1. Untitled vertices, Air Dodecahedron Icosahedron-20 irregular faces, 12 vertices, FrequencyGeodisk Strucure, Water Many more irregular polyhedra fractals Circle - Unity http://www.rhfweb.com/geomerty.html

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2. Fractals Apollonian Gasket, a fractal circle packing formed by packing smaller circles into each triangular gap formed by three larger circles. Area of the Mandelbrot set. Penrose tilings, Newtoniteration convergence-domain fractals, Schlegel diagrams of four-dimensional polyhedra, and more. http://www.swin.edu.au/astronomy/pbourke/fractals

F r a c t a l s , C h a o s -> M a i n I n d e x Fractal Dimension Calculator - FDC Box counting software for the Macintosh and Linux. Ruler or Compass Dimension Recurrence plots 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 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. Duffing Attractor Random Attractors Lorenz Attractor Rossler Attractor I believe the geometric proportion served the creator as an idea when He introduced the continuous generation of similar objects from similar objects. Johannes Kepler Henon Attractor Wallpaper Gingerbread man Henon Phase Diagrams The real world was full of larger structures, smaller structures, simpler and more complex structures than the tiny portion comprising sentient creatures and their societies, and it required a profound myopia of scale and similarity to believe that everything beyond this shallow layer could be ignored.

3. 3D Fractals A java applet which generates and displays your choice of 3dimensional fractal. Compatible with 3D glasses (optional). Written by Pete Gray. It generates 3-dimensional 'fractals' by superimposing polyhedra upon themselves recursively. 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!

4. Downloadable Polyhedra, Geometry, & Math Software (Science U) Software Download polyhedra, Geometry, Math. pff9709.zip (1572k) (Windows95)Paper Folding fractals Introductory lesson to fractals for the classroom or http://www.scienceu.com/library/software/download/polymath/

Browse By: mathlb95.zip (1095k) (Windows-95) MathLab 95: Draw any function in 2-D / 3-D and L-Systems. Currently, has no symbolic computation support. Geometry non9709a.zip (1657k) (Windows-95) NonEuclid: a Software Simulation offering Straightedge and Compass Constructions in Hyperbolic Geometry (a geometry of Einstein's General Relativity Theory and Curved Hyperspace) for use in High School and Undergraduate Education. pff9709.zip (1572k) (Windows-95) Paper Folding Fractals: Introductory lesson to fractals for the classroom or personal learning. Simple paper folding steps can give rise to varities of fractals. Explore the fractal rules of simple or complex patterns. Formulas convrtdg.zip (1860k) (Windows-95) Conversion Tables: A quick and simple way to convert between metric and standard measurements. mathbook.zip (955k) (Windows-95) MathBook: A mathematics handbook with different modules, including: integral calculus, advanced algebra, geometrya and trigonometry, random numbers, US/Metric conversions, and graph builder module. Info Center Geometry Center Library Observatory ... Science Me Page last updated Fri Oct 2 15:49:29 CDT 1998 Comments to webmaster@ScienceU.com

5. The Geometry Junkyard: Fractals Shawn's mathematical gallery. Penrose tilings, Newtoniteration convergence-domainfractals, Schlegel diagrams of four-dimensional polyhedra, and more. http://www.ics.uci.edu/~eppstein/junkyard/fractal.html

Fractals Apollonian Gasket , a fractal circle packing formed by packing smaller circles into each triangular gap formed by three larger circles. From MathWorld. Area of the Mandelbrot set . One can upper bound this area by filling the area around the set by disks, or lower bound it by counting pixels; strangely, Stan Isaacs notes, these two methods do not seem to give the same answer. Balanced ternary reptiles Cantor's hourglass reptile spiral reptile stretchtiles ... trisection of India , the three Bodhi problem , and other Fractal tilings by R. W. Gosper Circle fractal based on repeated placement of two equal tangent circles within each circle of the figure. One could also get something like this by inversion, starting with three mutually tangent circles, but then the circles at each level of the recursion wouldn't all stay the same size as each other. The Curlicue Fractal , Fergus C. Murray. The Dynamic Systems and Technology Project at Boston Univ. offers several Java applets and animations of fractals and iterated function systems. Expansions geometric pattern creation techniques by John S. Stokes III.

