21. The Math Forum - Math Library - Fractals This page contains sites relating to fractals. tessellations, polyhedra,limits, logs, fractals (with galleries), puzzles and problems, fun with numbers http://mathforum.org/library/topics/fractals/?keyid=14240478&start_at=51&num_to_

22. 3D Fractals It generates 3dimensional fractals by superimposing polyhedra upon themselvesrecursively. You get to pick the polyhedron, the level of recursion, http://www.houseof3d.com/pete/applets/wireframe/fractal/

3D Fractal Generator This is an extension of my wireframe applet. It generates 3-dimensional 'fractals' by superimposing polyhedra upon themselves recursively. You get to pick the polyhedron, the level of recursion, and the spaces between the individual polyhedrons. You can set the fractal spinning by dragging on it with your mouse and releasing. If you have 3D glasses, you can use those, too. Get a damn java enabled browser, would you? gently drag and release to start the model spinning window.onerror = null unit shape : tetrahedron pyramid sextahedron octahedron cube levels : DANGER!!! separation : no space a little space some space space lots of space tons of space 3D specs : no 3D glasses red lens right red lens left red right (Mac) red left (Mac) PLEASE increase levels gradually - see below.) Hint - spinning cubes look way cool with lots of separation. Use CAUTION when selecting how many levels of recursion you want. A recursive tetrahedron with 5 levels contains 1,536 lines, and 6 levels has 6,144 lines. A cube with only 4 levels also has 6,144 lines (if your browser can survive it, either of these looks wild spinning slowly (with tons of space for the cube, optional for the tetrahedron)). But BE CAREFUL!

23. What Is A Fractal? Fractal Is A Word Invented By Benoit Mandelbrot 3D IFS fractals are 3D analogous to 2D IFS fractals using polyhedra instead ofpolygons. ie Menger sponge is 3D analogous to Sierpinski Carpet. http://www.fractal3d.com/faq/faq.html

What is a fractal? Fractal is a word invented by Benoit Mandelbrot to specify the complicated phenomena of shapes with self-similarity. According Benoit Mandelbrot words in his book, The Fractal Geometry of Nature, "I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragents. It is therefore sensible - and how appropriate for our needs! - that, in addition to "fragmented" (as in fraction or refraction), fractus should also mean "irregular," both meanings being preserved in fragment." Other descriptions on fractals can be found from the following links: http://www.faqs.org/faqs/sci/fractals-faq/ sci.fractals FAQ (Q2: What is a fractal?) http://mathworld.wolfram.com/Fractal.html fractal: ERIC WEISSTEIN'S world of MATHEMATICS What is fractal3D? Fractal3D (www.fractal3D.com) is the world's first web site which provides information and products of great new finding 3D quasi-fuchsian fractals in pure mathematics. What is quasi-fuchsian fractals?

24. Amby's Education Site -- Math Puzzles & Learning Activities Applied Math / Simulations, Games Puzzles, Origami, polyhedra, fractals,Optical Illusions. Math Games Puzzles. GuessA-Number http://amby.com/educate/mathgame.html

Amby's Education Site: Learning Activities Applied Math / Simulations Origami Polyhedra Fractals ... Other Math Resources Applied Math / Simulations EduStock Edustock is designed to teach people of all ages about the stock market. It includes: Tutorials on the stock market How to pick good stocks Company Profiles: Information to help you research companies FREE realtime Stock Market Simulation Links to other sites with useful Stock Market info EduStock http://tqd.advanced.org/3088/welcome/welcomenf.html The Diet Problem: An Application of Linear Programming Check-off the foods you're willing to eat and the program will find the cheapest combination of them meeting a person's daily nutritional requirements. Provides numbers of servings, cost, percentages of nutrients from different sources, etc. in both table and pie chart form. PLEASE consult your doctor before embarking on any diet especially one, for example, including 10 servings of Air-Popped Popcorn!! Argonne National Laboratory Mathematics and Computer Science Division http://www.mcs.anl.gov/home/otc/Guide/CaseStudies/diet/index.html

