41. Amy's Geometry 6-8 Web Links What are fractals, anyway? Here's a site to help kids really understand the propertiesof fractals and how different types are formed. Virtual polyhedra http://teams.lacoe.edu/documentation/classrooms/amy/geometry/6-8/web/web.html

California Mathematics Standards California Mathematics Standards Links to adopted core academic content standards in the areas of English-language arts, mathematics, history-social studies and science. General Geometry Escher Paper Cutting Polyhedra ... Just For Fun General Geometry Resources Fractals What are fractals, anyway? Here's a site to help kids really understand the properties of fractals and how different types are formed. Interactive Mathematics The National Library of Virtual Manipulatives from Utah State University presents these online manipulatives to enhance student learning. They are indexed by math topic and grade level. Puzzle Parlor Twelve interactive online games from the puzzle makers at Kadon Enterprises, Inc. designed to give students practice with problem solving skills using mathematical and spatial sense intelligence. Virtual Polyhedra A list of virtual reality models, and links to further topics. Includes "Make Your Own Paper Models" with beautiful pictures and directions. Be sure to install a virtual reality player or plug-in, first. Return to the top Tangrams Math Forum - Tangrams This math unit on tangrams was created by Tom Scavo for the Math Forum at Swarthmore University.

42. Math Topic URL http//spanky.triumf.ca/ Spanky WelcomeThis site has all the information youneed for integrating fractals into your curriculum. Back to top. polyhedra. http://home.golden.net/~gvma/links_mt.html

GRAND VALLEY MATHEMATICS ASSOCIATION Gvma@golden.net Please E-mail GVMA to suggest additions to this list, or to comment about these links. Math Topic Links Calculus Fractals Geometry History of Math ... Trigonometry Calculus Links: URL: http://www.seresc.k12.nh.us/www/alvirne.html AP Calculus on the web URL: http://www-cm.math.uiuc.edu/ -This is an online calculus course combining technology with math. Mathematica offers real world applications. URL: http://www.hofstra.edu/~matscw/realworld.html Calculus Applied to the Real World -This web site is the home of three complete online calculus textbooks. These books are fully interactive and include exercises and quizzes. URL: http://www.math.psu.edu/dna/graphics.html Calculus Graphics-Douglas N. Arnold -This web site contains online multimedia representations of the basics of calculus. It covers topics such as the definition of the derivative, limits and differentiability, and integrating for volumes of 3-D surfaces. URL: http://dept.physics.upenn.edu/courses/gladney/mathphys/Contents.html Interactive Textbook for PEP '96 -This site combines the basics of calculus with the study of motion. There are detailed explanations of the derivative, as well as trigonometric and exponential functions.

43. Products fun! Tessellations, polyhedra, fractals, MC Escher, Tangrams dissections,Kaleidoscopes symmetry, Paper folding and more Buy http://www.hoagiesgifted.org/products.htm

Hoagies WWW Educational Products Don't miss Smart Toys - perennial favorites of gifted kids, Movies - featuring gifted kids in a positive light, and Nerd Shirts - those shirts we love to wear! The 24 Game Create the number 24 from the four numbers on a game card. You can add, subtract, multiply and divide. Use all four numbers but use each number only once. Available from simple addition and subtraction, right up to fractions, decimals and algebra. Click for 24 Challenge tournament schedule... Arbor Scientific Great science supplies, with good prices Calculus By and For Young People by The Mathman - Don Cohen and lots of related math materials... Carolina Supply Company Science (and some math) supply company, with products in a reasonable price range... Challenge Math For the Elementary and Middle School Student by Edward Zaccaro "Math is often taught as all scales and no music. This book contains the music!" I couldn't have said it better myself - tons of fun problems... Core Knowledge Foundation founded by E. D. Hirsch, Jr.

