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21. Products
fun! Tessellations, polyhedra, fractals, MC Escher, Tangrams dissections, Kaleidoscopes symmetry, Paper folding and more Buy
Hoagies Web
accepts no paid advertising. We appreciate your donations through PayPal Donate or Amazon Honor System, and your purchases through our affiliate links. Thanks!
##### Educational Products
Don't miss Smart Toys , perennial favorites of gifted kids, Materials for Gifted Classrooms , terrific educational materials, and Movies , featuring gifted kids (and adults) in a positive light. Enjoy!
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...

Spirolaterals were first encountered while investigating space curves and fractals. polyhedra in the Classroom http//mathforum.org/alejandre/workshops/unit14
##### Measurement
Magic Squares - Math Forum
http://mathforum.org/alejandre/magic.square.html

Late Primary - Intermediate
The Math Page - Skill In Arithmetic
http://www.themathpage.com/ARITH/arithmetic.htm

Multiplication: An Adventure in Number Sense
http://naturalmath.com/mult/index.html

This site illustrates patterns in multiplying 2 numbers up to 10 x 10. By using patterns and and 'twin facts' a 10 by 10 multiplication table is reduced to only 13 facts to be memorized.
Numbers: Facts, Figures and Fiction
http://richardphillips.org.uk/number/

Primary/Junior Numbers: Facts, Figures and Fiction Place Value http://www.sedl.org/scimath/compass/v03n02/place.html Primary The activities described in this lesson reinforce students' understanding of place value by using rhythm, physical action, and introspection. Simply Number Sense http://library.thinkquest.org/17888/opening.shtml

23. Chaos And Fractals
stellated and polyhedra. Uses tiny Java1.1 applets and VRML worlds. Fractal Geometry at Yale University http//classes.yale.edu/fractals/welcome.html Course
http://www.directory.net/Science/Math/Chaos_and_Fractals/
Chaos and Fractals Directory: Guide to Chaos and Fractals sites on the internet. Search Engines: Google Yahoo MSN FindWhat ... City Guides
##### Chaos and Fractals
Science Math Chaos and Fractals Categories Chaos
Fractal Art

Software

Websites Newton Basins http://aleph0.clarku.edu/~djoyce/newton/newton.html
Introduction to basins of attraction for Newton's method and the fractals they form.
Limit Sets of Kleinian Groups
http://klein.math.okstate.edu/kleinian/
Graphics, software.
Dynamical Systems and Technology Project
http://math.bu.edu/dysys/dysys.html
An educational site on chaos and fractals.
The Spanky Fractal Database
http://spanky.triumf.ca/
Large categorized index of software, information and links. Our Fractal Universe http://library.advanced.org/25810/ A site devoted to covering many different aspects of fractals, such as mathematics, music, applications, image compression, and advanced math. Fractal Adventures http://home.att.net/~novak.s/ The purpose of this site is to help the beginner to create fractal images. An overview of programs is provided along with a program-related fractal Gallery. Fractal Explorer http://www.geocities.com/fabioc

24. Educational
com/shopsite_sc/store/html/index.html Books, puzzles, posters and art gallery related to tessellations, polyhedra, fractals, anamorphic art, and MC Escher.
http://www.directory.net/Shopping/Toys_and_Games/Educational/
Educational Directory: Guide to Educational sites on the internet. Search Engines: Google Yahoo MSN FindWhat ... City Guides
##### Educational
Shopping Toys and Games Educational Websites Judaica for Kids http://www.judaicaforkids.com/
Judaica, Jewish oriented toys, games, puzzles and videos for kids.
Funtastic Learning
http://www.funtasticlearning.com/
Offering LeapPad, Odyssey III Globe, Music Blocks, Mr. MightyMind, Lauri puzzles, and other items to make learning fun.
Dragonfly Toys
http://www.dragonflytoys.com/
A comprehensive resource for parents and professionals, with toys, books, games, adaptive devices, alternative access computer devices, and sports equipment.
NetStarToys
http://www.netstartoys.com/
Quality pre-school and educational toys for children ages 3 and older.
FUNdaMENTALs
http://fun-da-mentals.com/
A shop full of fun science and creativity toys. From yo-yos and rockets to chemistry sets and posters.
Treasure Island Toy Store
http://www.treasuretoys.com/ Suppliers of toys, games, and activities including Brio, Playmobile, and Learning Curve. Also offers e-mail to santa, and coloring pages. PlayFair Toys http://www.playfairtoys.com/

works. polyhedra, fractals, chaos, curved surfaces, loops, interference, tilings big gallery with explanations of the works.
 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!:

