real time.
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 SIMMS IM LEVEL I  Research Project Help  T. DeBuff
High School freshmanlevel integrated mathematics research projects, to be used with the Systemic Initiative for Montana Mathematics and Science Integrated Math (SIMMS IM), curriculum Level I. View project descriptions and find links to sites that will
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 Sine of the Times: Dividing the Universe by Zero  Colin Graham
A collection of thoughts on math by an "eternal student" with an "eclectic educational background." Posts, which date back to June, 2010, have included "Creative ways of integrating more water in the body," "My first online manipulative  virtual origami,"
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 Sir Roger Penrose  World of Escher
Recreational math is the passion of Sir Roger Penrose, who was recently knighted for his outstanding contributions to mathematics. Penrose is a professor of mathematics at the University of Oxford in England, where he pursues an active interest in recreational
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 SketchMad  Nathalie Sinclair
An archived resource, not currently updated: A resource center devoted to using The Geometer's Sketchpad in the classroom. Includes tips, strategies, lesson plans, and sketches for beginning and intermediate Sketchpad teachers, stories from the classroom
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 Sliding Pi  Ivars Peterson (MathTrek)
"The entrance to the Downsview subway station in Toronto presents visitors with a striking vista. A vast mosaic of square tiles sweeps across a curved wall, inviting viewers to trace its lines and ponder intriguing irregularities
in its color scheme
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 Snow Activities  The Teel Family
A site that includes snow science (how to study snow crystals) and snow art (cutting sixsided paper snowflakes and a "selfsticking" snowflake).
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 Snowdays  Popular Front
Make your own snowflakes online: point and click to cut anywhere around or inside the folded piece of virtual paper. Preview the unfolded results in real time; when ready, save your design to the searchable gallery, where you may view and download plans
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 Snowflakes and Stars  Ignacio Dayrit
Graphic images of snowflakes with folding instructions to make from paper.
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 Snowflakes  Kenneth G. Libbrecht
"Your online guide to snowflakes, snow crystals, and other ice phenomena." Snow activities include instructions for using everday objects and dry ice to make your own snowflakes; for using glue to make plastic snowflake replicas ("snowflake fossils")
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 Soap Bubbles  Ron Hipschman; The Exploratorium
Why does a bubble form a sphere? Why not a cube, tetrahedron, or other geometrical figure? This site from San Francisco's Exploratorium provides information on the shape of bubbles and what happens when bubbles meet, with formulas for longlife bubble
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 Solving the Quintic by Iteration [PDF]  Peter Doyle, Curt McMullen
This paper, which appeared in Acta Mathematica, gives explicitly a new solution to the quintic polynomial, in which the transcendental inversion of the icosahedral map (due to Hermite and Kronecker) is replaced by a purely iterative algorithm. The algorithm
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 Some Problems & Puzzles  Nancy Casey
Problems inspired by math books (Instant insanity; Penrose rhombs) and friends Larry Copes and Mike Fellows (BikePed, Idaho  also known as Muddy City).
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 Spirals (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material on spirals of all kinds.
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 Student Tessellations  Memorial Middle School, Totowa, NJ
Memorial Middle School's Math 7 Class of 2004 completed a tessellation project using the Math Forum's Tessellation Tutorial instructions. The best work was selected for display on the website, with their accompanying poems.
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 Studies In Mathematics: Web Discussion  Breindel, Clair; University of Chicago
The University of Chicago's alternative course to calculus. The first quarter is number theory, and the second quarter is geometry and symmetry. When Bryan Clair taught the courses, he covered the history of number systems, different base number systems
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 studio:ludens  Wouter Walmink and Alexander Rulkens
Free online tools that "let you play with your own creativity," designed by two Dutch industrial and interaction designers in Eindhoven. Create your own patterns with Repper; your own tessellating coasters with Epa:kato; your own fractals and other complex
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 Style Line: Wheel Trim and Simulators for the 21st Century
Wheel covers to investigate for reflectional and/or rotational symmetry.
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 Superliminal  Daniel Green
Use the Tyler applet to create polygons of various sizes and attach them to edges of other polygons. Read about deltahedra, flexible polyhedra, and infinite polyhedra  figures made of an infinite number of polygons which would then fill all of space
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 The Superstring Theory  Paul Hoiland
A site that gives a broad background from a mathematical point of view on general relativity, quantum mechanics, supersymmetry, and string theory. It also contains the concepts and math behind a registered modification to MTheory that is nonperturbative.
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 SymmeToy  Hank Hufnagel
A Windows program for creating paint patterns, symmetry roses, tessellated art, and symmetrically decorated 3D polyhedron models. The program is easy to learn, fun to use and includes extensive online help. Appropriate for all ages, but especially for
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 Symmetric Patterns at the Alhambra, with Math Problems  Susan Addington, David Marshall
The Alhambra is a walled city and fortress in Granada, Spain, built during the last Islamic sultanate on the Iberian peninsula, the Nasrid Dynasty (12381492). The palace is lavishly decorated with stone and wood carvings and tile patterns on most of
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 Symmetrier, monstre & tesselationer med The Geometer's Sketchpad  Allan Bergmann Jensen, Kalundborg, Denmark
GSP Symmetries, Patterns and Tessellations. Contains a gallery of wallpaper designs constructed using the Geometer's Sketchpad with JavaSketchpad versions to explore, an illustrated paper on "The Math About Symmetries" (in Danish), a GSP guide, a links
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 Symmetries and the Millennium (Math Chat)  Frank Morgan; Christian Science Monitor
What are the symmetries of a cube in space? What about a hypercube in 4 dimensional space?
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