21. Math Forum: Alejandre: Math Lessons: Geometry draw reflections. Studying Mandelbrot fractals What is a fractal?A middleschoolers. Studying polyhedra What is a polyhedron? A http://mathforum.org/alejandre/geometrylist.html

Suzanne's Mathematics Lessons Geometry aligned to the NCTM Standards - Geometry: Grades 6-8 Bricks Activity Classroom activity (also called the Masonry Problem; a variation on polyominoes) to be explored through manipulatives (dominoes). Students explore different possibilities of making brick walls with and without fault lines, using diagram, process, and solution in their problem solving. Buying Tile Cooperative group activity designed to reinforce the concept of area. Two examples of student work are available. Chinese Tangram Puzzles Rompecabezas 'Tangrams' Chinos (Spanish version) In this activity from Frisbie Middle School's Multicultural Math Fair students solve puzzles using rotation, reflection and translation to fit all the pieces together. Circle Designs Includes some background on circles in Islamic cultures, and an activity exploring the geometry involved in some circle designs. Crystals Students studying polyhedra enjoy seeing the structures as they occur in the real world. Crystalline structures can be categorized into seven crystal systems: see photographs of beautiful crystals or make paper models of crystals by printing out nets of crystals and constructing the models. CrystalMaker software gives students experience with ball-and-stick, space-filling, wire-frame, stick, dot surface cloud, and polyhedral models of crystals. Also Java applets that show some of the major forms for the hexoctahedralclass (symmetry 4/m3bar2/m) of crystals; and links to Web sites about crystals.

22. Math Forum: Suzanne Alejandre: Math Lessons polyhedra Speech Using KaleidoTile Quadrilaterals Regular Tessellations Sonya's SymmetryStudying Mandelbrot fractals Studying polyhedra Tangrams Tessellation http://mathforum.org/alejandre/index1.html

Math Forum Web Units Suzanne's Mathematics Lessons organized according to the NCTM Standards - Grades 6-8 Other views: Suzanne's Mathematics Lessons 1995-2000 Suzanne's Mathematics Lessons 2001 Check out Web Workshop Ideas Suzanne's Home Page Content Standards Number and Operations Finding Sums Magic Squares Franklin Activity Lo Shu Activity ... Srinivasa's Spheres Algebra Algebraic Factoring Algebraic Thinking Dominoes Graphs to Go ... Rubric Measurement Anybody Know The Time? Area and Perimeter Building A Pyramid Buying Tile ... What is Area? Data Analysis and Probability How to Make a Bar Graph How to Make a Pie Chart How to Make a Pictogram Probability ... Three School Survey Project Geometry Bricks Activity Buying Tile Chinese Tangram Puzzles Circle Designs ... What is Area? Other Resources Curriculum and Standards 7th grade Mathematics Standards-aligned 7th Grade Math developed during 1999-2000 Middle School Math Curriculum Position Paper: Standards 2000 CA Mathematics Standards CA Mathematics Standards CA Mathematics Standards Tutorials Claris Works Tessellation Tutorial Claris Works Graphing Graphing Exponential Functions Graphing Linear Functions ... More LogoWriter Tessellations Multicultural Math Fairs 1999-2000 version aligned to CA Math. Standards

23. Ivars Peterson's MathTrek -Mathematical Art On Display store/html/index.html offers an introduction to mathematical art and a selectionof products related to tessellations, polyhedra, fractals, and much more. http://www.maa.org/mathland/mathtrek_11_4_00.html

Search MAA Online MAA Home Ivars Peterson's MathTrek November 4, 2000 Mathematical Art on Display The term "mathematical art" usually conjures up just one namethat of Dutch graphic artist M. C. Escher (1898-1972). Many people are familiar with Escher's endless staircases, hyperbolic tilings, Möbius ants, intricate tessellations, and other creations. They may also be aware of the intertwining of mathematics and art during the Renaissance, with the development of perspective painting and eye-teasing stagecraft. A few might add the names of Piet Mondriaan (1872-1944), Salvador Dali (1904-1989), Victor Vasarely, or Frank Stella. But the realm of mathematical art is far wider and more diverse than most people realize. A surprising number of contemporary artists count mathematicsfrom Fibonacci numbers and the digits of pi to tetrahedra and Möbius stripsas the inspiration for their creations. For those fortunate enough to be in New York City, the exhibit offers a rich sampling of artworks inspired by mathematics, ranging from the gracefully curved sculptures of Brent Collins and tensegrity structures of Kenneth Snelson to the playful polyhedra of George Hart and the wavy painted grids of Doug Pedens. The show runs from Nov. 7 to Dec. 15, 2000, at the Cooper Union for the Advancement of Science and Art, Humanities Gallery and Brooks Design Center, Albert Nerken School of Engineering, 51 Astor Place, New York City (