26. 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

27. Fractal Polyhedra
and VRMLArts 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.
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

28. TSG 10 - Research And Development In The Teaching And Learning Of Geometry
topics? (eg tessellations, polyhedra, fractals); Would they modify the learners notions about geometry and the ways they learn it?
http://www.icme-organisers.dk/tsg10/
 TSG 10: Research and development in the teaching and learning of geometry Team Chairs Iman Osta, Lebanese American University, Address: Beirut Campus, P.O.Box 13-5053, Chouran Beirut 1102 2801, Lebanon iman.osta@lau.edu.lb Harry Silfverberg, Department of Teacher Education, University of Tampere Address: P.O.Box 607, FIN-33014 University of Tampere, Finland harry.silfverberg@uta.fi Team Members David W. Henderson, Department of Mathematics, Cornell University, USA dwh2@cornell.edu VerÃ³nica Hoyos Aguilar, CAEMTIC, National Pedagogical University, Mexico vhoyosa@upn.mx Ewa Swoboda, Institute of Mathematics, Department of Mathematics and Natural Science, Rzeszow University, Poland eswoboda@univ.rzeszow.pl Aims and Focus Call for Papers Selection of Abstracts for Expected Papers ... Call for reaction / discussant papers Aims and Focus In Topic Study Group 10 (Research and development in the teaching and learning of geometry), we will examine and discuss recent research and developments in the teaching and learning of geometry at all levels for schooling from kindergarten to the university. The Group will incorporate short presentations on, and discussions of, important new trends and developments in research or practice, providing an overview of the current state-of-the-art in geometry teaching and learning, and expositions of outstanding recent contributions to it, as seen from international perspectives.

29. Fractal Resources
3D fractals (Using Java applets, this site allows a variety of forms such as fractal mountains, mandelbrot and julia sets, complex fractal polyhedra, etc.
http://home.att.net/~Novak.S/resources.htm
 Introduction to Fractals Chaos and Fractals: A Search for Order (A brief introduction to the history and basic concepts of the topic. Excellent for beginners!) Fantastic Fractals (an outstanding comprehensive site featuring tutorials on fractal images and music, an illustrated "Just for Kids" section, interactive online fractal generators, free fractal software, newsletter, and other resources.) Our Fractal Universe: Mandelbrot and More (in English, French, or Spanish) The Fractory (An outstanding and mathematically detailed comprehensive site with excellent explanatory graphics!) Robert L. Devaney's The Dynamical Systems Technology Project At Boston University (Funded by the National Science Foundation, this site was specifically developed to introduce these topics into secondary and college level courses. Also on this site is a section on Chaos, Fractals, and Arcadia which examines the use of concepts in chaos theory in Tom Stoppard's play "Arcadia" to facilitate interdisciplinary studies. This section also provides links to similar "Arcadia" related sites.) Michael Frame's, Benoit Mandelbrot's, and Nial Neger's site for their course on

30. Mainframe
3D fractals. Using Java applets, this site allows a variety of forms (eg, fractal mountains, terrains, mandelbrot and julia sets, complex fractal polyhedra, etc
http://home.att.net/~Novak.S/main.htm
"Contact"
##### Now That You're Here
The primary purpose of this site is to exhibit my images and music compositions. In addition, I wish to provide some personal suggestions for creating fractal images and music depending on different levels of technical knowledge. The "thumbnail" Galleries contain a number of fractal images or fractal-derived images created by programs requiring different levels of technical experience. The program used is indicated below each image and "clicking" on the image will link you to an enlarged version. Some enlarged images in Gallery-1 are also accompanied by fractal music compositions. In some instances, fractal images may have undergone additional processing with an image editor (i.e., post-processing) after being generated by the indicated program. The MIDI music for this page was composed with the full MusiNum program discussed below. Media Player (available on Windows Update) is the MIDI player used for Internet Explorer. Crescendo (operating in detached mode) is the MIDI player plugin I am using for . A basic version is available from the Crescendo homepage (freeware).

