Name: ____________________________________________ Per: _____ Date: ____________
Ms. Grenz – Geometry
Golden Ratio Project
1) Research the Golden Ratio – accurately fill in information sheet.
2) Research Fibonacci and Fibonacci’s Sequence – accurately fill in information sheet.
3) Derive the Golden Ratio – accurately complete the worksheet.
4) Find the Golden Ratio in a pentagram – accurately complete worksheet.
5) Find the Golden Ratio in the real world – accurately complete the worksheet.
6) Find the Golden Ratio in the human body – accurately complete the worksheet.
7) Construct a Golden Rectangle with the Golden Spiral inside it – accurately follow steps.
8) Present your construction on a poster (must be bigger than the size of a piece of computer
paper) that has dimensions (in centimeters) that represent the golden ratio. Include the
dimensions of your poster and your work of how you found the dimensions on the front of your
poster (neatly). Use pictures/drawings and color to make your poster fun and creative.
Parts 1 and 2 are tonight’s homework.
Part 3 and 4 will be due by the end of next class.
Parts 5 and 6 are due Friday, January 3rd.
Parts 7 and 8 are due Thursday, January 9th.
Be sure you are working on this project over the holiday break! I encourage you to complete the
parts of the project before they are due, bring them in to show me so I can check to make sure
you are completing it correctly. Feel free to come in for help on any of the parts. Ten points will
be taken off the total grade for each day any part is turned in late.
Golden Ratio Project Rubric:
Parts 1 and 2: Golden Ratio and Fibonacci info sheets.
Possible Points: Points Earned:
10
Parts 3 and 4: Derive Golden Ratio and find Golden Ratio in
Pentagram.
Part 5: Golden ratio in the Real World
10
Part 6: Golden Ratio in the Human Body
20
Part 7: Construction of Golden Rectangle and Golden Spiral
20
Part 8: Poster
20
20
Name: _______________________________________________ Per: _____ Date: __________
Part 1: Homework - The Golden Ratio
Directions: Look online to answer the given questions.
1) What is the value of the Golden Ratio? ___________________________________________
2) What is the Golden Ratio defined as? _____________________________________________
______________________________________________________________________________
3) What are some other names for the Golden Ratio (give at least four more)?
_____________________________________________________________________________
4) What mathematician discovered the Golden ratio? __________________________________
5) When did this mathematician live and where is he from?
_____________________________________________________________________________
6) Name at least three more mathematicians that have researched and used the Golden Ratio.
State when they lived and where they are from. _______________________________________
_____________________________________________________________________________
_____________________________________________________________________________
7) What structures have been built using the Golden Ratio (name at least three) and where are
they located? __________________________________________________________________
_____________________________________________________________________________
8) Name at least three pieces of art that use the golden ratio and state their artist.
_____________________________________________________________________________
_____________________________________________________________________________
9) Where else can the Golden Ratio be found in the real world? (Give at least three examples.)
_____________________________________________________________________________
Name: _______________________________________________ Per: _____ Date: __________
Part 2: Homework - Fibonacci and the Fibonacci Sequence
Directions: Look online to answer the given questions.
1) What was Fibonacci’s real name? ________________________________________________
2) Where was Fibonacci from? ____________________________________________________
3) What years did he live in? ______________________________________________________
4) What is the Fibonacci sequence? Describe how to get from one term to the next.
_____________________________________________________________________________
_____________________________________________________________________________
5) What are the first ten terms in the Fibonacci sequence?
_____________________________________________________________________________
6) If the 19th term is 4181 and the 20th term is 6765, what is the 21st term? _________________
7) How is the Fibonacci sequence related to the golden ratio?
_____________________________________________________________________________
_____________________________________________________________________________
8) What does the ratio of the 20th term to the 19th terms equal? Show your work.
9) What is the Fibonacci Spiral or the Golden Spiral? How is it drawn?
______________________________________________________________________________
______________________________________________________________________________
10) Where can the Fibonacci spiral be found in the real world? Give at least three examples.
_____________________________________________________________________________
Watch the first 13 minutes of Donald Duck in Mathmagic Land
(http://www.youtube.com/watch?v=WZH3PBdH5WQ)
Watch this video about the Fibonacci numbers by Arthur Benjamin:
http://www.youtube.com/watch?v=SjSHVDfXHQ4
Name: ____________________________________________ Per: _____ Date: _____________
Part 3: Classwork - Derive the Golden Ratio
Euclid defines the golden ratio as “A straight line is said to
have been cut in extreme and mean ratio when, as the whole
line is to the greater segment, so is the greater to the less.”
In other words, when you divide a line into two sections, the longer section divided by the
shorter section is equal to the sum of the two sections divided by the longer section.
Let a = x and b = 1. Then the segment will be
x
1
Now write the divine proportion as defined by Euclid using these values. Solve the proportion.
Show all work below (Hint: You must solve the quadratic equation using the quadratic formula.)
Name: ___________________________________________________ Per: _____ Date: ______
Part 4: Classwork - Golden Ratio in a Pentagram
Directions: Find the length of the given sides in centimeters. Then, find the ratio of the values.
AC = ________
PC = ________
𝐴𝐶
Now write it as a ratio and find its value: 𝑃𝐶 =
= _______
PC = ________
QC = ________
𝑃𝐶
Now write it as a ratio and find its value: 𝑄𝐶 =
= _______
What does this value represent? __________________________________________________
Name: _________________________________________ Per: _______ Date: ______________
Part 5: The Golden Ratio in the Real World
Directions: Find the golden ratio in each of the examples below. Fill in the measurements and the
ratio into the given chart.