44. Tessellations' Ordering Page Mathartfun.com carries a variety of books, puzzles, posters, videos and softwarerelated to tessellations, polyhedra, fractals, kaleidoscopes, anamorphic art http://members.cox.net/tessellations/Ordering.html

Ordering Tessellations' Products We encourage our customers to purchase our products from our retailers. Mail-order retailers that carry Tessellations' products include Jacobs Publishing, Educator's Outlet, Nasco, Delta Education, Mindware, and Timberdoodle. If you are interested in purchasing online, we recommend our sister company, Mathartfun.com. In addition to Tessellations' products, Mathartfun.com carries a variety of books, puzzles, posters, videos and software related to tessellations, polyhedra, fractals, kaleidoscopes, anamorphic art, and more! Click on the logo below to go to Mathartfun.com. mathartfun.com home You can also order direct from Tessellations. Click here for more information on this option. Click here to download a catalog in PDF format (960K). Click here to download an order form in PDF format (60K). If you would like a print catalog sent to you, click the e-mail link below and give us your name and street address. Go to Tessellations Home Webmaster: Robert Fathauer; rob@tessellations.com

45. This Page Is Your Entrance To Interesting Animations Applets - Iterated Function System fractals 2; Multiple Reflections in Mirrors. depicts anobject and Riggs and Jones Smith animations of 3D and 4D polyhedra rotations http://www.warren-wilson.edu/~fsolomon/MAT112IntroPage

This page is your entrance to interesting animations applets for Click on one of the underlined links below to start one. Once you enter an animation, you will also have to click on the buttons to start the motion. If your screen resolution is less than 800 x 600, you may have to scroll to reach the buttons. Golden Rectangle/Triangle/Spiral Series Pointsot Stars The Koch Snowflake a fractal example of recursion The next animation allows construction of Iterated Function System type fractals. The directions are figure-outable while running the applet, but you may have to experiment some. Ask Fred Solomon for a written handout that covers in more detail the notation. Once you enter the MAIN SCREEN, you can press the box on the far right of the MAIN SCREEN to retrieve predefined fractals. Two of them are by WWC's own Ginevra Clark. Iterated Function System Fractals #1 Yet another Iterated Function System Fractals applet: Iterated Function System Fractals #2 Multiple Reflections in Mirrors depicts an object and a viewer between two mirrors. The animation shows the images that the viewer sees of the object multiply reflected in the mirrors. You can change the orientations and positions of the mirrors, viewing point, and object location. The following is a variation on the last. A flashlight shines its beam in the space between two mirrors at an angle to one another. Will the light beam be trapped forever bouncing between the two mirrors? Or will it emerge?

46. GREEN EGGS REPORT For Sci.fractals Set I; Fractal polyhedra; Fractal Terrains in Bryce 2; Fractal TutorialIntroduction Amazing Seattle fractals! Fractal eXtreme Mandel http://ibd.ar.com/ger/sci/fractals/content.html

Green Eggs Report by alice at nospam.ar.com The Green Eggs Report is a list of URLs spotted within the sci.fractals newsgroup Refrenced Links 2-piece dissection 2-piece dissection - solution A list of Resources for Fractint Andrew Que's Fractals ... www.Schloesinger.de

47. Www.fractal3D.com FAQ 3D IFS fractals are 3D analogous to 2D IFS fractals using polyhedra insteadof polygons. ie Menger sponge is 3D analogous to Sierpinski Carpet. http://www.fractal3d.com/faq/faq.shtml

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?

