Below is a table of Fibonacci numbers up to the fifteenth term. Note how the sum of two adjacent numbers in the sequence is equal to the next number in the sequence. Also note that the Fibonacci sequence extends into negative numbers.
7
8
Take a look at the graph. The side lengths of each square is the sum of the lengths of the two previous squares. This is a geometric representation of the Fibonacci sequence. Also note how neatly each square fits with other neighboring squares.
9
Colored Squares Representing the Fibonacci Sequence.
10
The Fibonacci sequence can also be used to approximate the golden spiral. You can use quarter-circles, each quarter contained in each square, to approximate the golden spiral. Please note that the quarter circles may not be exact.
26
Quarter Circles Used to Approximate the Golden Spiral
27
Another rather queer thing about the Fibonacci sequence is that, when graphically represented, with the more terms you put in, the closer the ratio of the sides of that rectangle are to the golden ratio. In simpler terms, as the numbers in the Fibonacci sequence get larger, you can take two successive numbers in the sequence, divide the larger number by the other one, and you get a number close to the golden ratio. Below are the fifteenth and fourteenth term in the Fibonacci sequence. As you can see, the ratio of those two numbers is close to the golden ratio.
42
equals
43
44
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