The Golden Ratio Activity
When we talk about a person’s “beauty,” we often mention external aspects – makeup, hairstyle, clothing, figure, and many others. However, as far back as the Greek empire, mathematicians discovered an internal beauty to the human body, a beauty that they tried to mimic in their architecture, statues, and artwork. The goal of this activity is to discover more about this intrinsic beauty in ourselves.
Your procedure:
1. You will be put into groups of four or five. Each group will need a sheet of graph paper with four grids and the calculator information tables, one tape measure, and two calculators for the group, one logged into the Navigator network and one for general calculations.
2. Measure each of the traits for each member of your group and put them into the tables on the next two pages. Measure in centimeters, to be consistent.
3. Once all your group members have been measured, wait until all the other groups are finished measuring as well.
4. One trait at a time, we will put your collected data from the tables into the calculator. Follow my instructions for entering the data in the calculator.
5. Once you are sent the class’s data for each set of traits, follow the directions on the graphing pages to graph each set of data first by hand and then with the calculator.
6. We will repeat this for each of the four data sets.
7. Answer the questions on the final page about your graphs.
Instructions to graph by hand:
1. You should currently be in the Navigator screen. Hit 2nd Quit to get back to the main screen. Then hit 4 to return to the main blank calculator screen.
2. Hit STAT, then hit ENTER on the edit button.
3. In the column labeled L1, you should see the first set of measurements. In the column labeled L2, you should see the second set of measurements. These are all the measurements for the whole class.
4. One measurement per grid, graph each of your group’s data points, as well as five points from the rest of the class’s data (any five you want). This should give you about ten points on each graph. Make sure that the scale of your graph is appropriate, as each measurement will be different!
5. By hand, draw the “line of best fit.” This is the line that best conforms to your data, which probably don’t make a perfectly straight line! Make sure that your line of best fit goes right through the center of the data, with as many points above the line as below it.
6. Calculate the slope of this line by finding two points on the line and using the slope formula. Place it in the slope table at the bottom of the graphing page. List it as a decimal rounded to three places.
Instructions to graph on the calculator:
1. Now it’s time to the let the calculator do all the graphing. Follow my verbal instructions for sending your group’s data through the Navigator. Once collected, I will send you everyone’s data, so you have measurements from the whole class.
2. Hit Stat, so you are looking at the data from the class.
3. Look at the data, and note the highest and the lowest amount in each column. Put those values in the table on the graph page.
4. Hit WINDOW. Set Xmin to the lowest L1 data point (by typing the number and hitting ENTER), Xmax to the highest L1 data point, Ymin to the lowest L2 data point, and Ymax to the highest L2 data point. (Skip Xscl, Yscl, and Xres…you can just leave those alone!)
5. Now hit Y=, hit the up arrow to go up to Plot1, and hit ENTER. This will turn on the data you just entered.
6. Hit GRAPH. You should see a scatterplot of all of your data.
7. Now we want to make the trend line. To do this, hit STAT again. Then hit the right arrow to highlight the word CALC and hit 4, then ENTER. Your screen should show y=ax+b, followed by long decimals for a and b. Copy the first three digits of each number into the table on the graph page.
8. Hit Y=. Type your decimal for a, hit X (on the button that has X,T,θ,n on it), hit +, and then type your decimal for b. Hit GRAPH.
9. You should see your scatterplot again, but now it should have a line on it. That line is the best trend line the calculator can make.
10. Additionally, the decimal for a is the slope of the line you just graphed. Copy the decimal for a into the box labeled “Slope from calculator.”
11. Repeat this for each of the four measurements, one at a time. When you finish one measurement, wait until the rest of the class has as well, and then enter the data into the calculator for the next measurement, repeating the same steps to graph by hand and by the calculator.
Group Measurements
Measurement One: Facial Proportions
Name of Group Member / Distance from eyes tobottom of nose / Distance from eyes to chin
Measurement Two: Finger Proportions
Name of Group Member / Length of first segment of index finger (tip to first knuckle) / Length of second segment of index finger (first knuckle to second knuckle)Measurement Three: Arm Proportions
Name of Group Member / Length of hand(middle finger tip to wrist) / Length of forearm
(wrist to elbow)
Measurement Four: Body Proportions
Name of Group Member / Distance from head to naval/elbows / Total height, from head to ground
Graphs and Tables Page
Measurement One: Facial Proportions Measurement Two: Finger Proportions
Measurement Three: Arm Proportions Measurement Four: Body Proportions
Meas. / Slope of Graph by Hand / Lowest/HighestL1 / Lowest/Highest
L2 / a and b Values / Slope from calculator
One
Two
Three
Four
Questions to Consider:
1. Which measurement for the line of best fit do you believe is the more accurate, the one you did by hand or the one the calculator did? Why?
2. Look at the slopes you determined for each graph. Compare your lines by hand to the ones the calculator produced. Are they similar? Do all of the decimals rest between the same two whole numbers? Which two?
3. What does the slope represent in this activity? If it appears to the be the same in every measurement, why is that?
4. Do you think this is only true about those things we measured, or do you think there are other parts of the human body like this? Can you think of some possibilities?
GeoGebra Exploration:
The number that you came up with, about 1.618, is called the Golden Ratio in mathematics. It is a special number, and mathematicians have given it the name phi, or φ. This number has some amazing properties, so let’s explore one on GeoGebra.
The shape in front of you is called a Golden Rectangle. There doesn’t appear to be anything special about this rectangle, but let’s explore its properties for a while. First, click on the “First rectangle measurement” button to measure the length and width of the rectangle. What happens when you divide its longer side by its smaller one? Do you recognize that number? Maybe that’s why we call it the Golden Rectangle…
This rectangle has some pretty fascinating properties if divided correctly. First, click on the “First Division” box, to divide the rectangle into two quadrilaterals. Using the measurement tool, measure the four sides of the quadrilateral on the left. What shape is it?
Now look at the shape on the right. Click on “Second rectangle measurement” to measure its length and width. Again, try to divide the longer side by the smaller one. What do you notice?
Wow! Amazingly, if a Golden Rectangle is divided into a square and another rectangle, you create another, smaller Golden Rectangle. Click on second division to divide this new Golden Rectangle. We get a square on the bottom and a rectangle on the top. Click “Third rectangle measurement” to measure the newest rectangle’s sides. Verify that it is, in fact, another Golden Rectangle. Pretty neat.
Click “More divisions.” Using the same pattern we began above, the rectangle is now divided into many Golden Rectangles, one smaller than the next. Now comes the cool part. Eliminate all the measurements by unclicking each of the boxes labeled rectangle measurements. Then click “Spiral.”
The spiral is made by drawing a quarter of a circle between to opposite corners of each square. Does the shape of the spiral look familiar? It should. Several animals and plants use have this spiral in their anatomy, such as the shell of a spiral conch. Click “Shell” to see a picture of a spiral shell. See how the spirals are similar in shape.
The Golden Ratio is an amazing phenomenon in mathematics, with many interesting properties. Try to find other places where the Golden Ratio is present in nature and art. You’ll be shocked to see just how often it appears!