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

What are the Fibonacci numbers? The Fibonacci sequence is a recursively defined sequence where, F1 = 1 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

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

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 = 1.618033... Lisa

Consecutive Fibonacci numbers approximate the Golden Ratio: 8/5 = 1.6 13/8 = 1.625 With each successive pair, the quotient approaches the actual value of φ: 4,181/2,584 = 1.618034055727... φ = 1.6180339887... Lisa

Quirks of the Fibonacci Sequence 1 1 2 3 5 8 13 21 34 55 1 1 4 9 25 64 169 441 1056 3025 2 5 13 34 1+1+4=6=2ｘ3 1+1+4+9=15=3ｘ5 1+1+4+9+25=40=5ｘ8 1+1+4+9+25+64=104=8ｘ13 Lisa

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

Non-Example The “golden spiral”

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

Charlotte

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

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

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

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

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. http://community.wvu.edu/~krsubramani/courses/fa04/at/qen/fib.pdf Benjamin, A. (Jun. 2013). The magic of Fibonacci numbers. https://www.ted.com/talks/arthur_benjamin_the_magic_of_fibonacci_number s

Picture Links Leonardo Fibonacci: http://www.profitsrun.com/reports/fibonacci-secrets/ Golden ratio: http://relativestrengthadvantage.com/ideal-proportions/diagram1.jpg Rectangle of Spirals: http://1.bp.blogspot.com/-Pnos-jam2m8/VOo3h16wWoI/AAAAAAAAAhA/O_naNZAzvEI/s1600/PicMonkey%2BCollage.jpg Golden Spiral: 1-fibonacci-sequence1.jpg Pentagon: http://images.slideplayer.com/19/5918486/slides/slide_3.jpg Rectangle: http://goldenratiomyth.weebly.com/uploads/4/0/7/7/4077600/2488173_orig.jpg?304 Parthenon: The-Parthenon-Finding-the-Golden-Section-everywhere.jpg Violin:https://4.bp.blogspot.com/-AWTKYMpWTXc/VxhIuqaWjbI/AAAAAAAAAIw/YzmEG70ZFNAtFJ1ScOp0hY8uLu7G_IbuwCKgB/s1600/Stradavarius.JPG Raphael’s Madonna: 26-primary-0-440x400.jpg Bartok Fugue: http://solomonsmusic.net/fig20a.gif Stairs: https://i.stack.imgur.com/BPv2i.png Non-Example: https://www.pinterest.com/pin/210472982562012113/ Spiral Stairs: 5145b410ce4fc59ac06e6fe139c4143c.jpg Mona Lisa: leonardo-mona-lisa-golden-mean-beauty+web.jpg

Questions?

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