F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2) for n = 3, 4, 5, . .
.
Later, the French mathematician Edouard Lucas (1842-1891) gave the name Fibonacci numbers to this series and found many other important applications of them. .
It was also found if you take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, .), you will find the following series of numbers:.
1/1=1, 2/1=2, 3/2=15, 5/3=166., 8/5=16, 13/8=1625, 21/13=161538. .
As you can see, the ratio is beginning to become closer and closer to the previous ratio. (This is better seen in a graph.) The ratio begins to even out, and the average of the ratios is equal to 1.618034. This is termed the golden ratio, and is often represented by the Greek letter phi. .
It is believed that the Fibonacci numbers and the golden ratio both play a .
significant role in nature and architecture. It is also believed that famous artists and musicians such as, Mozart and DaVinci, followed the Golden Ratio when creating their works. We will first use the rectangle to lead us to some interesting applications in these areas. .
First, we will construct a set of rectangles using the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, and 34 which will lead us to a design found in nature. Start by drawing two, unit squares side by side. Next construct a 2-unit by 2-unit square on top of the two, unit squares. Next draw a square along the edge which borders both a unit square and the size 2 square (that is, a 3-unit square). The next square will border the 2-unit and the 3-unit squares, and each successive square will have an edge which is the sum of the two squares immediately preceding it. Continue until you have drawn a final square bordering the 13-unit and 21 unit squares. .
The construction will look like this: .
.
Now, we can draw a spiral by putting together quarter circles, one in each new square. This will form a spiral. .
.
This spiral construction closely approximates the spiral of a snail, nautilus, and other sea shells.