Day 20, MAT115

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Announcements:
- Spring break next week, hooray! A message for you, from Oscar Brown, Jr.
- PS: don't forget everything over the break...
  To help make sure that you don't, you have another assignment due Wednesday after the break.
- Keith Devlin examines Fibonacci and Golden Ratio mania, and covers what we're doing today. Please read! It's a fun reading, I think....
Question of the Day:

Where do Fibonacci numbers appear in nature?
- Fibonacci numbers start with Bunnies (although known known to the Indians circa 700 AD knew) in about 1200 AD Italy.
  - Leonardo Pisano Fibonacci wrote a story problem about bunnies in his book the Liber Abaci
  - Fibonacci number decomposition and Fibonacci Nim. To win:
    - If the starting number is a Fibonacci number, then it was best to go second (be player 2); if not Fibonacci, try to be player 1.
    - The play: every natural number is either
      1. Fibonacci, or
      2. can be written as a sum of non-consecutive Fibonacci's in a unique way.
    Break the number of counters into this sum. If you can take the smallest Fibonacci, do so. Otherwise, you're in danger. Take one, to slow down the game.
  - So suppose you're player one, and you're facing these counters: what do you do?
    1. 64
    2. 55
    3. 39
  - How Fibonacci numbers are defined: The Fibonacci numbers are given as a recurrence relation
    
    ${F_{1}=1$
    ${F_{2}=1$
    ${F_{n}=F_{n-1}+{F_{n-2}}}$
    
    (do it again, do it again, do it again!).
    The Fibonacci numbers have a number of interesting properties. Let's investigate one, as mathematicians do, by searching for a pattern:
    By experimenting with numerous examples in search of a pattern, determine a simple formula for
    ${(F_{n+1})^2+(F_{n})^2}$
    that is, a formula for the sum of the squares of two consecutive Fibonacci numbers. Try various values of n (at least five different values) to see what happens.
  - Fibonacci Numbers and Nature:
    - dandelions (Here's a scanned dandelion)
    - Spirals in daisies:
    - Similar things happen in artichokes, pineapples, etc.
    - Let's look at some pine cones....
  - As we move toward the so-called "golden mean" and "golden rectangle", we'll start with a nice logarithmic spiral (see this site)

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