Number 1: Pineapple

Photo Credit: Molley Shea

http://math125fall2014.blogspot.com/2014/09/fibonacci_12.html

In a pineapple, we observe the Fibonacci sequence in the numbers of “eyes” (or flowers) on the pineapple’s outer surface. The way they are naturally arranged is in sets in pairs, threes, fives, eights or thirteens (the Fibonacci sequence). A pineapple is formed by rows of “eyes” that typically end up being 5-8-13 or 8-13-21. In either case, a given row is the sum of the two “previous” rows (5+8=13, and 8+13=21). Therefore, the outer structure of pineapples exhibit a pure form of the Fibonacci sequence.

http://fcbs.org/articles/fibonacci.htm

Number 2: Stairs

Photo Credit: Ryan Farber

http://math125fall2014.blogspot.com/2014/09/blog-post_56.html#comment-form

When observing a staircase, how many ways can you climb it? For instance, if we break it down to three stairs.. How many different ways can you climb those three steps?
    One may leap two stairs at once, or take it one stair at a time. By combining those two actions, a person may step and then leap two at a time, or the reverse and leap initially and then follow it with a step. Therefore there are three ways to climb three steps: step-by-step (1+1+1), step-leap (1+2), and leap-step (2+1). However, whether we are analyzing three stairs or four stairs or five stairs, we will find that the way to climb the stairs calls for the same pattern of action: find a sum totaling n using only 1s and 2s. Since we may have any number of 1s and 2s, and the order of them in the sum matters, each solution is a composition of n with parts {1,2}.
    Looking at it in purely mathematical terms, if we are trying to solve for Sn and if n is 1 then the solution is Sn=1. If n=2, there are two ways. For n=3, there are three different ways. This sequence can be generalized in the following way for n>2: the number of possibilities for n stairs is equal to the sum of Sn-1 and Sn-2, so Sn= Sn-1 + Sn-2 (the Fibonacci sequence).


http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibpuzzles.html

http://books.google.com/books?id=Pq2AekTsF6oC&pg=PA40&lpg=PA40&dq=fibonacci+in+staircase&source=bl&ots=yA3XUWFPd9&sig=2nzMuppXzzzwHLbgrdl8h8gcrUM&hl=en&sa=X&ei=SH8pVLLVGveJsQTB2YDACw&ved=0CDwQ6AEwCQ#v=onepage&q=fibonacci 

Number 3: Bathroom Tiles

Post: Angela

http://math125fall2014.blogspot.com/2014/09/fibonacci-numbers_64.html

Commenter Credit: Jason Chaoran

Many Fibonacci numbers are present in these bathroom tiles. There is one green tile above the soap dispenser, a set of three green tiles further up, a set of five green tiles and the set of eight green tiles, and many more. One tile corresponds to the first and second fibonacci number [F(1) and F(2)], the three tiles correspond to F(4), the five tiles are F(5), and the eight tiles are F(6).