25. Some Common Themes In Visual Mathematical Art One of the common themes in mathematical art is polyhedra. His recognitionof the esthetic content of fractals is shown in the following quote. http://www.mi.sanu.ac.yu/vismath/fath/

Some Common Themes in Visual Mathematical Art Robert W. Fathauer Tessellations Company Tempe, AZ 85281, USA E-mail: tessella@futureone.com Abstract 1. Introduction Historically, mathematics has played an important role in visual art, particularly in perspective drawing; i.e., the means by which a three-dimensional scene is rendered convincingly on a flat canvas or piece of paper. Mathematics and art are two seemingly disparate fields according to contemporary views, the first often considered analytical and the second emotional. Mathematics does not play an overt role in most contemporary art, and in fact, many artists seldom or never employ even perspective drawing. However, there are a number of contemporary visual artists who make mathematics a focus of their work. Several notable figures in history paved the way for these individuals. This paper is intended to some degree to serve as an introduction to the Exhibit of Visual Mathematical Art held as part of Bridges 2001. The examples given here are taken for the author’s convenience from a limited set of artists. The choice of works is by no means intended to be a representative sampling of artists working in this area. 2. Notable figures in the history of mathematical visual art

26. Coolmath4kids -- GEOMETRY There a lots of cool polyhedra AND you can spin them around! Learn all aboutfractals. There are different types of fractals and the ones explained here 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

27. Books About Polyhedra (Science U) polyhedra Tiling Symmetry fractals General Interest. polyhedra, by Peter R.Cromwell. Build Your Own polyhedra, by Peter Hilton, Jean Pedersen http://www.scienceu.com/store/mathematics/polyhedra.html

Books: Buying Pictures: Polyhedra Polyhedra Fractals General Interest Polyhedra , by Peter R. Cromwell Build Your Own Polyhedra , by Peter Hilton, Jean Pedersen Regular Polytopes , by Harold Scott MacDonald Coxeter Polyhedron Models , by Magnus J. Wenninger Info Center Geometry Center Library Observatory ... Science Me Page last updated Sat Nov 23 17:06:41 CST 2002 Comments to webmaster@ScienceU.com Geometry Technologies 1999. All right reserved.

28. Books About Fractals (Science U) polyhedra Tiling Symmetry fractals General Interest fractals Everywhereis the most comprehensive, upto-date volume on fractals available today http://www.scienceu.com/store/mathematics/fractals.html

Books: Buying Pictures: Fractals Polyhedra Fractals General Interest Fractal Geometry of Nature , by Benoit B. Mandelbrot Fractals Everywhere , by Michael F. Barnsley The new edition features additional problems and tools emphasizing fractal applications, as well as a new answer key to the text exercises. Fractals Everywhere is the most comprehensive, up-to-date volume on fractals available today. [synopsis, amazon.com] Info Center Geometry Center Library Observatory ... Science Me Page last updated Sat Nov 23 17:06:40 CST 2002 Comments to webmaster@ScienceU.com Geometry Technologies 1999. All right reserved.

29. Zometool Geometry Workshop in the classroom to present interesting geometric ideas, including polyhedra,tessellations, fractals, space structures, the fourth dimension, etc. http://www.georgehart.com/zome-workshop/workshop.html

The following is reprinted from the Bridges 2001 conference proceedings. Zometool Geometry Workshop George W. Hart Zometool is a commercial plastic construction set which is well engineered to make a very wide range of beautiful 3D structures. George W. Hart and Henri Picciotto have co-authored the book Zome Geometry, which describes hundreds of mathematical models and other fascinating forms which can easily be made with Zometool [1]. It is designed for both classroom and self-study use. Detailed contents are given on the web page [2]. Hart has led Zometool workshops at conferences, museums, and schools. Figures 1-4 show a range of small and large models made by participants of the workshop he led at the MOSAIC 2000 conference at the University of Washington in Seattle, in August 2000. All photos are by Douglas Zongker. Fig. 1. A range of prisms, tetrahedra, icosahedra, dodecahedra, etc. made with Zometools by workshop participants. Fig. 2. A projection of a four-dimensional solid called the "120-cell" into our three-dimensional space. Fig. 3.

