 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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 Softronics Inc.: An Educational Software Company
MMLogic: A MultiMedia Logic Design System (free fully functional trial version available); MSWLogo: A free Educational programming language that brings Windows and Logo together, now supporting 3dimensional geometry.
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 Software Development  Dept. of Mathematics, University of Arizona
A page with links to mathematical and graphics packages developed within the University of Arizona Mathematics Department. Researchrelated and instructional software, with packages related to computer algebra and raster graphics; vector graphics and
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 Solid Gold Gnarly Math
A Windowscompatible CDROM for kids, homeschoolers, parents, and teachers, designed to teach algebra, geometry, numbers, trigonometry, topology, and probability. It contains games, magic tricks, and other fun things, along with a Math Lab where kids
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 Solving the Quintic with Mathematica  Wolfram, Inc.
Solving the Quintic  The impossibility of solving general quintics in radicals, Tschirnhaus's transformation, Klein's approach to the quintic, Solutions based on series, Solutions based on differential equations, Equations of higher degree. Steps to
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 The Soma Cube Puzzle  Modvix
Purchase the Soma Cube. Piet Hein, a Danish poet and inventor, visualized the set of all irregular shapes formed by no more than 4 cubes, each of the same size, and realized that the seven simplest irregular combinations of cubes can form a larger Soma
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 Sommer Cube  Michael Sommer
The faces of the modular Sommer Cube (S3) feed into twisted conduits that comprise a threedimensional, tunnelling maze. The website provides example arrangements and construction questions such as "Given that a ball must exit at the lowest S3 level and
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 Space Figures and Basic Solids  Mark Motyka, Math League Multimedia
Definitions and illustrations of "space figures" and information on crosssection, volume and surface area for the following: cube, cylinder, sphere, cone, pyramid, tetrahedron, and prism.
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 Spatial Geometry  Nima Bigdely; NIMA Software Group
3D display software, designed to help students visualize spatial geometry. Shareware for Win95/98/ME/2000/NT/XP. Download available online.
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 Sphere Formulas (Ask Dr. Math FAQ)  Math Forum
Formulas for finding the surface area and volume of a sphere, sphere sector, spherical cap, and segment, zone, and lune of a sphere.
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 Spidron System  Daniel Erdely
Erdely calls a planar, planefilling object a Spidron. Upon threedimensional deformation subject to specific constraints, the planar system generates a regular, dynamic, geometrical construction. Erdely's experiments have led to the idea of creating
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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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 Star Grades: The Best Tutor  Star Grades.com
Offers $20/hr online tutoring in multiple subjects, including SAT preparation. Package deals available.
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 Stella: Polyhedron Navigator  Robert Webb
"Small Stella" is a program for viewing polyhedra and printing their nets so you can build your own. It has over 150 builtin models, plus their duals. Models include Platonic solids, Archimedean, KeplerPoinsot, Johnson solids, "Near Misses," and Stewart
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 Stereograms: A COOL Application of Geometry  David Eger
Images that look like TV static, but if you look at them in a certain way, 3D pictures "pop out" at you. Download StereoGrinder (a Java program) to create stereograms; to see them do not look at the paper or screen on which the stereogram lies, but beyond
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 Stereographic Projection (Geometry and the Imagination)  Conway, Doyle, Gilman, Thurston; The Geometry Center
Let G be a sphere in Euclidean three space. We want to obtain a picture of the sphere on a flat piece of paper or a plane. Whenever one projects a higher dimensional object onto a lower dimensional object, some type of distortion must occur. There are
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 Stochastic Lab  Denis Kozyr
Shareware and freeware math tools include the Smart Monte Carlo Excel addin, the thirteen functions of which generate random numbers with a certain distribution using the Monte Carlo method; Archim, which graphs functions based on explicit parametric
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 The Storybook of Geometry  ThinkQuest 1996
Learn the basics of geometry and see how they relate to real world situations. Explore and read stories that help to uncover the mysteries of geometry. Story problems (with answers): The Great Space Ace; The Pharoah's Pyramid; The Ice Cream Cone; Filling
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 The Story of Pi  Project MATHEMATICS!, California Institute of Technology
A videotapeandworkbook module that explores a basic topic in high school mathematics in ways that cannot be done at the chalkboard or in a textbook. Although pi is the ratio of circumference to diameter of a circle, it appears in many formulas that
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 StradivariSecrets.com  Dmitry Badiarov
Violin drawing tutorial and software based on application of early acoustics to violin making.
