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Studying Mandlebrot fractals (applet with GOOD interactive info) http//forum.swarthmore.edu/alejandre/applet.mandlebrot.html. Geometer s Sketchpad. polyhedra.

Faculty

Course Schedule
...
Teaching Resources
##### Links to related topics in Informal Geometry
(Courtesy Dr. R Freese) Contents:
##### 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

82. SearchBug Directory: Shopping: Toys_and_Games: Educational
http//mathartfun.com/shopsite_sc/store/html/index.html Books, puzzles, posters and art gallery related to tessellations, polyhedra, fractals, anamorphic art
http://www.searchbug.com/directory.aspx/Shopping/Toys_and_Games/Educational/
 All Searches Auctions CompanyFinder Entertainment ... Travel Search the Web: Web Images Audio Files News Multimedia Shopping Open Directory Search: Featured Websites RobotBooks: Robot Kits, Books, Toys! - robotbooks.com Reviews of robotics books, plus robot kits, movies, magazines, and toy robots. Click here for an introduction to an interesting and fun new hobby! ALLEYCATSuperstore - ALLEYCATSuperstore.com SUPERSTORE with a Web Store, Auction Sites, Casinos, Marketing, Shopping Mall, and more. Products like bicycles, musical instruments, jewelry, camping equipment, tools, and more. Careers in Criminal Justice - careeruniversities.com Convenient Online Education. Offering Associates and Bachelors Degrees, and Certification in Criminal Justice and Law Enforcement. USA only. Special Education Software - aba.insightcommerce.net Special education software to teach children with learning and speech delays and difficulties. We sell to private buyers and schools. Educational Shopping Toys and Games Go to Directory Home Categories Board Games Consumer Information Science Teaching Stores Related Categories: Web Pages - ranked by popularity Kid Wizard.com

83. Blinded By Science
Refresh your knowledge of the basics, good logic, geometry and its formula junkyard, plane curves, polyhedra, fractals (more and more), cellular automata
http://www.dm.net/~lnh/science.html
##### Things of Science
Technically speaking, science isn't just random connections are important. The reverse is true of this page. Start by wandering the virtual American and Hooper natural history museums. Keep up with our evolving knowledge with science news , always being careful of bad astronomy and other forms of bad and junk science. For specific subjects, look out there , all around us , or inside us
##### Space
Start off your day with an astronomy picture Cosmology is the study of really large scales galactic collisions are dinky Bone up on it all with a tutorial , a guide to its evolution , and questions answered about the big bang , plus a nucleosynthesis calculator and other sites . This is the domain of general relativity , a subject often mystified and often explained , and sometimes even simulated . Don't forget the vexing questions of dark matter dark energy ... topological defects , the cosmological constant , and the value of the Hubble constant and other parameters After delving into the hows and whys of the whole universe , learn about its inhabitants galactic clus ters ... galaxies (such as our Milky Way ) as they evolve quasars and active galaxies stars (such as our sun ) as they (and it evolve binary double ... supernovae and supernova remnants nebulae planetary ... nebulae , the interstellar medium and our heliosphere cosmic rays gravity waves ... holes , and neutron stars . Catch up on the principles of astronomy or space physics ; for help, ask an

84. Science > Math > Chaos And Fractals
42 Interactive 3D fractals Shows how to create fractal mountains, 3D Mandelbrot and Julia sets, convex, stellated and polyhedra.
http://www.xasa.com/directorio/mozilla/Top/Science/Math/Chaos_and_Fractals/
 Results for Chaos and Fractals Open directory project Top Science Math Search in: Directory Web the entire directory only in Math/Chaos and Fractals Fractals as a branch of mathematics. See also: Web pages: Newton Basins Introduction to basins of attraction for Newton's method and the fractals they form. http://aleph0.clarku.edu/~djoyce/newton/newton.html Limit Sets of Kleinian Groups Graphics, software. http://klein.math.okstate.edu/kleinian/ Dynamical Systems and Technology Project An educational site on chaos and fractals. http://math.bu.edu/DYSYS/dysys.html The Spanky Fractal Database Large categorized index of software, information and links. http://spanky.triumf.ca/ Fractal Adventures 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. http://home.att.net/~Novak.S/ Fractal Explorer A fractal tutorial for beginners. Covers Mandelbrot and Julia sets, as well as 4D fractals. Also features an interactive fractal generator. http://www.geocities.com/fabioc

85. ► Chaos And Fractals [Science: Math] - WorldSearch.com
1, Interactive 3D fractals Shows how to create fractal mountains, 3D Mandelbrot and Julia sets, convex, stellated and polyhedra.
http://www.worldsearch.com/science/math/chaos_and_fractals/
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##### Chaos and Fractals
Web Directory Arts Business Computers Games ...