30. Hands-On Math Modules fractals; Gumdrop polyhedra; Mathematical Games and Brain Teasers Sonobe.pdf These are instructions for making the origami polyhedra. http://amanda.serenevy.net/GirlScouts/

31. Efg's Fractals And Chaos -- Von Koch Curve Lab Report Koch s Flakes in fractals polyhedra, Flakes Ltrees www.people.nnov.ru/fractal/VRML/3dLsys/3Dtree.htm. Koch Snowflake from EricWeissteins s World of http://www.efg2.com/Lab/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

32. Favorite Links Of Andrea And Friedrich Lohmueller - 3D Computergraphics, Mathema (en) About polyhedra, Stellations, fractals. Tymless by Julian Davies. (en)Tymless is a computer language for drawing and colouring shapes. http://www.f-lohmueller.de/links/index_se.htm

Home Links: Raytracing Art Raytracing + POV-Ray Raytracing + Science Mathematics + Geometry Chemistry, Biology Electronics, Astronomy, Physics Architecture ... Other Links last revised: 27-Aug-2005 to our Homepage to our Raytracing Galleries Deutsche Version Mathematics and Geometry Tsutomu Higo - Gallery Cyberbust One of the top galleries - a creative artist and a good POV-math-teacher! The source codes for many images show how he made this. (en,jp) Shuhei Kawashu An excellent gallery of creative mathematical and geometric art! (en,jp) Paul Bourke Astrophysics and Supercomputing at the Swinburne University of Technology, Australia. A great homepage on various things around rendering and visualization of all kinds, Geometry, Curves, Fractals, Polyhedra, platonic solids, polytopes, Star Sphere and Polar Sphere, etc. also about PovRay, MegaPov, MegaPovPlus (en) DivulgaMAT new ! Juno’s World All about Polyhedron - gallery and VRML - fascinating Puzzles! (jp, en) Erich Friedman's Puzzles Palace All about Puzzles! (en) Mechanical Puzzles Crafter and seller of original mechanical puzzles and other puzzle items (en) Erich Friedman's Packing Center About Tiling and packing problems! (en)

33. Fractal Polyhedra I ve started with these amazing Fractal polyhedra obtained by LaurensLapre s LParser pascal2 Hop David attracted my attention to Keplerian fractals. http://www.ibiblio.org/e-notes/VRML/Poly/Poly.htm

Fractal Polyhedra, Flakes and Webs I've started with these amazing Fractal Polyhedra obtained by Laurens Lapre 's "LParser": and . See also Encyclopedia of Polyhedra by George W. Hart Polyhedra Collection and VRML Arts by V.Bulatov Look at Fractal Polyhedra with interactive Java 1.1 applet, if you didn't install a VRML plugin yet. Kepler's (Stellated) Fractals Hop David attracted my attention to Keplerian Fractals . He replaces every horn of a stellated (non-convex) polyhedron by small self-similar polyhedron. The picture shows that we can make it in two different ways (we will obtain a II-type (solid) fractal if we combine I-type ones). Interactive Kepler's Octahedron is made of 4 Tetra Flakes (hit it to get next iteration). Hop David found out that it turns into a cube under iterations. He found "Cantor's Octahedron" beneath the cube's surface too. Kepler's Stellated Dodecahedron I-type II-type (solid) Kepler's Great Stellated Dodecahedron I-type II-type (solid) . Polyhedra with many vertises are too complicated :( Cantor's Webs You will get 1D Fractal Web (may be Laces or Hedgehog :) if you start at "skeleton" of a Polyhedron. The skeleton is made of rays from the Polyhedron center to its verteses. Look at Interactive

34. Mathematical Resources: Mathematical Art, Graphics, Chaos And Fractals (Math Lin Mathematical Art, Graphics, Chaos and fractals Mathematical Resources. Perspective Drawing, Moebius Strip, polyhedra, and Spreadsheets by Jan Garner http://mathres.kevius.com/art.html