48. MATH LINKS Resources fractals; JAVA 3D Viewer; Mandelbrot and Julia Set Explorer; MandelbrotExplorer; Origami Mathematics Homepage; Play a Piano; polyhedra Gallery at Coolmath http://www.emtech.net/math2.htm

Updated 3/3/03 Math Links General Math Algebra Calculus Geometry ... Algebra for Everyone Homepage - Carnegie Mellon University, PACT Center Algebra Online Algebra Review An undergraduate Linear Algebra course Commutative Algebra ... Calculus Applied to the Real World - Hofstra Calculus Computer Lab - Hawaii Calculus Graphics CALCULUS I WITH MAPLE V calculus@internet Damping Functions at Coolmath.com ... Aalborg University, Denmark - Mathematics and Computer Science. ActiveMath - Workshops and Seminars for Grades 3-12 Math Teachers. American Mathematical Society AMERICAN MATHEMATICAL SOCIETY (AMS) AMS gopher AMS: Combined Membership List (AMS, MAA, SIAM, AMATYC) ... Argonne National Laboratories - Mathematics and Computer Science Division. Artificial Life, Santa Fe Australian National University, Australia - School of Mathematical Sciences. Binghamton University / SUNY at Binghamton - Mathematical Sciences Department. Bowling Green State University - Department of Mathematics and Statistics. Brigham Young University - Mathematics Department. C I R L - Public Home Page Calculator and Techology Links for Math Education Calculators Online Caltech - Applied Mathematics Department Center for Statistical and Mathematical Computing, Indiana University.

49. Web Resources: Teens: Homework - MCPL Lessons on tesselations, polyhedra, fractals, etc. Also, supercool demos. andbrain-benders. Author CoolMath.com, Inc. (38 hits) Education Resources new! http://www.mcpl.lib.mo.us/Links/Teens/Homework/

Browse/Search New Links Popular Databases Web Resources Top Teens : Homework Related Categories: Reference Teens: Books and Writing 15 Links: AOL@School http://school.aol.com/highschool/index.adp AOL has put together helpful sites in all homework subjects, plus links to career and college information. Author: AOL@SCHOOL (70 hits) Awesome Library for Teens http://www.awesomelibrary.org/student5.html A cool collection of links on all sorts of homework topics. Author: Jerry Adams (74 hits) Big Chalk Online Tutoring http://www.tutor.com/bigchalk/ Connect with online tutors, for a fee, who specialize in more than 90 subject areas, 24 hours a day. Author: Big Chalk Corporation (38 hits) CoolMath http://www.coolmath.com/home.htm CoolMath is designed for the pure enjoyment of Math. Lessons on tesselations, polyhedra, fractals, etc. Also, super-cool demos. and brain-benders. Author: CoolMath.com, Inc. (44 hits) Education Resources http://www.educationindex.com/education_resources.html Directory of links to homework subjects online. The topic-by-topic breakdown makes this site very easy to use. Author: Education Index (8 hits) High School Hub http://highschoolhub.org/hub/hub.cfm

50. Citations: A Characterization Of Ideal Polyhedra In Hyperbolic 3-space - Rivin ( I. Rivin, A Characterization of Ideal polyhedra in Hyperbolic 3Space Hyperbolic Structureson the Complements of Selected fractals - Schwartz (2001) (Correct) http://citeseer.nj.nec.com/context/91412/0

6 citations found. Retrieving documents... I. Rivin, A Characterization of Ideal Polyhedra in Hyperbolic 3-Space , Annals of Math 143 (1996) Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: Hyperbolic Structures on the Complements of Selected Fractals - Schwartz (2001) (Correct) ....of one variable, a homology argument, and some cut and paste geometric topology. At the heart of our proof are two technical results, the Tiling Realization Theorem and the Tinkertoy Theorem. See x3.2 and x3.3. These results concern patterns of circles in the punctured plane. Compare [L] and [R] for some related results on circle patterns. This paper is organized as follows: In x2 we give several alternative definitions of horodisk quotients, and recall the material from [S3] which is needed for this paper. In particular, we give a heuristic proof of the Main Result from [S3] In x3 we .... I. Rivin, A Characterization of Ideal Polyhedra in Hyperbolic 3-Space , Annals of Math 143 (1996) Consider a Compact Convex Polyhedron - Intersection Of Finitely (Correct) In , Rivin classifies all ideal polyhedra in H 3 in terms of their combinatorics and of their dihedral angles.

