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1. Fibonacci Sequence2. Golden ratio
Lecture 4
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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.
Fibonacci Sequence
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Plants
White Calla LilyOne Petal
Euphorbia Two Petals
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Plants
Bloodroot8 petals are not very common
Shasta daisy with 21 petals
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Pine cone A pine cone’s petals spiral in two directions. The
number of petals to go around once is always a Fibonacci number.
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Sunflower Seeds on a sunflower also show the Fibonacci
spiral.
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Pineapple The sequence is found in pineapples.
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Fibonacci Spirals
Draw the spirals on the sheets provided.
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Fibonacci Spirals
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Fibonacci Spirals
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Fibonacci SpiralsDraw a spiral that matches the shell below:
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Fibonacci Spirals
http://www.shallowsky.com/blog/science/fibonautilus.html
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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
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Golden Ratio
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Golden RatioMona 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.
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Golden RatioThe 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.
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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.htmlhttp://jwilson.coe.uga.edu/emt669/Student.Folders/Lewis.Millard/fibonacci/Fib.htmlhttp://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htmhttp://4.bp.blogspot.com/_qC54jayKgko/SPtql92PbgI/AAAAAAAACH4/pn0vY4yIrm0/s1600-h/tiling8.gifhttp://library.thinkquest.org/trio/TTQ05063/phibeauty3.htm