1. The Fibonacci Sequence and The Golden Ratio By Reed Cogliano

2. Leonardo Pisano Bigollo  Lived 1170-1250  Born in Italy and traveled excessively  Arabic Numeral enthusiast  Liber Abaci  Lived 1170-1250  Born in Italy and traveled excessively  Arabic Numeral enthusiast  Liber Abaci

3. What is the Fibonacci sequence?  How do we get these numbers?  1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,9 87,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811,514229,832040,1346269,2178309,3524578,5702887, 9227465,14930352,24157817,39088169...  Add the first number to the number before it (0+1=1, 1+1=2, 2+1=3…) nowadays we use computer programs to generate large Fibonacci numbers  How do we get these numbers?  1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,9 87,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811,514229,832040,1346269,2178309,3524578,5702887, 9227465,14930352,24157817,39088169...  Add the first number to the number before it (0+1=1, 1+1=2, 2+1=3…) nowadays we use computer programs to generate large Fibonacci numbers

4. What is the Golden Ratio?  1,1,2,3,5,8,13,21,34,55…  F1 = 1, F2 = 1, F3 = 2…  To find the Golden Ratio we divide Fn/Fn+1 =   5/8 = 1.5, 8/13 = 1.625  1,1,2,3,5,8,13,21,34,55…  F1 = 1, F2 = 1, F3 = 2…  To find the Golden Ratio we divide Fn/Fn+1 =   5/8 = 1.5, 8/13 = 1.625

5. Fibonacci Sequence in Nature  Spirals let leaves have maximum sunlight

6. Golden Ratio  Angles of leaves

7. The Lucas Numbers  When a plant grows differently it tries to copy the Fibonacci sequence  Some plants like corn grow opposite each leaf  When a plant grows differently it tries to copy the Fibonacci sequence  Some plants like corn grow opposite each leaf

8. How we use it  Architecture  Painting  Crytography  Architecture  Painting  Crytography

9. The End