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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
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.
Every natural number is either
Examples?
Number of sticks | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Winner | X | P2 | P2 | P1 | P2 | P1 | P1 | P2 |
Note that the value is only approximately 1.618: is an irrational number, like , and its decimal representation goes on forever and never repeats:
Turns out, that if you do it over and over and over again, you'll get a rectangle that gets more and more golden.
Let's try: we'll use a piece of grid paper (looking short and squat), and a 3x1 rectangle (start it running the tall-way at the 12th row down, 8th cell over).
Now, let's figure out what the recurrence relation is for the dimensions of our rectangles, and then figure out if we're getting golden....
"A paper" is something you've no doubt encountered before: it's the long sheets we occasionly use (usually "A4" paper), which is far more common in Europe.
"A paper" is constructed so that, if folded in half (do we say the long way or the short way?), you get a sheet which has exactly the same shape -- that is, the ratio of its side lengths is the same as the original sheet.
Let's see what the dimensions of A paper must be so that this requirement is fulfilled....
Call the long dimension of the A1 paper above , and the short dimension 1.
In both cases there's "the world within the world". Both are fractals.