logging in or signing up Fractals Clown Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 5106 Category: Entertainment License: All Rights Reserved Like it (5) Dislike it (0) Added: June 26, 2007 This Presentation is Public Favorites: 2 Presentation Description No description available. Comments Posting comment... By: markbunting01 (4 month(s) ago) Enlightening! I'd like to download this presentation. Saving..... Post Reply Close Saving..... Edit Comment Close By: serkani (4 month(s) ago) May I download this presentatation. Saving..... Post Reply Close Saving..... Edit Comment Close By: serkani (4 month(s) ago) May I download this presentatation. Saving..... Post Reply Close Saving..... Edit Comment Close By: serkani (4 month(s) ago) good presentation Saving..... Post Reply Close Saving..... Edit Comment Close By: fazeela (11 month(s) ago) This slide show is very helpful to me . Please allow me to down load Saving..... Post Reply Close Saving..... Edit Comment Close loading.... See all Premium member Presentation Transcript Chaos and Fractals: Chaos and Fractals by John Como Benoit B. Mandelbrot: Benoit B. Mandelbrot Coined the term fractal from the latin word fractus meaning broken and irregular. Created the now famous Mandelbrot set in the late 1950’s with IBM. What is a fractal?: What is a fractal? A fractal has a very fine structure. Too irregular to be described by regular calculus or traditional geometry. Has self-similarity. Can have a non-integer dimension. Can be described by a simple iterative formula. Can often have a natural appearance. Simple Fractals: Simple Fractals M.C. Escher: M.C. Escher Circle Limits IV can be considered a fractal due to its self-similarity. If you zoom in on the edge of the circle, the same picture is repeated to infinity. Durer’s Pentagons: Durer’s Pentagons Start with a pentagon and divide it into six like pentagons (first iteration). Divide those into six pentagons etc. Koch Curve: Koch Curve The Kock snowflake is a fractal whose area is enclosed by an infinite perimeter! Slide8: The Kock curve area increases per iteration but converges at a finite number. Slide9: The perimeter is infinite! N is the number of sides L is the length of one segment P is the perimeter How to make a fractal…: How to make a fractal… Start with a recursion formula. Example: The different Julia sets are given by How to make a fractal…: How to make a fractal… Iterate different z values for the entire complex plain. If, after a set number of iterations, the iterates do not pass a critical magnitude, the point is colored black. If the iterates do pass the critical magnitude, the point is colored corresponding to the number of iterations it went through. Slide12: Julia sets for different c values: Julia sets for different c values The Mandelbrot Set: The Mandelbrot Set Acts as a ‘dictionary’ for the Julia sets. The Mandelbrot Set: The Mandelbrot Set Same iteration formula, however it contains all the Julia sets within it by connecting them. Slide16: What are fractals used for?: What are fractals used for? Fractal pictures bridge the gap between our real world and the mathematical world. Are used in Hollywood Ex. Were used in Star Trek II: The Wrath of Khan for the genesis planet. Are used in video graphic design to create landscapes. Make beautiful art. (M’art’hmaticians) Application: Application Fractals descibe this coastline. Application: Application These fractals look like real trees. Application: Application Fractal images of leaves and mountains. From chaos comes order and beauty.: From chaos comes order and beauty. Thanks to…: Thanks to… http://www.jracademy.com http://www.pen.k12.va.us/Div/Winchester/jhhs/math/lessons/calculus/escher.html http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Mandelbrot.html http://hypertextbook.com/chaos/23.shtml Falconer, Kenneth: Techniques in Fractal Geometry http://ecademy.agnesscott.edu/~lriddle/ifs/pentagon/Durer.htm http://en.wikipedia.org/wiki/Image:Cantors_dust_in_seven_iterations.png Devaney, Robert L.; Chaos, Fractals, and Dynamics http://www.fractaldomains.com/fractal-of-the-week/week.html http://www.mcgoodwin.net/julia/juliajewels.html http://aleph0.clarku.edu/~djoyce/julia/julia.html http://www.visualbots.com/tree_project.htm Barnsley, Michael; Fractals Everywhere You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.