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The Fabric of the Cosmos: Space, Time, and the Texture of Reality

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Fast Food Nation: The Dark Side of the All-American Meal

(Book) A Short History of Nearly Everything (Book) Guns, Germs, and Steel: The Fates of Human Societies (Book) more... Marketplace Arts Automotive Books and Media Clothing and Apparel ... ZoneAlarm Security Suite
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Chaos And Fractals On eBay Chaos And Fractals auctions on eBay. Register for free to begin buying and selling today.

 86. Directory - Science Math Chaos And Fractals Interactive 3D fractals Â· cached Â· Shows how to create fractal mountains, 3D Mandelbrot and Julia sets, convex, stellated and polyhedra.http://www.incywincy.com/default?p=26913

87. Saturday Morning Math Group: Previous Talks
(Zometools are what we used during the activity to build models of polyhedra). Nicholas Ormes Debra Carney, fractals In between dimensions There are many
http://www.ma.utexas.edu/users/smmg/PreviousTalks.html
Saturday Morning Math Group
##### Here is a list of talks and relevant links given in years past.
Spring 2003 Schedule:
Organizer: Heather Lehr
Fall 2002 Schedule:
Organizer: Heather Lehr

88. Erik D. Demaine And Martin L. Demaine: Planar Drawings Of Origami Polyhedra
line planar drawing. It displays a recursive structure in the polyhedra that may lead to interesting fractals. We introduce a ``zoom
http://theory.lcs.mit.edu/~edemaine/papers/DrawingExtremeTR/
##### Paper by Erik D. Demaine
Reference
Erik D. Demaine and Martin L. Demaine , ``Planar Drawings of Origami Polyhedra,'' Technical Report CS-98-17 Department of Computer Science, University of Waterloo , August 1998.
Abstract
We present a new infinite class of polyhedra based on a class of origami bases that we have developed. To understand these polyhedra and their underlying bases, we examine drawings of their graphs. We present an elegant linear-time algorithm to find a straight-line planar drawing. It displays a recursive structure in the polyhedra that may lead to interesting fractals. We introduce a ``zoom'' feature that allows one to interactively explore the details of the graph while displaying this recursive structure.
Length
The paper is 13 pages.
Availability
The paper is available in PostScript See information on file formats
Related papers
(Planar Drawings of Origami Polyhedra)
See also other papers by Erik Demaine These pages are generated automagically from a BibTeX file.
Last updated June 10, 2004 by Erik Demaine

89. Pop Quiz: Further Information For Question #5
on equilateral triangles are the pages about crystals; fractals, especially the Sierpinski Triangle 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

90. Educational Information From The Toy Collectible Directory
Mathartfun.com Books, puzzles, posters and art gallery related to tessellations, polyhedra, fractals, anamorphic art, and MC Escher. Arizona, USA.
http://www.toybox2000.com/directory/resource-subdirectory-114.html

Home
##### Educational - Toy and Game Directory
Location: Main > Educational
1st Place Publishing
Phonics game for spelling and reading. Acorn Educational Variety of toys designed for pre-school, early years and special needs children. Art-Matters Fine art, music, dance and theater educational products and toys for children of all ages including videos, games, compact disks, instruction, supplies, software and books. Baby Einstein Company Offering videos and toys to stimulate and develop a child's brain. Bella Luna Toys Waldorf toys, art supplies, co-operative games and wooden products. Best Kids Toys Puzzle mats for the enhancement of early childhood education. Billylids Brain teasers, construction, fine motor, puzzles and wooden. Australian site delivers internationally. Blue Heron Productions Therapeutic games for therapists, counselors, educators and other mental health professionals. Books by Barbara Sher Books of games for children, that enhance their social, motor and academic skills. Games use no materials or ones easy to find around the house.

 91. Hilbert R-tree An Improved R-tree Using Fractals - Kamel Hilbert Rtree An Improved R-tree using fractals} , booktitle = Proceedings (context) - Bially - 1969 30 Spatial search with polyhedra (context) - Jagadishhttp://citeseer.ist.psu.edu/kamel94hilbert.html

 92. Citations Spatial Search With Polyhedra - Jagadish (ResearchIndex search with polyhedra. Proc. Sixth IEEE Int l Conf. on Data Engineering, February 1990. Hilbert Rtree An improved R-tree using fractals - Kamel, Faloutsoshttp://citeseer.ist.psu.edu/context/75592/0