6. Mathartfun.com - Where Math, Art & Fun Come Together! Books, puzzles, posters and art gallery related to tessellations, polyhedra, fractals, anamorphic art, and M. C. Escher. Arizona, USA. Also order tollfree. http://mathartfun.com/shopsite_sc/store/html/index.html

7. Favorite Links Of Andrea And Friedrich Lohmueller - Raytracing, Fractals, Amimat A great homepage on various things around rendering and visualization of all kinds,Geometry, Curves, fractals, polyhedra, platonic solids, polytopes, Star http://www.f-lohmueller.de/links/index_se.htm

Our Links Raytracing Art Raytracing + POV-Ray Raytracing + Science Chemistry, Biology, Astronomy Architecture POV at University POV at School ... Other Links last revised: 25-Jan-2003 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) Juno’s World All about Polyhedron - gallery and VRML - fascinating Puzzles! (jp, en) Herman Serras From golden section and polyhedra to elliptic billiard tables - fascinating geometric games! (en) David Fontaine - Preternatural - Polyhedra Pages and Povray-Galerie . A intensive description of Polyhedra, Platonic solids and Stellated Icosahedron, also a very nice gallery with a creative variety of subjects. (en)

8. Index Page With Books Window Books related to math and art. Escher, tessellations, polyhedra, fractals and more. Browse the following types of books Tessellations. polyhedra. Perspective systems. fractals. M.C.Escher http://www.mathartfun.com/shopsite_sc/store/html/Bookindex.html

9. Amby's Education Site -- Math Puzzles & Learning Activities performance tasks) Geometry Collage, Logic Advertisement, Tessellations, ScaleDrawings, Building polyhedra, and Line Designs; Paper entitled fractals. 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

10. Polyhedra, Platonic Solids, Polytopes polyhedra, Platonic solids, Polytopes M a i n I n d e x The contents of this web site are © Copyright Paul Bourke or a third party contributer where indicated. You may view this site and its contents using your web browser. New Waterman polyhedra. The Buddhabrot. Self Similarity in fractals. Danzer aperiodic tiling http://www.astronomy.swin.edu.au/~pbourke/polyhedra

Polyhedra, Platonic solids, Polytopes -> M a i n I n d e x Waterman Polyhedra [Steve Waterman] Star Sphere and Polar Sphere Combined ) [Dan Thomasson] Rhombic Triacontahedron Coordinates for regular polyhedra Parallelohedron Polyhedra that fill space (without holes) using only translation Platonic solids and their measured properties To see a World in a Grain of Sand, And Heaven in a Wild Flower Hold Infinity in the palm of your hand, And Eternity in an hour William Blake Photographic contribution [Gayla Chandler] Build your own Polyhedra Use these templates (postscript and gif files) to build polyhedra out of cardboard. 3D Polyhedra in Super3D geometry format A large collection of polyhedra in a format that's easy to convert into your favourite format. Regular polytope (platonic solids) in 4D Geometric descriptions for all 6 plaonic solids in 4 dimensional space. 4D Objects in VEF format Index Geometry formats Data Formats Fractals, Chaos Rendering Radiance PovRay OpenGL Modelling Terrain Geometry Projections Curves Surfaces Polyhedra Stereographics Colour Textures Papers Analysis Skytower Albert Park Fun Puzzles Other for New: Fractal dimension calculator Chladni plate simulation Multiwall and offaxis projection Waterman Polyhedra Local Date/Time: Friday, 11-Apr-2003 10:13:56 EST

11. Mathematical Resources: Mathematical Art, Graphics, Chaos And Fractals (Math Lin Mathematical Art, Graphics, Chaos and fractals. Archive by Richard Hawkin; Directoryof /pub/pictures/art/MC.Escher; Encyclopedia of polyhedra Over 1000 http://www.abc.se/~m9847/matre/art.html

Mathematical Resources Mathematics Links by Bruno Kevius This list is continually under development Chaos and Fractals Newsgroups 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 ARS VIDA - Psychogenic Fugue In C Minor by Andrew Ho A study in the natural selection of ideas by Karen H. Parshall 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 Database for complex systems (Australian National University) Digital Archive by Richard Hawkin Directory of /pub/pictures/art/M.C.Escher Encyclopedia of Polyhedra - Over 1000 Virtual Polyhedra - 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 A Gallery of Interactive On-Line Geometry (Geometry Center) Gallery of Art from PFP'94 IDC students Gallery of Mathematical Surfaces - T. Nordstrand

12. MTH155 - Distance Learning - Topics In Mathematics (B Estes) Information Page 1) Using mathematical methods to solve riddles 2) The idea of number 3)Objects in space polyhedra, fractals and fractional dimensions. http://webtide.hccdl.org/public/MTH155BE/

