Day 19, MAT115

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Announcements:
- I hope that you've done your reading: Keith Devlin examines Fibonacci and Golden Ratio mania
- Each class had only one submission for Babylonian story, so you're both declared winners! Congratulations to Clara and Amber. Your each receive two get-out-of-homework free cards!
Question of the Day: Where do Fibonacci numbers appear in nature?
- Before we get to our question, however, let's recall a couple of things about Fibonacci numbers:
  - They're named after Leonardo Pisano Fibonacci, of 13th century Italy: they earned his name by being featured in a story problem about bunnies.
    - Start with a new-born pair of "immortal" bunnies (they never die).
    - A new-born pair requires a month to mature.
    - The following month, the pair produces a new pair, which is subject to the same rules.
    - Do it again, do it again, do it again!
  - As usual, they were known before Fibonacci, by known to the Indians prior to Fibonacci's time.
  - Fibonacci number decomposition and Fibonacci Nim.
    - If you looked at some special cases, what can we conclude?
      We concluded that, if the starting number was a Fibonacci number, then it was best to go second (be player 2); if not Fibonacci, try to be player 1.
    - What's the winning strategy in Fibonacci Nim?
      We concluded that it comes down to this fact:
      Every natural number is either
      1. Fibonacci, or
      2. can be written as a sum of non-consecutive Fibonacci's in a unique way.
    Two things:
    - If the number of counters starts as a non-Fibonacci (more common), try to go first. Otherwise, if Fibonacci, go second if you can. Then you're in the driver's seat.
    - If at any time you can apply this strategy, you are guaranteed to win:
      Write the number of counters as a sum of non-consecutive Fibonacci numbers, and take the smallest in the sum.
    Let's practice with a partner.
  - 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!).
  - Fibonacci Numbers and Nature:
    - Let's look at some pine cones....
    - dandelions (Here's a scanned dandelion)
    - Spirals in daisies:
    - Similar things happen in artichokes, pineapples, etc.
  - 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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