The Fibonacci Sequence
& the Golden Ratio
The Fibonacci sequence begins with 1, 1. From there on, each term is the sum
of the previous two terms. The first few terms of the sequence are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...
The Fibonacci sequence appears often in nature. The sequence is found in the
spiral patterns of pine cones and flowers. The number of spirals traced one way
will usually be a Fibonacci number. The number of spirals traced the other way
will be an adjacent Fibonacci number. Count the spirals in this pine
cone:
and in this flower:
The ratios of adjacent terms of the fibonacci sequence are also
interesting.
1 / 1 = 1
1 / 2 = .5
2 / 3 = .666667
3 / 5 = .6
5 / 8 = .625
8 / 13 = .615385
13 / 21 = .619047
21 / 34 = .617647
34 / 55 = .618182
55 / 89 = .617977
89 / 144 = .618056
144 / 233 = .618025
233 / 377 = .618037
377 / 610 = .618033
...
approaching
.61803398875....
|
1 / 1 = 1
2 / 1 = 2
3 / 2 = 1.5
5 / 3 = 1.666667
8 / 5 = 1.6
13 / 8 = 1.625
21 / 13 = 1.615385
34 / 21 = 1.619047
55 / 34 = 1.617647
89 / 55 = 1.618182
144 / 89 = 1.617977
233 / 144 = 1.618056
377 / 233 = 1.618025
610 / 377 = 1.618037
...
approaching
1.61803398875....
|
Do you notice anything strange about this sequences of ratios?
Do you notice
anything else strange about this sequences of
ratios?
|
