Are We Golden?
Investigating Mathematics in Nature
What do our skeletons, the Parthenon,
Greek statues, and the Fibonacci Sequence
have in common?
Do our bodies have mathematical
relationships in common with nature?
The Golden Ratio
(Phi or the golden number)
The Golden Ratio can be found:
• Greek Statues, urns, and artwork
• The Parthenon
• Leonardo da Vinci’s artwork
• All around us …. Windows, playing cards, book covers,
nature, and buildings
Leonardo da Vinci’s
Vitruvian Man
What do we notice about
this sketch mathematically?
• Circle
• Square
• Arm span = Height
Fibonacci Sequence
Leonardo of Pisa (Italy) lived about 1170-1250 and is
commonly called Fibonacci (meaning `Son of Bonaccio'). He
discovered what we now call the Fibonacci sequence, when he
published and solved his ``Rabbit Problem''. The problem is to
work out the number of rabbits alive after a given number of
generations.
The Rabbit Problem
The original problem that Fibonacci investigated (in the year 1202) was
about how fast rabbits could breed in ideal circumstances.
Suppose a newly-born pair of rabbits, one male, one female, are put in a
field. Rabbits are able to mate at the age of one month so that at the end of its
second month a female can produce another pair of rabbits. Suppose that our
rabbits never die and that the female always produces one new pair (one male,
one female) every month from the second month on. The puzzle that Fibonacci
posed was...
How many pairs will there be in one year?
Rabbits Diagram
Fibonacci Numbers and Plants
One plant in particular shows the Fibonacci numbers in the
number of "growing points" that it has. Suppose that when a plant puts out
a new shoot, that shoot has to grow two months before it is strong enough
to support branching. If it branches every month after that at the growing
point, we get the picture shown here.
A plant that grows very much like this is the "sneezewort": Achillea
ptarmica.
Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 56, ….
The ratio of any two consecutive Fibonacci terms (largest
divided by smaller) is approximately 1.6
Try:
34/21
144/89
1597/987
Rectangle Pageant
A
C
B
D
Rectangle Studies
In 1876 a German psychologist, Gustav Fletcher,
conducted similar studies with two groups of rectangles.
One rectangle in each group was a “golden rectangle” -that is, the proportion of the measure of the length of
the longer side to the length of the shorter side is the
golden ratio. An overwhelming number (75%) chose
those rectangles exhibiting the golden ratio. Why?
Investigating the Golden Ratio
Are you a “golden child?”
Activity Sheets
1. Sheet with directions on what lengths to measure
and ratios to calculate
2. Table to record fractional and decimal
representations of each ratio.
Materials Needed
1. TI – 108 Calculators (1 per pair)
2. Yard Stick (1 per pair)
3. Measuring Tape (1 per pair)
4. Activity Sheets and pencils
Investigate the Golden Ratio
Are we golden? Is the golden ratio somewhere in each of
us? Form groups of four or five and using the table and
directions given determine if you are golden.
X
Y
Step 1: Measure the height (B) and the navel height (N) of each
member of your group. Calculate the ratio B/N. Record them in
your table.
Step 2: Measure the length (F) of an index finger and the distance
(K) from the finger tip to the big knuckle of each member of your
group. Calculate the ratio F/K. Record them in your table.
A
B
E
Step 3: Measure the length (L) of a leg and the distance (H) from the
hip to the kneecap of everyone in your group. Calculate and record
the ratio L/H.
Step 4: Measure the length (A) of an arm and the distance (E) from
the finger tips to the elbow of everyone in your group. Calculate and
record the ratio A/E.
Step 5: Measure the length (X)of a profile, the top of the head to the
level of the bottom of the chin and the length (Y) from the bottom
of the ear to the level of the bottom of the chin. Calculate and record
the ratio X/Y.
Step 6: Select another pair of lengths (Q and P) on the body that you
suspect may be in the golden ratio. Measure those lengths. Calculate
the ratio (large to small) and record it.
N
Name
Express each ratio in its both its
fraction and decimal form.
1.
2.
3.
4.
5.
B/N
F/K
L/H
A/E
X/Y
References
• Beckmann, C., Thompson, D., Austin, R. (2004). Exploring
proportional reasoning through movies and literature, Mathematics
Teaching in the Middle School, 256-262, NCTM, Reston, VA.
• Garland, T. (1987). Fascinating Fibonaccis: Mystery and Magic in
Numbers, Dale Seymour Publications, NJ.
• Garland, T. (1997). Fibonacci Fun: Fascinating Activities with
Intriguing Numbers, Dale Seymour Publications, NJ.
• Serra, M. (1989). Special project: The golden ratio, Discovering
Geometry: An Inductive Approach, 475-480. Key Curriculum Press, CA.