51. Citations: Spatial Search With Polyhedra - Jagadish (ResearchIndex) search with polyhedra. Proc. Sixth IEEE Int'l Conf. on Data Engineering, February1990. Hilbert Rtree An improved R-tree using fractals - Kamel, Faloutsos http://citeseer.nj.nec.com/context/75592/0

29 citations found. Retrieving documents... H.V. Jagadish. Spatial search with polyhedra . In Proc. IEEE Intl. Conf. Data Engineering, pages 311319, 1990. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: First 50 documents On Indexing Large Databases for Advanced Data Models - Samoladas (Correct) ....The R tree was empirically shown to improve search performance by up to 50 , compared to the basic R tree. Also , space utilization was improved. The P tree In many R tree variants, the predicates describing tree nodes are (hyper) rectangles. A notable exception is the P tree of Jagadish [ ] In the P tree, a node s predicate can be an arbitrary polyhedron. This approach can substantially improve the approximation of the contents of a node. In the experiments reported in [Jag90] it was found that 10 gons are a good choice for two dimensional data. A drawback of the approach is that .... ....the predicates describing tree nodes are (hyper) rectangles. A notable exception is the P tree of Jagadish [Jag90] In the P tree, a node s predicate can be an arbitrary polyhedron.

52. DISCUSSIONS INDEX of irregular fractals and how the scale is indeterminite in these fractals. polyhedra,Questions about dice lead to a discussion of polyhedra and geometric http://www.shodor.org/interactivate/discussions/

These discussions are designed to give teachers ideas for how to introduce or explain a concept with a student or with a class. Informal and formal definitions of concepts as well as common student misconceptions are included in the dialogue. These discussions are best accessed from the lessons in which they are used, but are listed here for quick reference. Teachers who develop discussions similar to these for other topics are encouraged to submit them to the Project Interactivate. Contact the Project Interactivate Team for details. They are arranged according to the NCTM Principles and Standards for School Mathematics and the NCEE Performance Standards for Middle School Number and Operation Concepts (NCTM Content Standard and NCEE Standard M1) Geometry and Measurement Concepts (NCTM Content Standards and NCEE Standard M2) Function and Algebra Concepts (NCTM Content Standard and NCEE Standard M3) Probability and Data Analysis Concepts (NCTM Content Standard and NCEE Standard M4) Search Interactivate Number and Operation Concepts Discussion: Description: Integers Introduces the concept of an Integer.

53. Some Common Themes In Visual Mathematical Art These include polyhedra, tessellations, impossible figures, Möbiusbands, distorted or unusual perspective systems, and fractals. http://members.tripod.com/vismath6/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

54. Journal Of Electronic Imaging, July 1995, Vol. 04(03) anisotropy, polyhedra should be close to regular in shape; and (7) polyhedra shouldbe Paper 93064 Image encryption method using a class of fractals, pp.251 http://www.spie.org/web/journals/jei/jei_jul95.html

Journal of Electronic Imaging, July 1995, Vol. 04(03) Editor: Edward R. Dougherty ISSN: 1017-9909 Abstracts for the papers in this volume are located in this file immediately following the contents list below. All papers are copublished by SPIEThe International Society for Optical Engineering (P.O. Box 10, Bellingham, Washington, 98227-0010, USA) and (7003 Kilworth Lane, Springfield, VA 22151, USA). To order books from SPIE, call 360/676-3290, fax 360/647-1445, e-mail bookorders@spie.org , or telnet to spie.org, login SPIE and choose PUBLICATIONS SEARCH on the menu. To order papers, call SPIE Document Express at 360/676-3290 (note paper title, volume and page numbers) or e-mail doc_express@spie.org. SPIE Abstracts Online is a service provided free of charge by SPIE to members of the optics community. We are interested in your comments and suggestions. Please direct these, as well as any questions regarding the service, to net-abstracts@spie.org. for personal use only. No part of this database may be downloaded, reproduced, or distributed in any form, or by any means, or stored in or transferred to any other database or retrieval system for commercial purposes or for access by multiple users without written permission from SPIE. Contents: Performing color space conversions with three-dimensional linear interpolation (Paper #: 94-027) Image encryption method using a class of fractals (Paper #: 93-064) Application-specific integrated circuit based image processor and its simulated results (Paper #: 93-059)