Chaos and Fractals Newsgroups Svenska matematiklänkar Mathematical Art, Graphics, Chaos and Fractals not a complete list, only what I happened to see... 3D Strange Attractors and Similar Objects by Tim Stilson 5points art mathematics science - home page Algorithmic Mathematical Art Xah: Periodic Dosage Artlandia Thomas Banchoff's Project List Calculus graphics by Douglas N. Arnold The C*ODE*E Gallery The Chryzodes - - Les " Chryzodes " ou l'arithmétique en images Digital Archive by Richard Hawkin Digital Art, Digitale Kunst by Karin Kuhlmann Directory of /pub/pictures/art/M.C.Escher Encyclopedia of Polyhedra - Over 1000 Virtual Polyhedra by G.W. Hart ESCHER M.C. Escher Art FTP Directory @ ftp.sunet.se M. C. Escher: Artist or Mathematician? M.C. Escher - Interlocking Shapes and Tessellations, Math Art, ... Brendan's M.C.Escher page by Brendan DJ Murphy Mathematical Art of M.C. Escher a Platonic Realms Mini-Text Zvi Har'El's M.C.Escher Collection by Zvi Har'El Famous Curves Page Fun with cellular automata A Gallery of Interactive On-Line Geometry (Geometry Center) Gallery of Art from PFP'94 IDC students Gallery of Mathematical Surfaces by T. Nordstrand

35. Internet Resources For Use In Mathematics Classes Building polyhedra Project Students will work in groups to build six solids frompatterns Allows you to change the rules and make your own fractals. http://www.internet4classrooms.com/math_topic.htm

Daily Dose of the Web Links for K-12 Teachers On-Line Practice Modules Mathematics Topics General Math Elementary Secondary Topical Links verified 8/25/05 Mathematics activities related to: Number Sense, Concepts, and Operations Create a Graph - Sometimes, complicated information is difficult to understand and needs an illustration. At this National Center for Education Statistics site you will find four different graphs and charts for you to consider. EdHelper.com offers a wide range of fraction worksheets. Each time you request a worksheet an entirely new set of problems is generated. Fractions from SOS Math - This math review is targeted toward those individuals who have forgotten what they once knew about fractions. You may need the entire review or just portions of the review. Subjects covered are: Simple Fractions Complex Fractions Compound Fractions Decimals ... Percentage and Rules Multiplication: An Adventure in Number Sense - Multiplication tips; click on any underlined fact on the table to learn about a reason why you do not have to memorize it. You should also check out the multiplication index for other valuable information.

36. Modular Origami - Fractal Models Folded By Michaà ‚ Kosmulski Images of modular origami fractals folded by Michal Kosmulski. fractals Balls polyhedra Planars Spiky Balls Miscellaneous http://hektor.umcs.lublin.pl/~mikosmul/origami/fractals.html

Fractals IFS Models folded and photographed by Michaà ‚ Kosmulski. Modules designed by their respective authors. Click on images to enlarge them. Links in image titles lead to pages with more information about each particular object. Image thumbnails on this page and the full-size images they link to are licensed under a Creative Commons License (Free for non-commercial use with attribution; Ask me about other uses). Menger Sponge (level 1) Made from Sonobe module (72 modules). Menger Sponge Made from Robert Neale's penultimate module (144 modules). Thanks to the modules' shape, this model looks like a level 2 Menger sponge even though it is actually only level 1. Sierpinski Tetrahedron (level 3) Made from Nick Robinson's trimodule (128 modules and 126 links). The links were made from narrow rectangular straps of paper by folding two flat pentagonal knots on each. Koch Snowflake (level 1) Made from Nick Robinson's trimodule (12 modules). Koch Snowflake (level 2) Made from Nick Robinson's trimodule (72 modules).

