1 / 53

Fractal Cards

Fractal Cards. Mathematics & Science Center. What is this?. What is this?. Fractals:. A fractal is a mathematical object that exhibits self similarity — it looks the same at any scale. Coined the word “fractal” in 1975. Benoit Mandelbrot

jon
Download Presentation

Fractal Cards

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Fractal Cards Mathematics & Science Center

  2. What is this? Mathematics & Science Center

  3. What is this? Mathematics & Science Center

  4. Fractals: • A fractal is a mathematical object that exhibitsself similarity — it looks the same at any scale. Mathematics & Science Center

  5. Coined the word “fractal” in 1975 Benoit Mandelbrot Professor of Mathematical SciencesYale University http://www-gap.dcs.st-and.ac.uk/~history/PictDisplay/Mandelbrot.html

  6. Fractal Sequence: • A fractal sequence contains an infinite number of copies of itself, embedded within itself. • It is a pattern within a pattern within a pattern Mathematics & Science Center

  7. First wrote about Fractal Sequences in 1995 Clark Kimberling Professor of Mathematics University of Evansville http://faculty.evansville.edu/ck6/

  8. Fractal properties: Mathematics & Science Center

  9. Fractal properties: Iteration • Fractals are often formed by what is called an iterative process • to repeat exactly the same step over and over Mathematics & Science Center

  10. Step 0 Step 1 Step 3 Step 2 ? Mathematics & Science Center

  11. Paper Dragon • Sometimes called: Jurassic Fractal Stage 0: start with a segment Let’s add a dot to our strip to insure that we all fold the same way ! Mathematics & Science Center

  12. Paper Dragon Keep the dot facing down in your left hand. Mathematics & Science Center

  13. For Each New Stage: • Bend each segment in half • Open 2 new segments to 90 • degree angle Mathematics & Science Center

  14. iteration • Paper Dragon: • Stage 0: a strip of paper • Stage 1: right angle Mathematics & Science Center

  15. Paper Dragon • Stage 2 Mathematics & Science Center

  16. Paper Dragon • Stage 3 Mathematics & Science Center

  17. Paper Dragon • Stage 4 Mathematics & Science Center

  18. Paper Dragon • Stage 4 Let’s look at more computer-generated iterations Mathematics & Science Center

  19. Paper Dragon • Stage 5 Mathematics & Science Center

  20. Paper Dragon • Stage 6 Mathematics & Science Center

  21. Paper Dragon • Stage 7 Mathematics & Science Center

  22. Paper Dragon • Stage 8 Mathematics & Science Center

  23. Paper Dragon • Stage 9 Mathematics & Science Center

  24. Paper Dragon • 13 iterations Mathematics & Science Center

  25. Fractal properties: Self Similarity • Similar: same shape • Self similarity: same shape as itself Mathematics & Science Center

  26. Fractal properties: Fractal Dimension • Based on 2 things: • Number of new pieces • Ratio between lengths of consecutive steps Mathematics & Science Center

  27. Recap: Fractal Characteristics • Iteration: • to repeat the same process over and over • Self Similarity: • Within the whole you find smaller versions of the same shape • Fractal Dimension: • Non-integer values Mathematics & Science Center

  28. By: Corey Kisner and Cari Faszewski May 18, 2000 Sierpinski Carpet Mathematics & Science Center

  29. Golden Sectionby Georg Mogk 2004 http://astronomy.swin.edu.au/~pbourke/povray/povfrac/final/ Mathematics & Science Center

  30. Mathematics & Science Center

  31. Eight Tangent Reflecting Spheres by Dave Matthews 2004 http://astronomy.swin.edu.au/~pbourke/povray/povfrac/final/ Mathematics & Science Center

  32. Frost on my windshieldby Betsey Davis 2001 Mathematics & Science Center

  33. Tie-dyeby Betsey Davis 2005 Mathematics & Science Center

  34. Seven Zebra Poles by Betsey Davis 2005 Mathematics & Science Center

  35. Seven Yellow Feathers by Betsey Davis 2005 Mathematics & Science Center

  36. What patterns do you see ? Mathematics & Science Center

  37. Cross out each number in the first line only the first time it appears • On the line above the sequence, write the resulting sequence. Mathematics & Science Center

  38. Fractal Card Mathematics & Science Center

  39. Fractal Card Mathematics & Science Center

  40. Fractal Card Measure ¼ way along the fold from each end. Cut perpendicular to the fold. Each cut should be exactly ¼ of the length of the fold. Mathematics & Science Center

  41. Fractal Card Fold the center portion forward and down, creasing it parallel to the fold. Mathematics & Science Center

  42. Fractal Card Fold the center portion forward and down, creasing it parallel to the fold. Mathematics & Science Center

  43. Fractal Card Fold the center portion forward and down, creasing it parallel to the fold. Mathematics & Science Center

  44. Fractal Card Time for second iteration. Mathematics & Science Center

  45. Fractal Card Measure ¼ way along the fold from each end. Cut perpendicular to the fold Each cut should be exactly ¼ of the length of the fold. Mathematics & Science Center

  46. Fractal Card Fold the center portion forward and down, creasing it parallel to the fold. Mathematics & Science Center

  47. Fractal Card Fold the center portion forward and down, creasing it parallel to the fold. Mathematics & Science Center

  48. Fractal Card Fold the center portion forward and down, creasing it parallel to the fold. Continue for a total of 4 iterations ( 4 sets of cuts) Mathematics & Science Center

  49. Fractal Card Open carefully and re-fold steps Mathematics & Science Center

  50. Fractal Card Refold and put glue ONLY on the flat part Mathematics & Science Center

More Related