24. IDE Corp. - Innovative Designs For Education the mood ), AND several different downloadable software programs to enable studentsto challenge themselves and create their own fractals!!! polyhedra Lighting. http://www.idecorp.com/link_math.htm

Math Click here to return to Archive Links Table of Contents. Click here to see the current month's links. NOTE: The links provided here are suggestions only and have not been thoroughly reviewed for appropriateness in school. Educators should review a site before directing students to it. Remember that sites change frequently and might contain different content than on previous occasions. Authentic Math UBUYACAR This a site devoted to an authentic Problem-Based Learning activity about buying a car. There are details on implementing it, the teacher's role, questions to ask, evaluation materials, and an assessment model. Math in Daily Life There are many activities provided at this site that demonstrate how math affects everyday decisions. There are examples given for home decorating, cooking, population growth, universal language and more. Problems of the Week Problems of the week are creative and challenging for students in grades three through twelve. Space is provided for teachers to discuss these problems of the week and suggest different ideas for implementing them into the curricula. Calculators Calculator.com

25. Feltboard Sites ,Catalogs For Kits & Other Educational Products , Free Lewis & C http//mathartfun.com/shopsite_sc/store/html/index.html Tessellations, polyhedra,fractals, MC Escher, Tangrams dissections, Kaleidoscopes symmetry, Paper http://www.mindspring.com/~surgicalsteel/listarc/20020730.HTM

Feltboard Sites For those of you with young children thought these might be helpful. (and fun) http://www.fastq.com/~jbpratt/education/mypages/flannelstories.html http://www.bright.net/~double/feltbo.htm Steve - Without an enemy's action, there is no possibility for patience or tolerance to arise. - The Dalai Lama http://www.ies.co.jp/math/java/comp/itoi/itoi.html Just jump right in here and see what you find. Math, and it's fun. Steve - Without an enemy's action, there is no possibility for patience or tolerance to arise. - The Dalai Lama

26. Relevant Links polyhedra, Transformation. Program You can make simple fractals, (Digital Cats) Demonstration screen saver A program for creating fractal animation (Filmer V2 http://moscow.cityu.edu.hk/~ma4527/links.htm

Polyhedra Transformation Computer graphics Newsgroup Mesh Generation Robotics Fractal Convex Hull Voronoi Diagram Others In order to arouse your interest, this page covers a number of interesting links about computational geometry, such as convex hull, computer graphics, virtual reality, and robot motion planning. There is a brief introduction on every topic. You can simply click the links to open a new window. Newsgroup of professional computational geometry Fractal Introduction What is Fractal Dimension ? Simple example similar to lecturer notes What is the length of a coastline, or the border of this fern? Program You can make simple fractals, ( Digital Cats A program for creating fractal animation ( Filmer V2.0 beta ... Geometric Mappings in Fractals Applications and Others Full of Fractal Stuff,(Lystad's Fractal Top, Home of JavaQuat) Introduce some applications of fractal in computer world, ( Interated Systems Image Gallery I ... The Leap Fractal Game Convex Hull Introduction Convex Combinations, Convex Set, Convex Hull ( On-Line Computer Graphics Notes Full illustration of Graham Scan by using pictures ( Riot ... COMPUTING CONSTRAINED DELAUNAY TRIANGULATIONS IN THE PLANE Program Both Gift Wrap and Graham Scan used ( CG Drawing Board Show you the Graham's scan algorithm step by step ( Riot ... Incremental Convex Hull Application Quantitative Analysis of Tuberculosis ( Computer Research and Applications Convex Hulls: Some Algorithms and Applications Other stuff related to convex polygon Simple linear algorithm for intersecting convex polygons and the corresponding and the algorithm Test whether a point is inside / outside a polygon ... Hausdorff distance between convex polygon Voronoi Digram

