TASK #5
Math II - Golden Ratio Distributions
Euclid of Alexandria (300 B.C.) defined the golden ratio in his
book, “Elements.” Since then, artists and architects who deem this
ratio as being the most aesthetically pleasing ratio have used it as a
basis for their art and buildings. The Greeks used the golden ratio
in building the Parthenon in Athens.
a
b
It is thought that Leonardo Da Vinci may have used the golden rectangle
(having sides that are in the golden ratio) when painting the face of the
Mona Lisa. The dimensions of Salvador Dali’s painting, “Sacrament of the
Last Supper,” are also equal to the Golden Ratio.
Throughout time, psychologists have tried to determine what humans consider to be beautiful. Gustav
Theodor Fechner conducted an experiment during the 1860s and found that students preferred
rectangular shapes that had the golden ratio (approximately 1.62). Since then, similar experiments have
had conflicting results.
Some psychologists think that humans who have facial feature ratios closest to the golden ratio are deemed
as the most beautiful. Other psychologists think that the people with the most average measurements in
their facial features are considered to be the most beautiful. Still others believe that people who are not
average (have higher cheek bones, thinner jaw, and larger eyes than normal) are deemed as the most
beautiful.
Through the use of statistics, and using our class as a sample, we will investigate the average
dimensions of the face and calculate their ratios.
What question are we trying to answer with our investigation?
Make the following measurements for yourself and each student in your group.
You should use centimeters to be a little more accurate for the small areas of your face.
Gender:
Student Name:
6.
1. Top of Head to Chin cm
7. Hairline to Pupil
cm
2. Top of Head to Pupil cm
8. Nose Tip to Chin
cm
3. Pupil to Nose Tip cm
9. Lips to Chin 4. Width of Nose cm
cm
cm
10. Length of Lips 2.
5.5.
4.
3.
12.
11.
5. Distance Between Outer Edge of Eyes cm
11. Nose Tip to Lips
6. Width of Head cm
12. Pupil to Lip cm
7.
1.
8.
9.
cm
Original Task developed by Georgia Department of Education 2008 © Math 2 – Unit 4
Modified by M. Winking e-mail: wiseone@mindspring.com
p.1
Ratios
1. Top of Head to Chin
A.


D.
2. Top of Head to Pupil

12.
Pupil to Lip

E.
6.
B.
Width of Head
8. Nose Tip to Chin
C.
9.
Lips to Chin


F.
8.
Nose Tip to Chin
3. Pupil to Nose Tip


4. Width of Nose

11. Nose Tip to Lips
5. Outside Eye Dist.
7. Hairline to Pupil



1.
Were any of your ratios close to being golden?
2.
Did anyone in your group have ratios close to the golden ratio?
3.
Pool the class data. You should round your ratios to the nearest hundredth. Make 6 dot plots of the
class distribution below. Find the mean and standard deviation of each dot plot. Describe the shape of
the distribution for each ratio.
B.
A.
Mean :
Standard Deviation:
Mean :
Standard Deviation:
D.
C.
Mean :
Standard Deviation:
Standard Deviation:
Mean :
Standard Deviation:
F.
E.
Mean :
Mean :
Standard Deviation:
Original Task developed by Georgia Department of Education 2008 © Math 2 – Unit 4
Modified by M. Winking e-mail: wiseone@mindspring.com
p.2
The dot plots can be created using Fathom or TI-Nspire or a frequency histogram can be created using TI83/84.
TI-83/84: (example ‘A’ Chin to Top of Head : Width of head)
1) First it will be helpful to turn on additional diagnostic information in your calculator.
CATALOG SCROLL DOWN TO DianosticOn …….…
2) Press
. (This just resets the list menus)
3) Press
4) If there is OLD data already in the lists that needs to be cleared press the up
arrow,
to clear out the
, to highlight L1 and then press
old data. Do the same for L2 if it has OLD data that needs to be cleared.
5) Enter the ratios of each member of your class in L1.
6) Go to the STAT PLOT menu by pressing
to the
and press
7) Finally press
. Move the cursor
. Move the cursor to
and then press
and press
.
.
TI-Nspire: (example ‘A’ Chin to Top of Head : Width of head)
1) Start a New Document and Select “Add Lists & Spreadsheet”
2) Title the first column “ratioa” and type in all of the data in the first column.
3) Next, press the
button and select Data►6:Quick Graph.
4) You may need to drag the x-axis a bit to see the distribution appropriately.
In either of the distributions what seems to be the ‘center’ of the data?
To clear out OLD
data, first highlight
L1 and press
CLEAR, ENTER.
Measure the faces of Shrek and Barbie.
1. Explain why the size of the pictures below should NOT change the ratios of the face measures.
2. Add each of their ratios to your dot plot using a different color or symbol.
3. Where do their ratios fall with respect to the distribution of the ratios of your class? Are any of
their measurements beyond 2 standard deviations from the mean?
4. A typical description of an OUTLIER is a data point that is more than 3 standard deviations
away from the mean. Could any be classified as and OUTLIER?
5. Are any of their ratios close to the golden ratio?
RATIOS:
MEASURES:
6.
1. Top of Head to Chin A.
#1.

#6.
B.
#2.

#12.

C.
#8.

#9.

D.
#8.

#3.

E.
#4.

#11.

F.
#5.

#7.

2. Top of Head to Pupil 3. Pupil to Nose Tip 5.
2.
7.
4. Width of Nose 5. Distance Between Outer Edge of Eyes 4.
3.
1.
6. Width of Head 12.
7. Hairline to Pupil 11.
8. Nose Tip to Chin 8.
9. Lips to Chin 
9.
10. Length of Lips 11. Nose Tip to Lips 12. Pupil to Lip 6.
MEASURES:
1. Top of Head to Chin RATIOS:
A.
#1.

#6.
B.
#2.

#12.

C.
#8.

#9.

2. Top of Head to Pupil 3. Pupil to Nose Tip 4. Width of Nose 5. Distance Between Outer Edge of Eyes 

6. Width of Head 7. Hairline to Pupil E.
#4.

#11.

F.
#5.

#7.

9. Lips to Chin 5.
7.
1.
#8.
D.

#3.
8. Nose Tip to Chin 2.
4.
3.
12.
11.
8.
9.
10. Length of Lips 11. Nose Tip to Lips 12. Pupil to Lip Original Task developed by Georgia Department of Education 2008 © Math 2 – Unit 4
p.3
Modified by M. Winking e-mail: wiseone@mindspring.com