1) The Parthenon – Athens, Greece
2) The Mona Lisa – by Leonardo DaVinci
3) United Nations Building – New York
4) Shell
Object
Larger
Measurement (cm)
Smaller
Measurement (cm)
Ratio (written as Value of Ratio
a fraction)
(the quotient)
1.
2.
3.
4.
What do you notice about the ratios in each figure? ____________________________________
Now, let’s see if we can find the golden ratio represented in objects we use every day. Include
four of your own finds!
Object
Larger
Measurement (cm)
Smaller
Measurement (cm)
Ratio (written as Value of Ratio
a fraction)
(the quotient)
Math book
TV Screen
Computer
Screen
Single light
switch plate
Single window
pane
Why do you think the golden ratio appears so often all around us? Write a paragraph.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Do Now: Read the article below by Karen McVeigh from TheGuardian.com. (December 28, 2009).
Why golden ratio pleases the eye: US academic says he knows art secret.
From Leonardo da Vinci to Le Corbusier, the golden ratio is believed to have guided artists and
architects over the centuries.
Leonardo is thought to have used the golden ratio, a geometric proportion regarded as the key to
creating aesthetically pleasing art, when painting the Mona Lisa. The Dutch painter Mondrian used it
in his abstract compositions, as did Salvador Dali in his masterpiece The Sacrament of the Last
Supper.
Now a US academic believes he has discovered the reason why it pleases the eye. According to
Adrian Bejan, professor of mechanical engineering at Duke University, in Durham, North Carolina,
the human eye is capable of interpreting an image featuring the golden ratio faster than any other.
Bejan argues that an animal's world – whether you are a human being in an art gallery or an antelope
on the savannah – is orientated on the horizontal. For the antelope scanning the horizon, danger
primarily comes from the sides or from behind, not from below or above, so the scope of its vision
evolved accordingly. As vision developed, he argues, animals got "smarter" and safer by seeing
better and moving faster as a result.
"It is well known that the eyes take in information more efficiently when they scan side to side, as
opposed to up and down. When you look at what so many people have been drawing and building,
you see these proportions everywhere."
Many artists since the Renaissance have proportioned their work in accordance with the golden ratio
or "divine proportion", particularly in the form of the golden rectangle, which has informed
Leonardo's work. It describes a rectangle with a length roughly one and a half times its width.
Works most usually associated with it are the Mona Lisa and the Parthenon in Athens, although
Swiss architect Le Corbusier relied on it for his Modulor system for the scale of architectural
proportion and Dali explicitly used it in The Sacrament of the Last Supper. The Parthenon's facade is
said to be circumscribed by golden rectangles, though some scholars argue that this is a coincidence.
According to Bejan, these arguments are academic. Whether intentional or not, the ratio represents
the best proportions to transfer to the brain. "This is the best flowing configuration for images from
plane to brain and it manifests itself frequently in human-made shapes that give the impression they
were 'designed' according to the golden ratio," said Bejan.
"We really want to get on, we don't want to get headaches while we are scanning and recording and
understanding things," he said. "Shapes that resemble the golden ratio facilitate the scanning of
images and their transmission through vision organs to the brain. Animals are wired to feel better and
better when they are helped and so they feel pleasure when they find food or shelter or a mate. When
we see the proportions in the golden ratio, we are helped. We feel pleasure and we call it beauty."
http://www.theguardian.com/artanddesign/2009/dec/28/golden-ratio-us-academic.
Name: ______________________________________ Per: ______ Date: __________________
Part 6: The Golden Ratio in the Human Body
Mirror, mirror, on the wall, who is the most GOLDEN of them all?
Directions: Find the given measurements on your body. Then compare the measurements to see
if your body represents the golden ratio. (You are allowed to asked someone to help you.)
Larger Measurement
(cm)
1) Height – from head
to toe
Smaller Measurement Ratio (written as a
(cm)
fraction)
Top of head to finger
tips with arms at side
Value of Ratio (the
quotient)
2) Top of head to finger Top of head to elbow
tips with arms at side
with arms at side
3) Top of head to elbow Top of head to top of
with arms at side
pectorals (arm pit)
4) Top of head to top of
pectorals (arm pit)
Top of head to base
of skull
5) Top of head to chin
Width of head
6) Top of head to pupil
Pupil to lip
7) Nosetip to chin
Lips to chin
8) Nosetip to chin
Pupil to nose tip
Which of your ratios is the most golden? ___________________________________________
Name: ______________________________________ Per: ______ Date: __________________
Part 7: Construct a Golden Rectangle
Follow the given steps using ONLY a compass and protractor.
(1) Take the given square below.
(2) Find the midpoint of the base of the square
by constructing the bisector of the segment.
(3) Draw an arc with the center at the midpoint
and radius as the length from the
midpoint to the vertex across the square.
(4) Extend the base of the square to meet the arc.
(5) Construct a perpendicular to the extended
base where the base and the arc meet.
(6) Extend the top of the square to meet the
perpendicular.
______________________________________________________________________________
Golden Rectangle Construction:
Measure the length and width of your rectangle (in centimeters), and find the ratio (as a fraction
and a decimal).
Length = _____________
Width = ______________
Ratio = ---------- = __________ (Round to the thousandths place. It should be the golden ratio!!)
Continue to divide the Golden Rectangle into smaller and smaller golden rectangles using ONLY
a compass and protractor. Within this one large Golden Rectangle, there are six other Golden
Rectangles.
Draw in the Golden Spiral by connecting the opposite corner of one rectangle to the next with an
arc (using your compass).
______________________________________________________________________________
Part 8: Poster
Present your construction on a poster (must be bigger than the size of a piece of computer paper)
that has dimensions (in centimeters) that represent the golden ratio. Include the dimensions of
your poster and your work of how you found the dimensions on the front of your poster (neatly).
Use pictures/drawings and color to make your poster fun and creative.