27. Xah: Great Math Programs A collection of mathematical programs, with particular reference to geometry.Category Science Math Geometry Software...... explanations. Poly (v.1.05, 1999/12) Poly is another interactive 3D programthat does polyhedra. This Trudeau. (1994). fractals. fractals http://www.xahlee.org/PageTwo_dir/MathPrograms_dir/mathPrograms.html

back to Xah's home Great Math Programs This is a list of fun math programs i have played over the years. Most programs are for Macintosh, but Windows program is also included. They are mostly shareware or freeware. A heart means excellent. Email me if you think a program should be listed here. What's New 2002-11-25. 3D-FilmStrip for Mac OS X is out! It is now called 3D-XplorMath. This vesion includes some 50 PDF files explaining various math objects. Super. Thanks Richard Palais. 3D-XplorMath Home Page . Now there's also a Mathematica package that convert 3D-XplorMath surface files to Mathematica graphics. Please get it here: 3dxm.tar.gz (Use StuffIt or WinZip to decompress the folder.) Here's some screen shots: breather.png costa.png enneper.png inv_boy.png ... symmetric4noid.png 2002-11-21. I'm adding a few obscure computer algebra software for those in the computing industry: Octave is a high-level language, primarily intended for numerical computations. It is somewhat compatible with MatLab. FreeSoftware. Website: octave.org

28. Books On Special Topics He even relates fractals to art and architecture. Unit Origami (7up) Geometryof polyhedra; Graph Problems Tomoko Fusé; Japan Publications, 1990; $19 http://dimacs.rutgers.edu/Volumes/schools/paper82/node16.html

Next: Video Tapes Up: The Teachers's Bookshelf: Reference Previous: Periodicals Books on Special Topics The following sources give interesting background material not usually found in textbooks. We include grade-level suggestions for use of the material (or for student reading), where appropriate. Ethnomathematics: A Multicultural View of Mathematical Ideas Graph Theory and Applications Marcia Ascher; Chapman and Hall, 1991; $42. This is a fascinating book, which will give you a new perspective on mathematical ideas that one often takes for granted, such as the words used for counting. Of special interest is a chapter exploring Eulerian paths, a standard topic in discrete mathematics (e.g., see [ ]), as an artistic aid to story-telling in the South Pacific islands. This is the source of some of the material in the module Drawing Pictures with One Line , described on page . The following is an excerpt from a review by Susan Picker LP `90. But Ethnomathematics explores more than the topic of graph theory as it presents the mathematical ideas of number, kin relations, games of chance and strategy, and symmetric strip decorations. ...It provides a comprehensive look at the meaning and use of similar mathematical ideas in different cultures, illuminating both the mathematics and the culture in which it appears, and through this showing the value of the study of mathematics in a multicultural setting. Alan Turing: The Enigma Codebreaking Andrew Hodges, Simon and Schuster, NY, 1983. This is a popular book that tells the story of Alan Turing and his role in breaking the German ENIGMA code during WWII. It makes good background reading for teaching cryptography or computer science.

29. Index By Grade Level. Intro to Tessellations, *; Build Your Own polyhedra, *; UnitOrigami, *; fractals for the Classroom, *; Math/Human http://dimacs.rutgers.edu/Volumes/schools/paper82/node32.html

Next: Index by topic. Up: Recommended Resources for Teaching Previous: Web Sites Index by grade level. In this section, for each of the grade-level ranges K-2, 3-5, 6-8, and 9-12, we list the titles of resources described in the text whose content seems especially appropriate. Within each subsection, titles are ordered (approximately) by reading level. Each title is followed by the number of the page that contains its description. See Appendix B for unabbreviated titles. Grades K-2 Texts Math Their Way Insides, Outsides,... Math on the Wall Literature Dr. Seuss books One Hundred Hungry Ants Grandfather Tang's Story The Tangram Magician Three Hat Day Sam Johnson ...Quilt Cloak for the Dreamer Two of Everything Anno's ...Jar Periodicals Teaching Children Math Wonderful Ideas Elementary Mathematician Web Sites MegaMath Grades 3-5 Texts Math on the Wall Insides, Outsides,... Math Coloring Book Intro to Tessellations Build Your Own Polyhedra Literature Sam Johnson ...Quilt Cloak for the Dreamer Two of Everything Anno's...Jar Math Curse Periodicals Wonderful Ideas Elementary Mathematician Software Logo Tesselmania Web Sites MegaMath Grades 6-8 Texts Insides, Outsides,...

