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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

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Fractal Cards

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  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

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