1. Emma Stephens, Charlotte Evans, Kenneth Mcilree, Lisa Yuan
Fibonacci Numbers
Emma Stephens, Charlotte Evans, Kenneth Mcilree, Lisa Yuan
Charlotte - intro, ask if anyone knows what they are, list 4 and have them guess the rest

2. What are the Fibonacci numbers?
The Fibonacci sequence is a recursively defined sequence where,
F1 = and F2 = 1
Fn = Fn-1+ Fn-2 where n>2
{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...}
Another form is
{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,...}
¯\_(ツ)_/¯
Emma

3. History of the Sequence
Leonardo Fibonacci was an Italian mathematician. He published many books about mathematics and in one, Liber Abaci, he introduces what are later named Fibonacci numbers. He did this in a problem finding the pairs of rabbits in a population every month. The list of monthly populations ended up being the sequence of Fibonacci numbers.
Emma

4. The Golden Ratio
The golden ratio is the number is found by dividing a line into two parts, the longer part divides shorter part is as same as the sum of the two part divides the longer part. Expressing it in algebra, where a > b > 0,
a/b = a+b/a
The Greek letter φ represents the golden ratio and the value is,
φ = (1+ √5)/2 =
Lisa

5. Consecutive Fibonacci numbers approximate the Golden Ratio:
8/5 = /8 = 1.625
With each successive pair, the quotient approaches the actual value of φ:
4,181/2,584 =
φ =
Lisa

6. Quirks of the Fibonacci Sequence
1+1+4=6=2ｘ3
=15=3ｘ5
=40=5ｘ8
=104=8ｘ13
Lisa

7. Fibonacci Numbers in Nature
Male Bees
Pineapple Spirals
Pine Cones
Many types of flowers
Nautilus Shells
Snail Shells
Emma

8. Non-Example
The “golden spiral”

9. Fibonacci Numbers in Art and Architecture
“Golden rectangle” has a ratio of its length to its width of l / (w+l) , or (1/φ), which is seen as aesthetically pleasing
Charlotte

10. Charlotte

12. Musical Applications
Bartok: Music for Strings, Percussion, and Celesta
Charlotte

13. The ratio of the side of a regular pentagon to the diagonal is φ (irrational number)
Charlotte

14. Other Instances Every 4th Fibonacci number is divisible by 3
A number n is a Fibonacci number iff 5n2±4 is a perfect square
Two consecutive Fibonacci numbers are relatively prime (no common factors)
The number of waves and subdivided minor waves of a stock market
The number of ways to climb n stairs taking them one or two steps at a time
Stock market - to use it to predict price swings and market extensions
Charlotte

15. Connections to the Course
Inductive proofs, specifically strong induction
Recursively defined sequences
Pascal’s triangle
Relatively prime
Emma

16. References
Lakins, T.J. (2010). The tools of mathematical writing. Providence, RI: American Mathematical Society.
Posamentier, A.S., Lehmann, I. (2007). The (fabulous) Fibonacci numbers. Amherst, NY: Prometheus Books.
Subramani, K. Proof of Fibonacci sequence closed form. Morgantown, WV: West Virginia University.
Benjamin, A. (Jun. 2013). The magic of Fibonacci numbers. s

17. Picture Links
Leonardo Fibonacci:
Golden ratio:
Rectangle of Spirals:
Golden Spiral: 1-fibonacci-sequence1.jpg
Pentagon:
Rectangle:
Parthenon: The-Parthenon-Finding-the-Golden-Section-everywhere.jpg
Violin:
Raphael’s Madonna: 26-primary-0-440x400.jpg
Bartok Fugue:
Stairs:
Non-Example:
Spiral Stairs: 5145b410ce4fc59ac06e6fe139c4143c.jpg
Mona Lisa: leonardo-mona-lisa-golden-mean-beauty+web.jpg

18. Questions?

19. Homework Questions What are the first 12 Fibonacci numbers?
How is the Fibonacci sequence defined?
Give an example of where the Fibonacci sequence appears in the non-math world.
Ken