93. AnsMe Directory - Science > Math > Chaos And Fractals
26. Interactive 3D fractals Shows how to create fractal mountains, 3D Mandelbrot and Julia sets, convex, stellated and polyhedra.
http://dir.ansme.com/science/26913.html
 Search: Web Sites Dictionary - Define Dictionary - Sounds Like Dictionary - Relations Dictionary - Rhymes Dictionary - Translate Domains - WHOIS Sponsors Web Directory Chaos and Fractals Web Directory Science Math Category Description Fractals as a branch of mathematics. Sub Directories Chaos Fractal Art Software Newsgroups relating to this Category alt.binaries.pictures.fractals Google Groups sci.fractals Google Groups Related Categories Science > Math > Applications > Complex_Systems Science > Math > Differential_Equations > Dynamical_Systems This category in other languages French Korean Spanish Web 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. http://www.3dfractals.com/ A Sketchbook on L-systems Modeling and visualization of plants using parametric L systems. http://coco.ccu.uniovi.es/malva/sketchbook/ An Introduction To Fractals Gives definition and explains the different types of formulas used. Includes illustrations and bibliography. http://www.swin.edu.au/astronomy/pbourke/fractals/fracintro/

94. Math Education Resources
A good website for instructors who would like to introduce their students to fractals. polyhedrals Welcome to my personal pavilion of practical polyhedra!
http://libweb.sonoma.edu/search/web/math.html
Web Searches

Library Collections
... University Library Search for Information Subject Guides to Web Resources Math Education Resources
##### Math Education Resources
The world wide web sites listed below were collected by Mary Anne Sobieraj and Jeff Silverman for the Math Colloquium Series at Sonoma State University. Use these sites as a starting point for your own web research.
##### General Math Sites and Subject Indexes
Mathematical Association of America - MAA organizational information as well as links to professional development workshops via the Web, their Interactive Teaching Exchange, and much more. Mathematics by Nerd World Media (tm) - A great subject index and links to topics in calculus, geometry, statistics, etc. Math Forum - wonderful resource for all aspects of math education, including specific lesson plans for K-12 through college and advanced math, as well as links to issues of interest and concern for math educators and the mathematics community in general. Includes Dr. Math, which allows students (elementary school through college level and beyond) to e-mail questions to Dr. Math and receive answers in response. The math doctor(s) are in reality math students from Swarthmore college who enjoy answering math questions. World of Mathematics - a "comprehensive, and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers."

95. Mathematics Archives - Topics In Mathematics - Geometry
Generalizations, Ford s touching circles, Isoperimetric Theorem and Inequality, fractals, Napoleon s Theorem, Ptolemy s Theorem, Regular polyhedra and Euler s
http://archives.math.utk.edu/topics/geometry.html
 Topics in Mathematics Geometry Alice Meets the 4th Dimension ADD. KEYWORDS: Four Dimensional Cube AMS's Materials Organized by Mathematical Subject Classification ADD. KEYWORDS: Electronic Journals, Preprints, Web Sites, Databases AskERIC Lesson Plans - Geometry Border Pattern Gallery ADD. KEYWORDS: Symmetry Cabri Java Project ADD. KEYWORDS: Examples, Animations, 3D geometry, Loci, Conics, Hyperbolic Geometry SOURCE: TECHNOLOGY: Java applets ADD. KEYWORDS: Cabri Geometry resources Cercle d'Euler ADD. KEYWORDS: Euler Circles, Malfatti Circles, Poncelet Triangles, Morley Triangles, Podaires Triangles, Fermat Points, Feuerbach Points, Nagel Points, Gergonne Points Combinatorial Tiling Theory: Theorems - Algorithms - Visualization ADD. KEYWORDS: Two Dimensional Tilings, Three Dimensional Tilings, Face-Transitive Tilings, Periodic Delone Tilings, Orbifold Graphs, Classification of 3-Dimensional Tilings, Tiling Space by Platonic Solids, Enumeration of Polyhedral Crystal Structures The Computation of Certain Numbers Using a Ruler and Compass ADD. KEYWORDS:

96. Kepler's Fractals
Kepler's solids can be fractalized in the same manner the Star of David is made into a Koch Snowflake Keplerian fractals. - A Koch snowflake To see his drawings, go to Keplerian fractals -
http://www.clowder.net/hop/Keplrfrct/Keplrfrct.html
 Keplerian Fractals A Koch snowflake. Start with a Star of David and place a smaller Star of David within each point. Repeat as many times as you like. In a similar fashion a five pointed star can be made into a fractal. Now to use this idea in three dimensions This is Escher's rendering of a stellated dodecahedron, the polyhedron which the next fractal is based on. Again, I do something similar but in three dimensions. Below is Kepler's stellated dodecahedron. Smaller stellated dodecahedra can be placed within the stellated dodecahedron's pyramids. An article on Keplerian solids appears in "Computers and Graphics" Vol. 19, No. 6, pp. 885-888, 1995 This Escher drawing is a wonderful depiction of two interpenetrating tetrahedra, aka a stellated octahedron, the polyhedron which the next fractal is based on. Kepler was able to stellate the octahedron by extending its faces. The stellated octahedron is also two interpenetrating tetrahedra, kind of a three dimensional version of the Star of David Just as with the Koch flake, place a smaller version of this solid within each pyramid. I was amazed when this critter started approaching the shape of a cube.