37. Modular Origami - Balls And Polyhedra Folded By Michaà ‚ Kosmulski Images of modular origami balls, polyhedra and other models with high symmetryfolded by Modular origami main page fractals Intersecting planes http://hektor.umcs.lublin.pl/~mikosmul/origami/balls.html

Models folded and photographed by Michaà ‚ Kosmulski. Modules designed by their respective authors. Click on images to enlarge them. Links in image titles lead to pages with more information about each particular object. Image thumbnails on this page and the full-size images they link to are licensed under a Creative Commons License (Free for non-commercial use with attribution; Ask me about other uses). Icosahedron Made from Francis Ow's 120 degree module (30 modules, pointing inwards). Icosahedron Made from Francis Ow's 120 degree module (30 modules, pointing outwards). Truncated octahedron Made from Robert Neale's penultimate module (36 modules). Truncated tetrahedron Made from Francis Ow's 60 degree module (scroll down the linked page for unit folding instructions) (42 modules). Rhombicuboctahedron Made from Robert Neale's penultimate module (48 modules). Dodecahedron with pentagonal pyramids on all faces and inverted spikes on pyramids' side faces Made from Nick Robinson's trimodule (90 modules).

38. Generation Of 3D Fractals For Web A wide variety of amazing 3D fractals can be obtained by iteration of very Fractal Trees and polyhedra. It turns out that very realistically looking 3D http://www.people.nnov.ru/fractal/VRML/Web3D/Web3D.htm

Generation of 3D fractals for Web Introduction A wide variety of amazing 3D fractals can be obtained by iteration of very simple rules many times. The simple rules lead to very small (1-3kb) algorithms and scripts realized in Java, JavaScript or VRML. The scripts are flexible and can be combined in a library of fractals for application in complex scenes. It is well known, that as since fractals possess self-similarity on scaling, therefore they can be made by iterations of scaling transformations. In Figure 1 it is shown how the famous Koch's snowflake is obtained by repetition of simple transformations: scaling, rotations and translations. Figure 1. Generation of the Koch’s snowflake. The initial segment 1 is shrunk three times, then three small copies are translated upward and two of them rotated by +-60 degrees. Then this procedure is applied again to the entire curve 2 to get the next curve 3 and so on. Unlike the well known L-systems fractals, this algorithm does not need a rather complicated string parser and 3D turtle for its realization. It uses "natural" 3D operations: scaling, rotations and translations and it can be used with any "3D engine" (VRML, Java3D or Java1 applets). Unfortunately, there is no such "built in" 3D engine in present-day browsers. Fractal Trees and Polyhedra It turns out that very realistically looking 3D Trees and other Plants can be obtained just as the Koch’s snowflake by repetition of scaling, rotations and translations. Generation of a simple tree is shown below.

39. Math Surfaces, curves, fractals, and polyhedra, platonic solids, polytopes. Topics covered Geometry formats, curves, surfaces, polyhedra, fractals, http://mainelearns.org/materials/math

ALL CONTENT AREAS MATH RESOURCES Comprehensive Sites Reinforcement Games and Quizzes Start here before going on to other resources! Numbers and Number Sense Computation Data Analysis and Statistics Probability ... Discrete Mathmatics Mathematical Reasoning Mathematical Communication Advanced Applications MaineLearns Math Resources Welcome to MaineLearns Earth Science Resouce links. Included on this page are websites chosen for their quality, interactivity, and potential abililty to extend learning in ways that go beyond the traditional classroom. The "comprehensive" collections include valuable resources for teaching many of the earth science content standards. Navigation All earth science resources are linked from this page and can be reached by clicking on the links in the navigation bar on the left or by scrolling down through the page. Each catergory has related sub categories. The indicates sites that stand out in content, quality, interactivity, and ability to extend learning. Use the navigation menu to the left return to MaineLearns by clicking on the icon or navigate to addtional content specific areas. Other: COMPREHENSIVE MATH SITES - Resources for Multiple Standards. Start here before going on to other resources!

40. 3D Inversion Fractals 3D Inversion fractals. By Anders Sandberg. This page deals with the limit sets of One can also use other things than polyhedra for sphere centers. http://akira.nada.kth.se/~asa/Klein/

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