The Golden Ratio
The Physics of Everyday Things
Students will collect, graph, and interpret experimental data.
Comprehension Strategies Emphasized
Activating and Connecting Prior Knowledge
 This comprehension strategy involves you connecting your learning to your experiences,
events in the world, and to other learnings you may have in and out of school.
 You may say statements like…
 I used my background knowledge to…
 I learned…
 I already know…
Asking Questions
 This strategy involves you actively wondering about topics, and questioning facts and
information.
 You may say statements like…
 I wonder…
 Why…
 My big question is…
Making Inferences
 This comprehension strategy involves you predicting, hypothesizing, interpreting, and
making conclusions.
 You may say statements like…
 I’m guessing that…
 I predict…
Before you explore…
Have you ever wondered what made Bach’s
compositions so appealing or why some works of
art or architecture take your breath away with their
beauty? For thousands of years artists and
architects have been using “the golden ratio” to
help construct beautiful works of art.
Mathematically, the golden ratio has been determined to be
~1.618. This means that, in music, one note may have a
frequency of 1.618x that of the previous note. In architecture, it
may mean that the width of a building may be 1.618x its height.
In fact, this ratio is also prevalent in nature, like the seashell that
you see to the left. You can see that it resembles the spiral
created by ‘The Golden Rectangle’ shown on the right. Inside of
the big rectangle is another rectangle that has an exact ratio of 1:1.618. Inside of that
rectangle that is also a perfect ratio of 1:1.618. More rectangles can be added on forever.
Notice that if you draw a curved line from the bottom left vertex of the outer rectangle to
the top right vertex of the outer rectangle and then to the bottom right vertex of the first
inner rectangle and so on, you will get a spiral that continues to approach the center, but in
theory will never touch itself.
The Golden Ratio is represented by the Greek letter phi (φ). Some say that this “golden
ratio” was used in constructing pyramids in both Egypt and Mexico and it is also found in
ancient Greek art. Most importantly, this ratio is also found in the human body. This lab
will have you measuring and comparing in a ratio certain parts of your body and finding out
where the golden ratio exists.
In your lab group, activate your prior knowledge by completing a ‘Place Mat’
Activity to summarize what you know about ratios, how they are written,
what they represent, and where you might see them in the world.
Purpose
The purpose of this lab is examine nature by exploring the mathematical
relationships that exist in the human body.
Predictions
Many people believe that the length of your foot is equal to the distance
between your elbow and your wrist.
Q:
Ask the question…are there other ratios that exist between parts of
the human body? What are they?
Accessed from: _______________
While you explore…
Materials



Meter stick
Sticky notes
2 or 3 different colored pens/pencils
Procedure
PART A: Collecting the data
1.
2.
3.
4.
5.
Work in groups of 2 or 3. Have one partner stand against the wall.
Place the first sticky note on the wall at the height of the student’s head.
Draw a horizontal line indicating the student’s height (beware of parallax issues!).
Measure the height of the student. Record the height in the table below.
Stand under the sticky note height marking so that your arm is directly below the
marking. Place another sticky note and mark fingertip position. Record the
measurement in the table below.
6. Repeat step #5 to measure the distance between top of the head and navel, and the
top the head to the armpit.
7. Measure the distance from the inside of your elbow to your wrist as well as the
length of your hand from fingertip to wrist.
8. Repeat with your other partner(s) and reuse the sticky notes, this time, using a
different colored pen or pencil.
Total height
(cm)
Student
Distance
from head to
fingertips
(cm)
Distance
from head to
navel (cm)
Distance
from head to
armpit (cm)
Distance
from floor to
navel (cm)
Distance
from inside
of elbow to
wrist (cm)
Length of
hand (cm)
1
2
3
PART B: Finding the golden number.
9. Find the following ratios with the data that you collected (please assume the smaller
measurement has a value of 1)
Ratio Calculated
total height : distance
from top of head to
fingertips
distance from top of head
to fingertips : distance
from top of head to navel
total height : distance
from the floor to your
navel
distance from top of head
to navel : distance from
head to armpit
distance from top of head
to armpit : distance from
elbow to wrist
#1
#2
#3
Class Avg
Q:
How do the ratios that you calculated compare to those with the rest
of your group?
Q:
Find the class average for each of the ratios calculated. Enter these numbers
into the table above.
Q:
Did you find a ratio in your body that is similar to the golden ratio? Use the
value of 1.6180339887 to calculate the percentage error for the ratio(s) you found.
Q:
Make a prediction…Invite someone to participate who is not in your
class. Take one body measurement identified above. Using the golden ratio,
calculate what their other body measurement should be. Measure the actual length and
compare your calculated with your measured values.
After you explore…
Conclusion
Write a conclusion that includes at least the following:
 terms: ratio, golden ratio
 a summary of what you discovered in this lab
 a comparison of the results of this lab with your predictions
 a comparison of your results and the actual value of φ
C:
Extension Activities
1. If you were an artist painting on a canvas and you wanted it to be as aesthetically
pleasing as possible, what would the dimensions be if you wanted the short side to have a
length of 3m? What if you wanted the long side to have a length of 3m?
2. Research different examples of the Golden Ratio in:
a) nature
b) art
c) architecture
d) music
3. Compare pictures of famous scientists with current beauty pageant winners. Which
faces follow the golden ratio?
4. Graph the class values for one of the ratios studied, for example height vs distance from
floor to navel. Does this create a linear relationship? Calculate the correlation coefficient
to determine this.