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 A Stranger from Spaceland  Ivars Peterson (MathTrek)
An essay about Flatland, Edwin A. Abbott's 1884 "doorway to the fourth dimension and beyond for many explorers of geometry," and "a sharply delineated satire that reflects widely debated social issues in Victorian Britain," particularly women's rights.
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 Student of Hyperspace  Eric Swab
A series of computer generated projections of analogs of the platonic solids in 4space; a projection of a regular polytope that exists only in 4space. Pages: Hypercube analog of the cube with sections and envelopes; Simplex analog of the tetrahedron
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 StudyBeat: Math  StudyBeat
Educational video clips on topics in highschool mathematics, including algebra, basic math, calculus, differential equations, geometry, prealgebra, precalculus, statistics, and trigonometry. Subscription required; sample videos available.
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 Studying Dimensionality through a StairStep Fractal  Burrill, Frost, Anderson, Burkett, Gribble, Ryerson, & Williams
A background paper and a related classroom activity about a lesson studyproduced activity on using a stairstep fractal to study other dimensions.
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 Success for All
Curriculum driven by cooperative learning that focuses on individual pupil accountability, common goals, and recognition of team success, all with the aim of getting learners "to engage in discussing and explaining their ideas, challenging and teaching
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 Surface Area of a Cone  NASA Lewis Learning Technologies Project
What is the approximate surface area of a cone whose slant height is 6 inches and whose radius is 3 inches? A detailed answer is provided. From NASA's 9th Grade Math Proficiency Test.
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 Surfaces Beyond the Third Dimension  Thomas Banchoff
A site that documents an art exhibit hosted by the Providence, Rhode Island Art Club in 1996, featuring computergenerated artwork designed by Tom Banchoff in collaboration with Davide Cervone (now at Union College) and student associates at Brown University.
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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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 Synergetics on the Web  Kirby Urner, 4D Solutions
R. Buckminster Fuller's "explorations in the geometry of thinking" is detailed in two thick volumes, Synergetics and Synergetics 2. Synergetics thoroughly permeates the rest of Fuller's writings as well. The invention for which Fuller is most famous,
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 SysteMATHics  Sigurd Andersen
Based on a discipline started by John G. Bennett, systematics is an attempt to understand the meaning implicit in number. The three areas of mathematical exploration covered are triadic numbers; ngrams, number patterns expressed graphically using the
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 Tangencies of Circles and of Spheres: Radii Solutions  Eugene F. Dearing; Dearing Aviation
New methods of finding the radii of circles tangent to three given circles, (Apollonius' Problem), and the radii of spheres tangent to four given spheres, (Apollonius' Problem in three dimensions). Download the article in zipped format, or view it online.
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 Teacher Exchange  the Math Forum
Web units, lessons, and activities organized by NCTM Standards and grade band level. Created by Math Forum staff members, educators, and collaborations between the two. Teachers may contribute their own lessons and ideas as well.
