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1. Fibonacci Sequence 2. Golden ratio

Lecture 4. 1. Fibonacci Sequence 2. Golden ratio. Fibonacci Sequence. Recap: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Finding Patterns in Nature that exhibit numbers in the sequence: your project. Plants. White Calla Lily One Petal Euphorbia Two Petals. Plants. Bloodroot

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1. Fibonacci Sequence 2. Golden ratio

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  1. Lecture 4 1. Fibonacci Sequence 2. Golden ratio

  2. Fibonacci Sequence Recap: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Finding Patterns in Nature that exhibit numbers in the sequence: your project.

  3. Plants • White Calla Lily One Petal • Euphorbia Two Petals

  4. Plants Bloodroot 8 petals are not very common Shasta daisy with 21 petals 4

  5. Pine cone A pine cone’s petals spiral in two directions. The number of petals to go around once is always a Fibonacci number. 5

  6. Sunflower Seeds on a sunflower also show the Fibonacci spiral. 6

  7. Pineapple The sequence is found in pineapples. 7

  8. Fibonacci Spirals Draw the spirals on the sheets provided. 8

  9. Fibonacci Spirals 9

  10. Fibonacci Spirals 10

  11. Fibonacci Spirals Draw a spiral that matches the shell below: 11

  12. Fibonacci Spirals http://www.shallowsky.com/blog/science/fibonautilus.html 12

  13. The Golden Ratio Your project report: discuss your findings with your group. Give a short presentation on your findings. Golden Ratio - Phi φ = 1.618033989 http://library.thinkquest.org/trio/TTQ05063/phibeauty1.htm 13

  14. Golden Ratio 14

  15. Golden Ratio Mona Lisa Mona Lisa's face is a perfect golden rectangle, according to the ratio of the width of her forehead compared to the length from the top of her head to her chin. 15

  16. Golden Ratio The Great Pyramid at Giza Half of the base, the slant height, and the height from the vertex to the center create a right triangle. When that half of the base equal to one, the slant height would equal to the value of Phi and the height would equal to the square root of Phi. 16

  17. The Golden Ratio Activity: Find the ratio of the length to the width of your credit card. What value did you get? Find other objects that exhibit the Golden Ratio or Golden Rectangle. Internet Sources: http://www.shallowsky.com/blog/science/fibonautilus.html http://jwilson.coe.uga.edu/emt669/Student.Folders/Lewis.Millard/fibonacci/Fib.html http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm http://4.bp.blogspot.com/_qC54jayKgko/SPtql92PbgI/AAAAAAAACH4/pn0vY4yIrm0/s1600-h/tiling8.gif http://library.thinkquest.org/trio/TTQ05063/phibeauty3.htm 17

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