55. DISCUSSIONS INDEX Dimension for Irregular fractals, Looks at random fractals and how selfsimilarityis generalized. polyhedra, Questions about dice lead to discussion of polyhedra. http://www.ecu.edu/si/cd/interactivate/discussions/

These discussions are designed to give teachers ideas for how to introduce or explain a concept with a student or with a class. Informal and formal definitions of concepts are given. Common student misconceptions are pointed out as well. Teachers might have students read them or, more commonly, use ideas from them to guide class discussions. They are arranged according to the NCTM Principles and Standards for School Mathematics and the NCEE Performance Standards for Middle School Number and Operation Concepts : Includes working with fractions and decimals, clock arithmetic, and finding number patterns. (NCTM Content Standard and NCEE Standard M1) Geometry and Measurement Concepts : Includes basic notions of lines, rays and planes, working with tessellations, fractals. (NCTM Content Standards and NCEE Standard M2) Function and Algebra Concepts : Includes an introduction to functions, special properties of linear functions, graphs and the coordinate plane, and reading graphs. (NCTM Content Standard and NCEE Standard M3) Probability and Data Analysis Concepts : Includes an introduction to probability, conditional probability, sampling, expected value, statistics, histograms, boxplots, and the normal curve. Also includes statistical simulations.

56. POLYHEDRA Returning this series to an octohedron three times the edge length of thestarting octohedron. Selfrepeating, like fractals. MPEG- 141 kbytes. http://euch3i.chem.emory.edu/proposal/www.pandragon.com/polyhedra/tetraocto.html

Tetrahedra and Octohedra MPEG- 141 kbytes AVI(1.1meg) of Close Packing Tetra/Octo VRML2.0 (1.4 megabytes) Animation From Pandragon Design Continuing to added octohedron to tetrahedron and tetrahedron to octohedron produces a quasi Cubo-octohedron (missing square faces). Returning this series to an octohedron three times the edge length of the starting octohedron. Self-repeating, like Fractals. MPEG- 141 kbytes AVI(1.1meg) of Close Packing Tetra/Octo VRML2.0 (1.4 megabytes) Animation

57. Mathematics Special Interest Groups Fractal microscope; Fractint; The fractory; fractals a la INRIA; Julia and Papermodels of polyhedra; Pavilion of polyhedreality; polyhedra; Regular Polytopes http://math.haifa.ac.il/math-special.html

Special interest groups and pages in mathematics Abstract state machines ACM special interest groups relating to mathematics SIGACT special interest group on algorithms and computation theory SIGNUM special interest group on numerical mathematics SIGSAM special interest group on symbolic and algebraic manipulation Algebraic Approach to Performance Evaluation of Discrete Event Systems (ALAPEDES) ... Categories Cellular automata resources Cellular automata FAQ Cellular automaton Evolving cellular automata interest group Mirek's Cellebration ... Nada page Combinatorics Combinatorics net Design theory on the web Encyclopaedia of combinatorial structures Commutative algebra resources ... Complex analysis home page in Japan Complexity resources Complex systems virtual library Complexity on line Stas Busygin's NP-completeness page Computational fluid dynamics Computational geometry resources Computational geometry on the web Computational geometry resources Duke University computational geometry pages Minimal surfaces ... Descent and category theory connections Differential equations resources Consortium for ordinary differential equations experiments (CODE-E) TMR network differential equations in industry and commerce Discrete event simulation Dynamical systems resources Dynamical systems and technology Dynamical systems homepage Outils de simulation des systemes dynamiques Economic and game theory ... Electronic Colloquium on Computational Complexity Elliptic curve theory Elliptic curves and cryptography Elliptic curves and related topics Entropy on the World Wide Web Erich's combinatorial geometry page ... Fixed point theory on the web The Four Color Theorem

