Session 9, Part C:
Fibonacci Numbers

In This Part: The Fibonacci Sequence | Ratios of Fibonacci Numbers
The Golden Mean and the Golden Rectangle

 As you saw in the previous problems, as n increases, the ratio of any Fibonacci number Fn to the previous Fibonacci number Fn - 1 approaches one particular number, approximately 1.618. This number, called the golden mean, is referred to by the Greek letter phi (ø). To explore this concept, let's start with a square, size 1 • 1, which is the first Fibonacci number. Then put a square above it with a side equal to the next Fibonacci number (which is also 1). Then put a square next to them with a side equal to the next Fibonacci number (2): You are now approximating what is known as a golden rectangle. A golden rectangle has the property that a square constructed on its longer side will make a new configuration that is also a golden rectangle -- one that is similar to the first in that its sides have the same ratio as the original rectangle. If you continue this process, each rectangle you create will be closer to the golden rectangle, just as the ratio of consecutive Fibonacci numbers gets closer to the golden ratio. The ratio of the sides of a golden rectangle is ø, the golden mean.

 Problem C4 Can you use proportions to compute the value of ø from this information?

Set up a proportion. The two golden rectangles have the same shape (they are similar), so their sides will have the same ratios.   Close Tip

 Like the Fibonacci numbers, golden rectangles also have their place in nature. The spiral chambers of a nautilus shell can be traced into the growing squares of a golden rectangle.

 Video Segment Why do we study the golden rectangle? What applications could it have? In this segment, architect Ed Tsoi explains how the golden rectangle has been an important architectural element throughout history, from ancient Greek architecture to contemporary modern buildings. If you are using a VCR, you can find this segment on the session video approximately 21 minutes and 10 seconds after the Annenberg Media logo.

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 Session 9: Index | Notes | Solutions | Video