30. Coolmath4kids -- GEOMETRY There a lots of cool polyhedra AND you can spin them around! What are fractals? byCynthia Lanius http//math.rice.edu/~lanius/frac/ Learn all about fractals. http://www.coolmath4kids.com/geometrymain.html

When you click on a link, you will be taken to another website. The little black stripe will always stay at the top and will help you get back to this page. Try it now! Links to Other Sites These sites have all won the Virtual Polyhedra: http://www.georgehart.com/virtual-polyhedra/vp.html You cannot miss this site! There a lots of cool polyhedra AND you can spin them around! Michael's Kaleidoscope: http://www.michaels.com/main/kaleidoscope/kscope-vert.html This is just plain cool... Put your pointer on the kaleidoscope and drag it around to change things. Snowflake Designer: http://www.explorescience.com/snwflak6.htm This is one of the most awesome sites I've even seen... You really get to make snowflakes! Hoberman Associates: http://www.hoberman.com/fold/hoberman.htm These guys make my favorite toys! You've got to see what a Hoberman Sphere is. What are Fractals? by Cynthia Lanius: http://math.rice.edu/~lanius/frac/ Learn all about fractals. There are different types of fractals and the ones explained here are not the kind you can make with the next four links. You'll need to wait until Intermediate Algebra to start to understand the math behind those. The Fractal Microscope: http://storm.shodor.org/cgi-bin/mandy/cnew.pl

31. Quantum Jumps, EEQT And The Five Platonic Fractals 5. Transition probabilities. 3. The Five Platonic fractals 1. Tetrahedron; 2.Octahedron; 3. Cube; 4. Icosahedron; 5. Dodecahedron; 6. Other polyhedra. 4. Notes1 http://www.cassiopaea.org/quantum_future/papers/qfract/

Next: 1. Introduction Quantum Jumps, EEQT and the Five Platonic Fractals A. Jadczyk and R. Oberg Abstract: It is shown that symmetric configurations of fuzzy spin direction detectors generate, through quantum jumps, IFS fractals on the sphere OpenSource Java project accompanies the paper. 1. Introduction 2. EEQT - Quantum Fractals 1. Geometry 2. The algorithm ... Contact Us Site maintained by www@cassiopaea.org ark@cassiopaea.org

32. Efg's Fractals And Chaos -- Von Koch Curve Lab Report Fraktaler www.e.kth.se/~e97_llj/fraktal.html. Koch's Flakes in fractals polyhedra,Flakes Ltrees www.people.nnov.ru/fractal/VRML/3dLsys/3Dtree.htm. http://homepages.borland.com/efg2lab/FractalsAndChaos/vonKochCurve.htm

Fractals and Chaos von Koch Curve Lab Report Neils Fabian Helge von Koch's "Snowflake" Purpose The purpose of this project is to show how to create a von Koch curve, including a von Koch snowflake. Mathematical Background Swedish mathematician Helge von Koch introduced the "Koch curve" in 1904. Starting with a line segment, recursively replace the line segment as shown below: The single line segment in Step 0, is broken into four equal-length segments in Step 1. This same "rule" is applied an infinite number of times resulting in a figure with an infinite perimeter. Here are the next few steps: If the original line segment had length L, then after the first step each line segment has a length L/3. For the second step, each segment has a length L/3 , and so on. After the first step, the total length is 4L/3. After the second step, the total length is 4 L/3 , and after the k th step, the length is 4 k L/3 k . After each step the length of the curve grows by a factor of 4/3. When repeated an infinite number of times, the perimeter becomes infinite. For a more detailed explanation of the length computation, see [ , p. 107] or

33. Full Site History to gallery. Grouped collections into subgalleries (lego sculptures,objets d'art, twysted polyhedra and snowflake fractals). http://davidf.faricy.net/history.php