31. Catalogs For Kits & Other Educational Products , Free Lewis & Clark Map! , Feltb
shopsite_sc/store/html/index.html Tessellations, polyhedra, fractals, MC Escher, Tangrams dissections, Kaleidoscopes symmetry, Paper folding and more
http://homeschool.resource.ods.org/20020730.HTM
 This list comes from a gifted students' resource, but these are not just for gifted students. Every student is gifted in some way. Check them out: Educational Products Don't miss Smart Toys (Smart Toys for Gifted Kids) http://www.hoagiesgifted.org/smart_toys.htm Lists perennial favorites of gifted kids The 24 Game http://www.math24.com/ 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 http://www.arborsci.com/ Great science supplies, with good prices Calculus By and For Young People http://www.shout.net/~mathman/ by The Mathman - Don Cohen and lots of related math materials... Carolina Supply Company http://www.carolina.com/ Science (and some math) supply company, with products in a reasonable price range... Challange Math For the Elementary and Middle School Student http://www.challengemath.com/

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://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

33. Efg's Mathematics Page
html. polyhedra Database www.netlib.org/polyhedra/index.html. html. F. fractals and Chaos Also see efg s fractals and Chaos Projects Page.
http://www.efg2.com/Lab/Library/mathematics.htm
 Mathematics in look for Mathematics Fractals and Chaos Contents A. Reference D. Functions G. Wavelets E. Fourier Analysis, FFTs ... F. Fractals and Chaos also see efg's Delphi Math Info and Links Delphi Math Functions Math Projects A. Reference Calculators The Calculator Home Page. Calculator.org has a number of calculator related resources, including an online scientific calculator, units conversion and constants database, and other information. www.calculator.org Calculators Online Center. Part II. Mathematics http://www-sci.lib.uci.edu/HSG/RefCalculators2.html Geometry Geometry Algorithms http://geometryalgorithms.com Geometry http://mathworld.wolfram.com/topics/Geometry.html GÃ©omÃ©trie http://perso.wanadoo.fr/jpq/geometrie/index.htm Famous Curves http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html 2D Curves www.2dcurves.com Geometry, Solid The 5 Platonic Solids and the 13 Archimedean Solids www.scienceu.com/geometry/facts/solids Math Reference Tables math2.org (formerly Dave's Math Reference Tables) General, Algebra, Geometry, Trig, Discrete/Linear, Other, Stat, Calc, Advanced

34. Tessellations' Ordering Page
Mathartfun.com carries a variety of books, puzzles, posters, videos and software related 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

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

35. Shiki.archive.9512: Fractals
fractals are the opposite of optimal. Intervals, straight lines, planes, smooth surfaces, smooth curves, polyhedra are not fractals.
http://haiku.cc.ehime-u.ac.jp/~shiki/shiki.archive/html/9512/0172.html
##### fractals
Wlodzimierz Holsztynski ( everest@netcom.com
Mon, 4 Dec 1995 22:16:50 -0800 (PST)
Alexey's blinding knowledge of fractals:
Before anybody would use the above text for a fractal
metaphor let me correct it, since Alexey's text is absolutely
A fractal has *nothing* to do with covering space optimally.
No fractal covers space. And no fractal surrounds area optimally.
In the plane circles and only circles do but they are not fractals.
In the euclidean space spheres and only spheres (this time with
respect to volume, which is the 3-dim analog of area) do but they are
not fractals. In that sense and in most any natural, naive sense fractals are the opposite of optimal. Intervals, straight lines, planes, smooth surfaces, smooth curves, polyhedra are not fractals. There are two notions of dimension in geometry (in the theory of so called metric spaces): topological dimension and Hausdorff dimension. For regular sets like polyhedra the two coincide with

36. Mathematical Resources: Mathematical Art, Graphics, Chaos And Fractals (Math Lin
Pavilion of Polyhedreality images and geometry links by GW Hart; Perspective Drawing, Moebius Strip, polyhedra, and Spreadsheets by Jan Chaos and fractals.
http://mathres.kevius.com/art.html
Chaos and Fractals
Newsgroups
Kazohina
##### Mathematical Art, Graphics, Chaos and Fractals
not a complete list, only what I happened to see...

