Are We Golden?Are We Golden?

Investigating Mathematics Investigating Mathematics in Nature 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 SequenceLeonardo 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

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 Numbers and Plants

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 1.6Try: 34/21

144/89

1597/987

Rectangle Pageant

CA

B D

Rectangle StudiesIn 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?”

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 Needed1. TI – 108 Calculators (1 per

pair)

2. Yard Stick (1 per pair)

3. Measuring Tape (1 per pair)

4. Activity Sheets and pencils

Activity Sheets

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.

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.

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.

B

N

A

E

XY

NameExpress each ratio in its

both its fraction and decimal form.

B/N F/K L/H A/E X/Y

1.

2.

3.

4.

5.

ReferencesReferences

• 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.