1. The Golden Ratio In this chapter, we’ve been discussing ratios and proportions. Remember, that a ratio is simply a comparison of two numbers. For the next two days, we are going to be studying a special ratio, called The Golden Ratio. It appears many places in nature, music, architecture and art. First, we are going to define the golden ratio and then we are going to look at two of its special applications.

2. Defining the Golden Ratio 1. You and a partner should pick up the worksheet titled “Where Are the Golden Rectangles?” and a ruler from your teacher. Before beginning step 2, you and your partner should pick one rectangle from each set that you like the best, or find the most pleasing. 2. Measure the length and width of each rectangle to the nearest millimeter. You can write the results directly on the worksheet. 3. One person should open Spreadsheet 1, while the other reads the directions on the next page. Spreadsheet 1

3. Completing Spreadsheet 1: Finding the Golden Rectangle Step 1) Fill in the length of each rectangle in Column B. Step 2) Fill in the width of each rectangle in Column C. Step 3) In cell D2, type in the formula: =B2/C2 Step 4) Click on cell D2 again. Under the Edit menu at the top of the screen, choose Copy. Your cell is now outlined. Step 5) Click on cell D3. Holding your left mouse key, drag down the remainder of column D (through row 13). The column should now be shaded black. Step 6) Under the Edit menu, choose Paste. What has happened in each of the cells? Was this expected? Step 7) Repeat steps 4-6 for the E column. Your formula for cell E2 should be: =C2/B2

4. Defining the Golden Ratio Looking at Spreadsheet 1, which rectangles in column D have the same value? What about column E? Golden Ratio Rectangles B, H, and J are called Golden Rectangles because their ratio of length to width (or width to length) approximates the Golden Ratio. (Were these the rectangles that you and your partner preferred from each set?) The Golden Ratio is an irrational number (like Pi) and is denoted by the Greek letter, Phi. When dividing the length by the width, the approximate value of the golden ratio is 1.618. When the width is divided by the length, the value is approximately 0.618. Now that we’ve defined the golden ratio, we are going to look at two of its applications. (Your partner should close Spreadsheet 1 by clicking on the “x” at the very top right corner of the screen and proceed to this slide.)

5. Application 1: Architecture The ancient Greeks believed that the Golden Rectangle was one of the most pleasing shapes to the eye. Thus, many of their buildings, such as the Parthenon (pictured above center) are composed of golden rectangles. Likewise, many modern works of architecture, such as the United Nations Headquarters, are also made up of golden rectangles. For our next project, we are going to explore the golden rectangles that make up the Parthenon in a little more detail.

6. Application 1: Architecture Pick up the worksheet titled “Find the Parthenon’s Golden Rectangles” from your teacher. Your job is to outline as many golden rectangles as you can find in the architecture of the Parthenon. To do this, you will want to measure the length and width of each rectangle and then find the ratio of length divided by width. To make this process go more quickly, you may want to type the lengths and widths into a spreadsheet and then calculate the ratio using a formula. I’ve started a spreadsheet for you. You just need to type in the appropriate formula in column C. See Spreadsheet 2.Spreadsheet 2 How many golden rectangles did you find?

7. Application 2: Art In addition to architecture, the golden ratio can be found in many places in the human body. When one measures various body parts in relation to the whole (such as the length measured from the top of the head to the floor divided by the length measured from the navel to the floor), a golden ratio is found. Mathematically, the ratio looks like this: Many artists are aware of this fact and use mathematics and the golden ratio when planning the composition of their paintings (or sculptures) to make the work seem more realistic.

8. Application 2: Art Pick up the worksheet titled “My Body Measurements” and a tape measure from the teacher. You and your partner should measure each other and fill in sections A and B for each part on the Data Recording Sheet. You will then use a spreadsheet to calculate the remaining pieces. Use the metric system when choosing your units. Please note that the units used for the entire height of the person (such as meters or centimeters) do not have to be the same as those used to measure the parts of the finger (millimeters). Just make sure that you are using a common measure for each separate part. Once you have filled out sections A and B, type in the appropriate formulas to complete the spreadsheet below. (There are two areas to complete, one for your measurements and one for your partner’s.) Spreadsheet 3 For our last project, we are going to explore the golden ratios that occur in human beings.

9. Application 2: Art Did you and your partner discover any golden ratios? In which parts were your measurements pretty close to the golden ratio? Are there any areas that were way “off the mark”? In conclusion, we have looked very briefly at the Golden Ratio and some of its applications. If you would like more information about the Golden Ratio, please feel free to visit the following website: http://mathforum.org/dr.math/faq/faq.golden.ratio.html The site provides an introduction to the topic, including its relationship to the Fibonacci sequence, as well as links to many other websites. Homework Complete the “Beautiful Faces” worksheet that was handed out in class. (Teachers: you may access this worksheet at the Illinois State Board of Education website: http://www.isbe.net/ils/math/stage_I/7A_7B_7C_8BI.pdf http://www.isbe.net/ils/math/stage_I/7A_7B_7C_8BI.pdf