The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the two numbers before it.
The 2 is found by adding the two numbers before it (1+1),
The 3 is found by adding the two numbers before it (1+2),
The 5 is (2+3), and so on.
If we made squares using those widths, you get a nice spiral:
Fibonacci Spiral
Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.
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