- How Fibonacci numbers are defined: The Fibonacci numbers are given as a recurrence relation
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$F_{1}=1$
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$F_{2}=1$
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$F_{n}=F_{n-1}+F_{n-2}$ (we would translate this as "the next Fibonacci number is the sum of the two preceding Fibonacci numbers.)
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(do it again, do it again, do it again!).
- Fibonacci
Numbers and Nature:
- dandelions (Here's a scanned
dandelion -- a scandelion)
- Spirals in daisies:
- Similar things happen in artichokes, pineapples, etc. I
just lost my cactus skeleton that shows Fibonacci
spirals:(, or I'd show you that.
- Let's take a look at a particularly nice pine cone.
Why does it happen? Some folks believe that it's because of the way things grow, suggesting something like this.
- Time out for a card trick!
- As we move toward the so-called "golden mean" and "golden
rectangle", we'll start with a nice Fibonacci spiral (see this site)
We build the Fibonacci spiral, by building bigger and bigger rectangles. The shapes of the rectangles were changing, in such a way that the ratio of side lengths were Fibonacci numbers. Let's look at the sequence of the ratios....
- Start with a piece of graph paper (the wide way), and darken the
square in the 10,10 spot (ten over, ten down). You might want
several colors, if you've got them -- might as well
make this pretty!
- Rule: attach the largest square you can, in either a
clockwise or counter-clockwise fashion, to make a larger
rectangle. We'll go counter-clockwise.
- Do it again, do it again, do it again! (You could do it forever....)
- What do you notice about the dimensions of the rectangle? (We can use
Excel to help us.) It's the ratio of those dimensions that
interests us most -- the ratio of dimensions (larger to
smaller) dictates the shape of the rectangle.
- The thing we are approaching is the so-called golden rectangle,
whose side-lengths are in the ratio called "the golden
ratio". But we will derive the golden rectangle in a
different way -- the way that the Greeks did it!
- One last thing, though: a current
application of the Fibonacci spiral...
- Now we'll derive what the Greeks called "the golden rectangle." When we're
done, we'll have also defined the "golden mean", or the
"golden ratio".
- Begin with a rectangle.
- Remove the largest square possible from the rectangle, to leave a rectangle.
- If the rectangle that results has the same shape as the original
rectangle, then that rectangle is golden.
What this means is that the golden rectangle preserves its shape under
this operation, of being reduced in size by having the
largest square removed.
Now if we look back at the Fibonacci spiral sequence of rectangles, we
see that they're tending toward a golden rectangle.