- Image.php is finally working again! - Added to Gallery - I'm getting tired of spam, so when you click on email link, you may now have to replace '[AT]' with '@'. - Greatly simplified Calculating the Platonic Solid Measurements in the polyhedra section - Pointed links back to mathworld in the references page of the polyhedra section - New polyhedra section - Fixed image.php code to use forward slash in image URLs. - Added Black Seas Barracuda at Sunset to Gallery - Pictures now load against a black background. - I recently had an IBM 75GXP drive failure resulting in total data loss. My most recent backup was September 2001, so I lost the tori and civilization source. - Removed WADStrip utility from the Doom II section - Added Ancient Civilization to Gallery - Added CHG Digital Landscapes to Links - Added Photonic Tori and Isotoroidophotophilia to Gallery - Added MIT's Kismet page to Links - Added Newton Method Fractal Generator source and binary - Added Newton Method Fractals to Fractals - Added focal blur version to Lego castle gallery - Comprehensive clean-up of Links section; fixed/deleted broken links etc.

34. David's Links Page works. polyhedra, fractals, chaos, curved surfaces, loops, interference,tilings big gallery with explanations of the works. http://davidf.faricy.net/links.html

If you find any broken links or wrong information, kindly notify me 3D graphics: Programs: POV-Ray homepage -The Persistence of Vision freeware raytracer. Check out the links section where Ken Tyler has organized hundreds thousands of useful resources both POV-related and for raytracing in general. MegaPOV -Nathan Kopp has combined uvPov, the SuperPatch, and the MultiPatch. sPatch homepage -sPatch is a freeware spline-based modelling program that exports POV, RenderMan, DXF and VRML. POVLAB homepage -Freeware wireframe modeller for POV. Moray homepage -Moray is a wireframe modeller for POV and PolyRay. Registration is about $80 US. Resources: POV-Ray VFAQ -A list of Very Frequently Asked Questions and answers about POV-Ray. Uncle Tyler's links page -1200+ graphics links. By the time I get this up it will be 1300. By the time you read this is will be 10000. John Vansickle's POV-Ray resources -Including the Thoroughly Useful Macros. TabsNet -Large POV render time benchmark database! (buy AMD!) Galleries!:

35. Mathematiques Et Geometrie En VRML VRML with Fields amp;amp;amp;Operators Lars Schumann VRML 2.0 - polyhedra polyhedrahttp//library.thinkquest.org/19387/3D.html fractals Newton's Basins VRML http://apia.u-strasbg.fr/vrml/ressources/domaines/maths.html

Vladimir Bulatov, VRML corner V.Bulatov VRML corner http://www.physics.orst.edu/~bulatov/vrml/ Polyhedra Collection http://www.physics.orst.edu/~bulatov/polyhedra/index.html Divers exemples Maths, geometrie VRML Fractal Monoliths V.Bulatov's Home Page V.Bulatov's Polyhedra Collection Virtual kaleidoscopes ... Java Arts E

36. Hello Homework! Mathematics an introduction to fractals for middleschoolers. Paper Models - Paper polyhedrahttp//www.georgehart.com/virtual-polyhedra/paper-models.html Shows how to http://www.tds.lib.mn.us/HMath.htm

Mathematics Web sites to help you with your homework! Gathered by librarians of the Traverse des Sioux Library System Arithmetic History of Mathematics ARITHMETIC: Create a Graph http://nces.ed.gov/nceskids/graphing/ Use this graph generator to print or insert into web pages: area graphs, bar graphs, line graphs, and pie charts. Education 4 Kids sponsored by WellCenter[a free educational resource] http://www.edu4kids.com/math/ What is 2 - 6 + 3? Practice addition, subtraction, multiplication, and division with these electronic flashcards for kids. Figure This! Math Challenges for Families - Table of Contents http://www.figurethis.org/challenges/toc01-04.htm Eighty word problems - in English and Spanish - for middle schoolers from the National Council of Mathematics Teachers and the U.S. Department of Education that don't just give the answers to the problems, but explain how to think them through and solve them. The problems focus on measurement (eg. "Line Up") , numbers (eg. "Beating Heart"), statistics and probability (eg. "I Win!"), algebra (eg. "Which Way?), and geometry (eg. "Popcorn"). FunBrain.com - Kids Center