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 Teachers' Lab  Annenberg Media
Labs based on professional development series and workshops broadcast on the Annenberg/CPB Math and Science Project Channel. Each combines online activities with background information and interactive polls or worksheets for use in classrooms, with links
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 Teaching Geometry with Google SketchUp  Bonnie Roskes
3DVinci's resources for teaching geometry with this 3D modeling application include projects of the month, videos, teacher resources and guide, 2D and 3D project books, a blog, geometry tips and projects, and much more. Google originally created SketchUp
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 Teaching mathematical thinking through origami  Daniel Meyer, Cornell University; Jeanine Meyer, Pace University
A paper that offers general and specific strategies for using activities based on origami to invoke mathematical thinking. With background on origami, including its development and critical aspects of practice in the United States; how origami can relate
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 Teaching Math to Visually Impaired Students  Susan Osterhaus, Texas School for the Blind and Visually Impaired (TSVBI)
Questions and answers offering teaching strategies for improving the mathematical
understanding of visually impaired students. Challenges include solving quadratic equations; geometric constructions; transformations, line symmetry, and tessellations;
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 Teaching  Peter Alfeld, University of Utah
A page of teaching resources intended primarily for Alfeld's University of Utah students, but useful to others as well. Includes a passage from Richard Feynman on teaching; Understanding mathematics  a study guide; current and past courses (Analysis
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 Tensegrity Structures  Paul Flavin
"Compression and Tension are good; Torque's a Killer. Plato, Fuller and the three little pigs." "Cubes (constructed with beams) and houses can and do collapse when a sufficient horizontal (shearing) force is applied above the foundation. The walls act
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 The Tesseract  Cut the Knot!, Alexander Bogomolny, with Don Greenwell
A hypercube is a multidimensional analogue of a 3dimensional cube in that each coordinate of a point in a hypercube is restricted to the same 1dimensional (line) segment. The Tesseract is a 4dimensional hypercube. With an applet designed to demonstrate
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 Tesseracts: Cubes Get Hyper (MatheMUSEments!)  Ivars Peterson (Math Muse for Kids)
Madeleine L'Engle, who wrote A Wrinkle in Time, used tesseract to mean a shortcut through space and time. In her story, spacetime wrinkles, or folds onto itself, creating new paths that allow characters to tesser, or travel from one end of the galaxy
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 The Theorem of Pythagoras  Project MATHEMATICS!, California Institute of Technology
A videotapeandworkbook module that explores a basic topic in high school mathematics in ways that cannot be done at the chalkboard or in a textbook. Several animated proofs of the Pythagorean theorem are presented, with applications to reallife problems
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 Thinking Mathematics!  James Tanton
Tanton writes wordless volumes of creative and surprising math puzzles, accessible approaches to calculus suitable for standardized test preparation, and curriculum "tidbits." Download his free guides to fractions and "to everything quadratic," and watch
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 ThinkQuest
An international contest designed to encourage students from different schools and different backgrounds to work together in teams toward creating valuable educational tools on the Internet while enhancing their ability to communicate and cooperate in
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 Tiling and Packing results  Torsten Sillke
Packing problems, mostly, in English, with some in German. Packing Polycubes and Polyspheres; Tilings with Matching Conditions; links.
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 Tiling Plane and Fancy  Steve Edwards
The 17 wallpaper groups identify the plane symmetry groups and link to background patterns illustrating them. Tiling Definitions define, with examples, the terms fundamental region, isohedral tiling (isohedral and nonisohedral), isogonal tiling, lattice,
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 Time, The Fourth Dimension  Jim Loy
Because of Einstein, we often call time the fourth dimension. Special relativity shows that time behaves surprisingly like the three spatial dimensions... Pretend, for a moment, that there are more than three spatial dimensions. What is a four or fivedimensional
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 TLC  The Learning Company  Broderbund
Search for math to find product names (Reader Rabbit, Schoolhouse RockMath Rock, and more).
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 Tomoko Fuse Origami Quilts  Jeff Kerwood
These origami quilts were designed and made by Jeff Kerwood, using the modules developed by Tomoko Fuse, a Japanese origami artist universally acknowledged as the master of modular or unit origami. In modular origami, the model is built up from multiple
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 Tony Phillips  Dept. of Mathematics, SUNYStony Brook
Ongoing Web math projects: Honors Calculus; Visualizing Women in Mathematics, the Physical Sciences and Technology; Calculus Reform; Geometry of the Binary Tetrahedral Group in SU(2); Mazes; Tides; Integrating Research and Education (RAIRE); Math Photo
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 Topics in High School Mathematics  Martin Cohen
Lesson ideas on geometric series, mathematical induction, Pythagorean triples, combinatorics, continuity, modular arithmetic, rotations and revolutions, and more.
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