Fractals Clown Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 5106 Category: Entertainment License: All Rights Reserved Like it (5) Dislike it (0) Added: June 26, 2007 This Presentation is Public Favorites: 2 Presentation Description No description available. Comments Posting comment... By: markbunting01 (4 month(s) ago) Enlightening! I'd like to download this presentation. Saving..... Post Reply Close Saving..... Edit Comment Close By: serkani (4 month(s) ago) May I download this presentatation. Saving..... Post Reply Close Saving..... Edit Comment Close By: serkani (4 month(s) ago) May I download this presentatation. Saving..... Post Reply Close Saving..... Edit Comment Close By: serkani (4 month(s) ago) good presentation Saving..... Post Reply Close Saving..... Edit Comment Close By: fazeela (11 month(s) ago) This slide show is very helpful to me . Please allow me to down load Saving..... Post Reply Close Saving..... Edit Comment Close loading.... See all Premium member Presentation Transcript Chaos and Fractals: Chaos and Fractals by John Como Benoit B. Mandelbrot: Benoit B. Mandelbrot Coined the term fractal from the latin word fractus meaning broken and irregular. Created the now famous Mandelbrot set in the late 1950’s with IBM. What is a fractal?: What is a fractal? A fractal has a very fine structure. Too irregular to be described by regular calculus or traditional geometry. Has self-similarity. Can have a non-integer dimension. Can be described by a simple iterative formula. Can often have a natural appearance. Simple Fractals: Simple Fractals M.C. Escher: M.C. Escher Circle Limits IV can be considered a fractal due to its self-similarity. If you zoom in on the edge of the circle, the same picture is repeated to infinity. Durer’s Pentagons: Durer’s Pentagons Start with a pentagon and divide it into six like pentagons (first iteration). Divide those into six pentagons etc. Koch Curve: Koch Curve The Kock snowflake is a fractal whose area is enclosed by an infinite perimeter! Slide8: The Kock curve area increases per iteration but converges at a finite number. Slide9: The perimeter is infinite! N is the number of sides L is the length of one segment P is the perimeter How to make a fractal…: How to make a fractal… Start with a recursion formula. Example: The different Julia sets are given by How to make a fractal…: How to make a fractal… Iterate different z values for the entire complex plain. If, after a set number of iterations, the iterates do not pass a critical magnitude, the point is colored black. If the iterates do pass the critical magnitude, the point is colored corresponding to the number of iterations it went through. Slide12: Julia sets for different c values: Julia sets for different c values The Mandelbrot Set: The Mandelbrot Set Acts as a ‘dictionary’ for the Julia sets. The Mandelbrot Set: The Mandelbrot Set Same iteration formula, however it contains all the Julia sets within it by connecting them. Slide16: What are fractals used for?: What are fractals used for? Fractal pictures bridge the gap between our real world and the mathematical world. Are used in Hollywood Ex. Were used in Star Trek II: The Wrath of Khan for the genesis planet. Are used in video graphic design to create landscapes. Make beautiful art. (M’art’hmaticians) Application: Application Fractals descibe this coastline. Application: Application These fractals look like real trees. Application: Application Fractal images of leaves and mountains. From chaos comes order and beauty.: From chaos comes order and beauty. Thanks to…: Thanks to… http://www.jracademy.com http://www.pen.k12.va.us/Div/Winchester/jhhs/math/lessons/calculus/escher.html http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Mandelbrot.html http://hypertextbook.com/chaos/23.shtml Falconer, Kenneth: Techniques in Fractal Geometry http://ecademy.agnesscott.edu/~lriddle/ifs/pentagon/Durer.htm http://en.wikipedia.org/wiki/Image:Cantors_dust_in_seven_iterations.png Devaney, Robert L.; Chaos, Fractals, and Dynamics http://www.fractaldomains.com/fractal-of-the-week/week.html http://www.mcgoodwin.net/julia/juliajewels.html http://aleph0.clarku.edu/~djoyce/julia/julia.html http://www.visualbots.com/tree_project.htm Barnsley, Michael; Fractals Everywhere