37. Math Links Knots on the Web (Peter Suber); fractals An Introduction; Beauty of fractals;3d fractals! Number Theory. The Prime Pages. polyhedra other Geometry. http://www.cs.stedwards.edu/math/links.html

38. Hyperlinks For Geometry Studying Mandlebrot fractals (applet with GOOD interactive info) http//forum.swarthmore.edu/alejandre/applet.mandlebrot.html.Geometer's Sketchpad. polyhedra. http://euler.slu.edu/teachmaterial/hyperlinks_for_geometry.html

Hyperlinks to related topics in Informal Geometry (Courtesy Dr. R Freese) Contents: Area and Volume Circles Crystals Drawing ... Reference (NCTM) Area and Volume Experiment with Volume (simle worksheet, estimating volume) http://math.rice.edu/~lanius/Geom/cyls.html Volume Functions: (Worksheet)Calculating volumes http://math.rice.edu/~lanius/Geom/cyls2.html Geoboard Area : a worksheet exploring area. (Based on the geoboard) Circles Designs with Circles: construction of circles (rotation and reflection symmetry) http://forum.swarthmore.edu/alejandre/circles.html Percentage Circle http://forum.swarthmore.edu/~sarah/shapiro/gsp.percent.circle.gif Crystals Nets of crystals: paper construction of crystals, polyhedrahttp://forum.swarthmore.edu/alejandre/workshops/crystalnet.html Systems of Crystals (type of crystals, and physical examples) http://forum.swarthmore.edu/alejandre/workshops/chart.html Real World - Viewing Crystals (applications of Tesselations in the real world) http://forum.swarthmore.edu/alejandre/workshops/crystal.html Drawing Isometric Drawing http://forum.swarthmore.edu/workshops/sum98/participants/sanders/Isom.html

39. Pop Quiz: Further Information For Question #5 on equilateral triangles are the pages about crystals; fractals, especially the SierpinskiTriangle and the Koch Snowflake (see below); and polyhedra (try the http://www.hamilton.net/subscribers/brian/popquiz/0005.html

Further Information Question #5 An equilateral triangle is simply a triangle in which all three sides are of equal lengths. This, for example, is an equilateral triangle: Another interesting thing about equilateral triangles is that they are also equiangular , meaning all three angles are equal as well. Since there are 180 degrees in a triangle, and all three angles are the same in an equilateral triangle, each angle of an equilateral triangle measures 60 degrees. An equilateral triangle is known as a regular polygon , since all its sides are equal and all its angles are equal. Equilateral Triangles in Everyday Life Equilateral triangles are found nearly everywhere. Examples include many everyday objects, landmarks, and symbols. Yield signs Tetrahedra (Pyramids) Triangles (Musical instruments) The Greek letter delta, and our own letter A Formulas for Equilateral Triangles Several formulas exist which are useful in finding measurements of equilateral triangles and polyhedra which contain equilateral triangles. A few are given below. Area of a Triangle The area of any triangle is given by A bh where A is the area

40. Math 8 - Frisbie Middle School Building Math Power, polyhedra, Graphing and Functions, Linear and ExponentialFunctions, fractals and Chaos Theory, Applied Data Analysis. http://www.rialto.k12.ca.us/frisbie/math8.html

Math 8 As part of the Rialto Unified School District, Frisbie Middle School is using the Glencoe instructional materials to teach mathematics. Math 8 is a general mathematics course for eighth grade students not enrolled in Algebra I. Under the Curriculum Links the alignment of the course can be checked against the RUSD Math Matrix, California Framework Standards, and the NCTM Standards. Links are provided to support the Math 8 curriculum for students, teachers and parents. Some of the links are to Internet sites with information, others link to sites with interactive lessons which could be used at home as a supplement or during class and still others could be used as enrichment for students willing to work before or after school. A similar Math 7 page is available for the general mathematics course for all seventh grade students. As you use these pages, comments and/or suggestions would be appreciated. Please send them to Suzanne Alejandre Curriculum Links: Math Matrix - RUSD RUSD Course of Study CA Academic Standards Commission California Mathematics Framework Strands Unifying Ideas for the Middle Grades NCTM Standards Reference Links: Dr. Math FAQ - The Math Forum

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