Mathematical Ideas Welcome to Mathematics 155: mathematical ideas Mathematical ideas will introduce ways of thinking about things and ideas that may be new to you. The methods will NOT work for every type of problem in our lives, but ... in certain situatuions ... mathematical modes of thought prove very effective. The methods include reasoning and imagining. Imagination is used to ask questions, to understand questions, and to conceive of possible answers. Reasoning can then be used to test what we have imagined. (The first few weeks' assignments are given at the bottom of this page.) A list of the topics we will study: Some Platonic solids doing their homework for MTH 155 1) Using mathematical methods to solve riddles 2) The idea of number 3) Objects in space: polyhedra, fractals and fractional dimensions. 4) Why and how things happen (chaos and probability). Topics in Mathematics 1) Instructor: Bart Estes 2) Topics in Mathematics MTH155 3) general course description (from college catalog): An exploratory course in mathematics for the Liberal Arts student. Topics may include mathematical logic, algebra of sets, Boolean algebra, permutations and combinations, probability, statistics, transfinite and finite numbers, bases other than 10, group theory, history of mathematics, puzzles and paradoxes, the mathematics of finance, or others chosen by the instructor.

13. 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.people.nnov.ru/fractal/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. 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

14. Math Sites games, and activities for teachers and students. Topics include polyhedra, fractals, probabilty and statistics, etc. http://jeffcoweb.jeffco.k12.co.us/isu/math/links.html

Your JeffcoNet Location is: Home Instructional Services Math Online Classroom Activities ... Teacher References Updated 10/7/02 Online Classroom Activities Description Grade Level Go Math.com Online Help Students of algebra, geometry, physics, chemistry, and those studying for the SAT will find this site very helpful. K -12 Online Metric Converter What do you need to convert? Weight, measurements, temperature, speed, area, length... This site offers a handy little online conversion tool for these and other units, with instant conversions available. K - 12 Fun Mathematics Lessons This site by Cynthia Lanius and Rice University offers interesting on-line mathematics problems and activities. It is well worth looking into it for children of all ages. K - 12 Michigan's Teacher's Network Access a world of math sites by topic. An extensive list, you'll find resources you never dreamed existed. K - 12 AOL@school Visit the School Wide Web. Activities for students and teachers. Includes homework helper sites and other resources. K - 12 The World of Trotter Math This site offers lessons, games and classroom activities for elementary and middle school math topics.

15. Generation Of 3D Fractals For Web Unlike the well known Lsystems fractals, this algorithm does not need a rathercomplicated string parser and 3D turtle for its Fractal Trees and polyhedra. 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.

16. Books About Polyhedra (Science U) polyhedra. 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.

17. Fractal Links Fractal Music Software from Fractovia. Fractal Music Software; fractals; MUSIC FROM TheFractal Music Project; Fractal polyhedra; Fractal polyhedra without any VRML http://www.apu.ac.uk/music/rhoadley/links/links_fractals.html

links Fractal Links Fractal Bibliography (text) Fractal Music URLs (text) Fractal URLs (text) More Fractal URLs (text) ... www.fractaldomains.com other MAC programs: hypertextbook.com/chaos spanky.triumf.ca/pub/fractals iq.tvsoft.com www.swin.edu.au/astronomy/pbourke For an exhaustive list of fractal generators, galleries featuring fractals, and other fractal related information, go to the following: Paul.N.Lee/Fractal_Software Paul.N.Lee/Fractal_Links Paul.N.Lee/Fractal_Census Some additional information relating to "Fractal Art": F-art-faq and more... www.ultrafractal.com Or you could get Kai's Power Tools (version 6)as a plug-in for Photoshop if you use it.If you must have freeware then I suggest Fractal Zplot from: terry Other useful/interesting fractal links: www.fractalus.com www.crosswinds.net library.advanced.org Paul.N.Lee ... Juan Luis: Third.Apex.to.Fractovia Fractal Music Software from Fractovia Fractal Music Software Fractals MUSIC FROM CHAOS index SSEYO homesite Koan generative music software sequencer ... The Fractal Music Project mersenne numbers blank Richard Hoadley Make a comment or report a fault Site questionnaire

18. IUSB Math Club and Bipyramids (Dipyramids) Dual polyhedra Constructing polyhedra MiscellaneousGeometry and Palindromic Numbers Game Theory Codes fractals, Image Compression http://www.iusb.edu/~mathclub/UsefulLinks.html