58. MATH3050.06 Projects. ,. Other topics which are now being investigated polyhedra with connections Hereis an interesting web site; fractals, with a series of web site as auxiliary http://www.math.yorku.ca/Who/Faculty/Whiteley/projects.html

Guidlines for Projects Check the information on the next page Areas and resources . One project on the geometry of Origami was presented in December. Core resources for people who want more information are: Robert Geretschlager: Euclidean Constructions and the Geometry of Origami, Mathematics Magazine 68 (1995); Michael Serra: Patty Paper Geometry, Key Curriculum Press, 1944. A web site Joseph Wu's Origami Page which has information on history, instructions, photos, etc. Another web site: http:/www.planetj.com.origami/origami.html (I have not yet gotten this to work but ... ), Other topics which are now being investigated: polyhedra: with connections to duality, generation etc. Here is a web site of tesselation pages at the Math Forum. symmetry in music (composers of modern music ... ) symmetry in architecture, design, nature .... One web resource is the pages of Goodman-Strauss central place theory (an geometric pattern in the location of towns, villages, etc. studied in geography); the geometry of string patterns and envelopes of curves;

59. Fall, 2002 Classes - Delene Perley Topics selected from voting methods, apportionment, money, paths and networks, tilingsand polyhedra, fractals, codes, game theory, environment, and math as http://faculty.millikin.edu/~dperley/fall02.htm

Fall, 2002 Classes - Delene Perley Math 125 - Math in the World Math in the World surveys and explores a broad spectrum of mathematical topics with an emphasis on observing the many practical uses of mathematics in our modern society and their connection to other disciplines. Concentrates on mathematical literacy, not manipulative techniques. Topics selected from voting methods, apportionment, money, paths and networks, tilings and polyhedra, fractals, codes, game theory, environment, and math as observed in other cultures. Textbook: Mathematics Beyond the Numbers by Gilbert and Hatcher. Prerequisite: Placement test score of 3 or better. Fulfills "Quantitative Reasoning" or "Semiotic Systems" in the MPSL. Return to: [ Delene Perley Homepage Math/Computer Science Department

60. Thomas Ruedas - Mathematics Link List Chaos/fractals fractals ; Topics in mathematics fractals; The Chaos Homepage;The Chaos Numbers and the Golden Section; The Uniform polyhedra; Symmetry, Crystals http://www.geophysik.uni-frankfurt.de/~ruedas/links-math.html

Thomas Ruedas - Mathematics links Link pages: LaTeX GNU/Linux Programming Music ... Miscellaneous Chaos/fractals "Fractals" Topics in mathematics - fractals The Chaos Homepage The Chaos Metalink ... Fractal picture link list at CNAM, Paris Essays and Tutorials SOS Math Mathematical Formula Collection (german) Analysis WebNotes by John Lindsay Orr Books on numerical methods for eigenvalue problems and iterative methods for linear systems by Yousef Saad The Matrix Manual Matrix Analysis and Applied Linear Algebra by Carl D. Meyer The Statistics Homepage Introductory Statistics: Concepts, Models, and Applications by D.W. Stockburger Hypertext Maths Dictionary Eric Weisstein's World of Mathematics Biographies of great mathematicians Biographies of Women Mathematicians ... Unsolved Mathematical Problems , a collections of nice short essays about classical mathematical problems The MacTutor History of Mathematics Archive Pages on Logic, Math, and Reasoning by Christer Blomqvist Introduction to Probability by C.M. Grinstead and J.L. Snell The Kepler Conjecture Web Resources for the History of Mathematics Alan Turing Home Page The Geometry Junkyard ... Symmetry, Crystals, and Polyhedra

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