97. Fractals
Index.
http://astronomy.swin.edu.au/~pbourke/fractals/
 F r a c t a l s , C h a o s Index Geometry Formats Curves Surfaces Polyhedra Fractals, Chaos Projection Stereographics Rendering Radiance PovRay OpenGL Modelling Terrain Colour Textures Other Data Formats Analysis Fun Puzzles Old Stuff Papers POVRay Fractal Raytracing Contest Fractal Dimension Calculator - FDC Box counting software for the Macintosh and Linux. Ruler or Compass Dimension Multifractal spectrum Lacunarity Recurrence plots ... Rendering Wada-type basins of attraction Here's a party trick for over Christmas. Get 4 large shiny Christmas balls, some coloured wrapping paper, some lights from the Christmas tree, and you have you own fractal laboratory to amuse yourself with over the festive season. Self similarity Examples of self similarity in fractals with examples from mathematics and photos of the physical world. I wonder whether fractal images are not touching the very structure of our brains. Is there a clue in the infinitely regressing character of such images that illuminates our perception of art? Could it be that a fractal image is of such extraordinary richness, that it is bound to resonate with our neuronal circuits and stimulate the pleasure I infer we all feel? P. W. Atkins

98. The Math Forum - Math Library - Polyhedra
The Math Forum's Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites relating to polyhedra. Regular Convex
http://mathforum.com/library/topics/polyhedra
 Browse and Search the Library Home Math Topics Geometry Higher-Dimensional Geom. : Polyhedra Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category Geometric Formulas (Ask Dr. Math FAQ) - Math Forum Formulas for two-dimensional figures: triangle (equilateral, isosceles, right, scalene); quadrilateral (general, square, rectangle, parallelogram, rhombus, trapezoid, kite, cyclic and cyclic-inscriptable); regular polygon; circle (arc, segment, sector); ellipse; parabolic segment; and three-dimensional figures: rectangular parallelepiped (cube); prism; pyramid (square pyramid, frustum); regular polyhedron (tetrahedron, cube, octahedron, dodecahedron, icosahedron); cylinder (right circular); cone (right circular, frustum); sphere (sector, cap, segment, zone, lune); ellipsoid; circular torus; spherical polygon. Includes Defining Geometric Figures. more>> Naming Polygons and Polyhedra - Math Forum, Ask Dr. Math FAQ What do you call an 11-sided polygon? A 14-sided polygon? Is there a list of names of polygons? What about polyhedra? more>> Polyhedra Study - Ha Le An attempt to bring out the beauty of polyhedral study in particular and of mathematics in general via the use of the geom3d package in Maple and VRML (Virtual Reality Modelling Language). Categories: regular polyhedra; Archimedean solids; duality; facet; stellate; transformation.

99. Redirect
Plese look at Generation of 3D fractals for Web
http://www.people.nnov.ru/fractal/VRML/3dLsys/3Dtree.htm
 Plese look at Generation of 3D fractals for Web

100. Suzanne Alejandre - Fractal Links
Basic fractal information for the math teacher or interested user, with Internet resources to expand your fractal knowledge.
http://mathforum.com/alejandre/workshops/fractal/fractal3.html
##### Suzanne Alejandre
Magic Squares Multicultural Math Fair Polyhedra ... Tessellations
##### From the Fractal FAQ:
"A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale."
"There are many mathematical structures that are fractals; e.g. Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor. Fractals also describe many real-world objects, such as clouds, mountains, turbulence, and coastlines, that do not correspond to simple geometric shapes." This Fractal FAQ was created and edited by Ken Shirriff through September 26, 1994. The current editor is Ermel Stepp.
##### Alan Beck writes:
"Basically, a fractal is any pattern that reveals greater complexity as it is enlarged. Thus, fractals graphically portray the notion of 'worlds within worlds' which has obsessed Western culture from its tenth-century beginnings."
Beck further explains that when we look very closely at patterns that are Euclidean, the shapes look more and more like straight lines, but that when you look at a fractal up close you see more and more details.

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