37. 6. Other Polyhedra
Dodecahedron 6. Other polyhedra. In principle our algorithm should create quantum fractals for each of the regular polyhedra. The
http://www.cassiopaea.org/quantum_future/papers/qfract/node16.html
Next: 4. Notes Up: 3. The Five Platonic Previous: 5. Dodecahedron
##### 6. Other Polyhedra
In principle our algorithm should create quantum fractals for each of the regular polyhedra. The only restriction on the array of vectors is that they are all of unit length, and their sum is a zero vector. We added, for comparison with the Platonic solids configurations, two additional simple yet regular figures: double tetrahedron and icosidodecahedron. Notice that tetrahedron is self-dual, while dodecahedron and icosahedron are dual to each other. Double tetrahedron array is obtained by combining with - that is with the inverted configuration. Figure: Quantum Double Tetrahedron. Icosidodecahedron has particularly simple and elegant expression for its 30 vertices: they are of the form: and its cyclic permutations, and and its cyclic permutations, where is the golden ratio. All of its edges are of length . For its 30 vertices was needed to resolve the atrractor's fine structure. Figure: Quantum Icosidodecahedron.
Next: 4. Notes

38. Generation Of 3D Fractals For Web
Unlike the well known Lsystems fractals, this algorithm does not need a rather complicated string parser and 3D turtle for its Fractal Trees and polyhedra.
http://www.people.nnov.ru/fractal/VRML/Web3D/Web3D.htm
##### 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 KochÂ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 KochÂs snowflake by repetition of scaling, rotations and translations. Generation of a simple tree is shown below.

39. Web Resources: Teens: Homework - MCPL
Lessons on tesselations, polyhedra, fractals, etc. Also, supercool demos. and brain-benders. Author CoolMath.com, Inc. (71 hits) Create a Graph new!
##### TopTeens : Homework
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 (159 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 (138 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 (72 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. (74 hits)
Create a Graph new!
http://nces.ed.gov/nceskids/graphing/ Using this source enables students and others to quickly devise printable bar graphs, pie charts, line graphs, or area charts using the colors and values of their choosing. Straightforward instructions and a simple interface make this an easy way to illustrate and expand reports or assignments. (2 hits)
Education Resources
http://www.educationindex.com/education_resources.html

40. WebGuest Directory - Math : Chaos And Fractals
Interactive 3D fractals Shows how to create fractal mountains, 3D Mandelbrot and Julia sets, convex, stellated and polyhedra.
http://directory.webguest.com/Science/Math/Chaos_and_Fractals
##### ScienceMath : Chaos and Fractals
Related categories:
Sites:
• 3D Fractals using Bicomplex Dynamics - The Tetrabrot is the bicomplex generalization of the Mandelbrot set as realized by Dominic 'Ramdam' Rochon. Articles, pictures, news and other downloads. A Sketchbook on L-systems - Modeling and visualization of plants using parametric L systems. An Introduction To Fractals - Gives definition and explains the different types of formulas used. Includes illustrations and bibliography. Chaos Theory Resources - Fractals - Directories - Links - A Directory of Internet resources on chaos theory and fractals. Chaos Theory and Fractals - History of chaos as well as extensive information on Chaos Theory and fractals. Contains several pictures of fractals and links to other Chaos Theory and fractals pages. Chaos, Fractals And Attractors In Science - Chaos theory is a popular name for the theory of Dynamical Systems (DS), mainly for non-linear systems. The theory covers many aspects of the life of DS, all of them have the same characteristics, they were born, they lived and they died. Climate Dynamics - Explains a general systems theory for chaos, quantum mechanics and gravity as applied to weather patterns.

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