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In This Part: The Fibonacci Sequence
For the final activity in this session, we’ll look at an interesting application of ratios that again demonstrates the amazing patterns that emerge when we examine mathematics.
Fibonacci was the nickname of Leonardo de Pisa, an Italian mathematician. He is best known for a sequence of numbers that bears his name. The Fibonacci sequence begins with 1, 1. Each new number is then found by adding the two preceding numbers:


The Fibonacci numbers are found in art, music, and nature. You can find them in the number of spirals on a pine cone or a pineapple. The numbers of leaves or branches on many plants are Fibonacci numbers. The center of a sunflower has clockwise and counterclockwise spirals; the numbers of these spirals are consecutive Fibonacci numbers.
Problem C1
Examine a pineapple, looking for its three different sets of spirals. Use a toothpick to mark a starting place, and hold a pencil at the bottom of one spiral. Count the number of spirals of this type, moving the pencil as you count. Stop when you get back to your starting place. Now count the spirals in a different direction. See if you can find the third direction. Record the number of spirals in each of the directions. What do you notice about these numbers?
If you don’t notice anything, try again, or try it with a different pineapple. The pattern should be apparent on most pineapples. If you don’t have access to a pineapple, try this with a pine cone, or examine a nearby tree to see if you can find a Fibonacci pattern.
In This Part: Ratios of Fibonacci Numbers
Problem C2
Here again is the Fibonacci sequence:


Using a calculator, find the decimal value of the ratio of the first 10 consecutive Fibonacci numbers. What pattern do you find?


Problem C3
Make a conjecture about the ratio of the 100th to the 99th Fibonacci number.
In This Part: The Golden Mean and the Golden Rectangle
As you saw in the previous problems, as n increases, the ratio of any Fibonacci number F_{n} to the previous Fibonacci number F_{n – 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.
TAKE IT FURTHER
Problem C4
Can you use proportions to compute the value of ø from this information?
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.
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.
You can find this segment on the session video approximately 21 minutes and 10 seconds after the Annenberg Media logo.
Problem C1
On most pineapples, all three numbers will be consecutive Fibonacci numbers: 8, 13, and 21.
Problem C2
The ratios seem to be approaching one number, which is about 1.618, to three decimal places.
Problem C3
If the pattern continues, this ratio should be fairly close to the ratios found in the table in Problem C2; it should also be very close to the ratio of the other consecutive Fibonacci numbers around it.
Problem C4
Consider the ratio of sides in each golden rectangle. In Rectangle 1, the ratio is ø to 1. In Rectangle 1 + 2, the ratio is ø + 1 to ø. These are similar rectangles, so the ratios must be equal:
Crossmultiplying and simplifying the equation gives us a quadratic equation: ø^{2} – ø – 1 = 0. This equation does not factor, so we must use the quadratic formula to find the value of ø; the two possible values are
.
Since the side of a rectangle can’t be negative,
.
Evaluating this on a calculator gives us the decimal 1.618 to three decimal places, as expected.