Useful Links General Search Engine General Math Sites Geometry and Topology Tilings (Tessellations) ... Other Math Related Links GENERAL SEARCH ENGINE http://www.go2net.com/search.html The best general search engine around GENERAL MATH SITES http://forum.swarthmore.edu/~steve/ A Mathematics Search Engine http://www.mathpuzzle.com/ Math Puzzles http://math.rice.edu/~lanius/Lessons/ Nifty math lessons http://www.cs.uidaho.edu/~casey931/mega-math/menu.html Mega Math http://www.ics.uci.edu/~eppstein/junkyard/ The Geometry Junkyard http://csr.uvic.ca/~mmania/ Great unsolved problems and lots more! http://archives.math.utk.edu/k12.html#other Listing of lots of other math resources GEOMETRY AND TOPOLOGY http://www.ushistory.org/betsy/flagstar.html Betsy Ross' method for cutting a 5-pointed star in one snip http://www.cs.uidaho.edu/~casey931/mega-math/workbk/map/mpactiv.html Map Coloring http://www.cs.uidaho.edu/~casey931/mega-math/workbk/knot/knot.html Math with Knots TILINGS (TESSELLATIONS) http://www.mathacademy.com/platonic_realms/minitext/escher.html The Mathematical Art of M.C. Escher http://www.cs.uidaho.edu/~casey931/puzzle/penrose/penrose.html

19. Hop's Gallery Fantasy and math influenced art, and related links.Category Arts Visual Arts Personal Pages D...... asteroids Spiral Tessellations Spiral Tessellations and a Helicoid - Kepler fractalsSome Koch like fractals based on Kepler's polyhedra - Perspective Study http://www.clowder.net/hop/

Hop's Gallery Changoros Nellie Jo Tunnel Virgin of Guadalupe ... Crab Dandelin Spheres A study of conic sections Asteroid Towns Colonizing the asteroids Spiral Tessellations Spiral Tessellations and a Helicoid Kepler Fractals Some Koch like fractals based on Kepler's polyhedra Perspective Study Including some of M. C. Escher's observations Curl Ups An essay on Escher's invented animal Frank Zubek's Construction Blocks Frank's Elusive Cube Parabolas Eccentricity of Conics Circles? Drawings of Numbers Coefficients of (a+b)^n ln(2) pi/4 (pi^2)/12 Escher Sphere Spirals Drawings of Equations Angles of a triangle total 180 degrees Pythagorean Theorem Paper Blackhole Models http://clowder.net/hop/etc./wormhol2.html Quadpods Tetrahedral Spaceships An interesting curve Tiles The Table Top Tessellations 17 Wallpaper Groups Twistohedra A family of polyhedra starting with the icosohedron Me (taken by William Chow) My family and the Chow family Outside Links e-mail hopspage@tabletoptelephone.com Hop David P. O. Box 39, Ajo, Arizona 85321

20. Lifesmith Classic Fractals Gallery 1988-2002 Lissajou Figures(100) 3D Spherical Harmonics (215) Knots(100) polyhedra ConicSections (115) Stellated polyhedra(45) Starface polyhedra(45) Affine http://www.lifesmith.com/gallery.html

New for 2002! The Year 2003 "Y" Line 156 Images The Year 2002 ... 270 Images This feature is under reconstruction...please bear with us... Displayed to the left is a random image from the 2002 line of the largest fractal image library in the world. Magnification Series 2 Up to 10^46 Magnification! 24 images added to the 2000 X line! 22 images added to 1996 J line! 12 images added to 1988-90 A line! Selected Fractal Imagery by Series/Years (1988-2002): 2003(Y) Line 156 Images 2002(P) Line 237 Images 2001 (G) Line 270 Images 2000 (X) Line 175 Images ... Two-dimensional imagery 1988-1996 (B) 320 Images Thumbnails are 72 pixels by 72 pixels where Series images are 216 pixels by 216 pixels and up. Earliest two-dimensional fractal images in the gallery were generated originally at a resolution of 900 by 900, then 1800 pixels by 1800 pixels and are stored in our six+ terabyte archive. Recent 2D images are done at 2880 pixels square (and up) whereas 3D scenes have evolved (as my tools improved) from a mere 640 by 960 to now 6000 by 9000 pixels, nearly a ninety-fold increase in overall resolution Thumbnails are 24-bit JPEG; larger images are 8-bit GIF.

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