Graphing calculator technology makes it easy to study the golden ratio, one of the most interesting graphical and numerical concepts in algebra. The golden ratio can be found by dividing a line segment into two parts such that the length of the smaller part divided by the length of the larger part is the same as the length of the larger part divided by the length of the line segment.

Let s represent the length of the shorter part, and let the unit length 1 represent the length of the longer part. The following is a sketch of the line segment:

Algebraically, the golden ratio is

.

This equation will be solved later in the text, but for now just accept it that one of its solutions is .

Since the ratio of the shorter part to the longer part is s to 1, or  to 1, it follows that the shorter part is times the length of the longer part.

Part I

Many philosophers, artists, mathematicians, architects, musicians, and others have been intrigued by the golden ratio and have used it in their undertakings. As an example of how you might use it, suppose you want to construct a rectangle whose length and width are pleasing to the eye. Furthermore, suppose the perimeter of the rectangle is fixed at 50 centimeters. Complete the following steps to determine the length and the width of the rectangle needed:

1. Define the variable x to be the length of the rectangle - the longer side.

2. Write an expression for the width of the rectangle - the shorter side - using s as defined above and x.

3. Draw a rectangle and label its length and width in terms of x.

4. Write an expression for the perimeter of your rectangle (using the expressions for length and width
    developed in steps 1 and 2), and set the expression equal to the value given in order to obtain an equation.

5. Solve the equation you found in step 4 for the length, x.

6. Approximate the value of the length to one decimal place (by substituting the value for s).

7. Find the width using the expression from step 2. Substitute the values for s and x, and then round your
    answer to one decimal place.

8. Check your answer to see that the perimeter is in fact 50 centimeters. If it is not, explain.

Next, construct a rectangle with the dimensions you have determined. Is the rectangle's shape pleasing to the
eye? Does the perimeter check?

Part II

Here is another interesting fact: If you mark off a square in your rectangle, with a side measuring the same as the width of the rectangle, the resulting inner rectangle also has dimensions in the golden ratio. You can continue marking off squares in each inner rectangle to obtain another golden rectangle. The resulting picture should be an aesthetically pleasing modern work of art. To do this, hold your rectangle with the width, or shorter side, up. Mark off a square across the top of the rectangle. Turn the rectangle clockwise, and mark off the next square across the top of the remaining rectangle. Continue this procedure and mark off at least five squares. Do you like the pattern? The golden ratio can be tried with other geometric shapes as well.

Part III

Now let's try to generate a table of dimensions of rectangles whose lengths and widths are in the golden ratio. Complete the following table, rounding your answers to one decimal place:
 

(Hint: You can use your calculator to fill in the table. Begin by storing the expression for the golden ratio, s, under the letter S in the calculator. Then set Y1, the width, equal to SX and Y2 equal to the formula for the perimeter, in terms of S and x. Finally, use the table feature of the calculator to obtain the values needed for the table.)

Part IV

As stated earlier, the golden ratio has a long history of use because of its aesthetic properties. The Egyptians thought that the golden ratio was sacred, and it can be found in the design of their temples, pyramids, and artwork. Even some of Egyptians hieroglyphics have proportions based upon the golden ratio. Leonardo da Vinci's drawings often have overlays of rectangles with the golden ratio. The golden ratio also may be seen in many of the rectangles used by Piet Mondrian in his form of art called neoplasticism. The golden ratio can be found in many of the dimensions of the Parthenon, the famous Greek temple. The design of the United Nations building in New York City is said to have windows in the shape of the golden ratio. The music of Beethoven and Mozart are said to have pieces that divide into parts exactly according to the golden ratio. Renaissance writers called it the "divine proportion."

However, there is also controversy about the golden ratio. Is it really as pleasing as is claimed? Does some of the architecture, such as the Greek and Egyptian, conform with the golden ratio as a result of erosion?

As a final task in this project, find a reference on the golden ratio. You may go to the library to search, or use the Internet. Write a short summary of your findings. Be sure to document your reference sources.