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This week's LabZone activity
March 10, 2004
The Golden Ratio
Things you will need:
- paper
- ruler
- protractor
- pencil
- drawing compass
- picture of Parthenon
- buildings and pictures of buildings
- measuring tape
- a number of different people
Pythagoras was a Greek philosopher who believed that all beauty and knowledge was to be found in numbers and their relationships. Musicians will tell you that the ratio of the frequencies of two notes, a topic first investigated by Pythagoras, determines whether or not they are harmonious. Early architects claimed that the key to beauty in architecture is the golden ratio.
The golden ratio, so named because objects with dimensions in this ratio were believed to have great beauty, was discovered through a geometric construction. You can discover the golden ratio for yourself by using a ruler, a protractor, and a sharp pencil to draw a square 10 cm on a side like the one shown in the scaled drawing in Figure a (below).
Extend the base line WX to a point near the edge of the paper. Divide the square you have drawn into two equal rectangles, UMNW and MVXN, as shown by the dotted line. Then use a drawing compass to draw the arc of a circle with a radius equal to the diagonal of the rectangle MVXN as shown in b (above). The arc should meet the extended base of the original square you drew. Next, construct a rectangle VXYZ as shown in Figure c (above), whose base is the extended base of the square that intercepts the arc. Use the original height of the square as the height of the new rectangle.
The ratio of the height of the rectangle you just drew to its base (VX ÷ XY) is defined as the golden ratio, and the rectangle VXYZ is a golden rectangle. According to your measurements, what is the golden ratio? How closely does your value agree with the actual value, which is very close to 1.618, or about 8/5?
There are a number of reasons why Greek mathematicians and architects regarded the golden ratio with such reverence. The ratio of the length of the original square and its extended base, WX + XY, to the original base, WX, is the golden ratio. Furthermore, if you construct such a square within a golden rectangle—ABYX in Figure d (above)—the rectangle that remains, ABZV, is a golden rectangle. And a square constructed in that rectangle will leave a new, smaller golden rectangle. The process can go on indefinitely, giving rise to smaller and smaller golden rectangles (below).
Notice, too, if WX = 1.618 and XY = 1, then
(1.618 + 1) ÷ 1.618 = 1.618.
Also, look at a series of numbers where each number is the sum of the two preceding ones, as, for example:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55 . . . .
The golden ratio forms a series of numbers such that when raised to increasing powers, each term is the sum of the two preceding terms.
(1.618)0, (1.618)1, (1.618)2, (1.618)3, (1.618)4, (1.618)5 . . . .
Try it using the yx key on a calculator. For example, to obtain (1.618)3, enter 1.618, press the yx key, and then press 3, followed by the equal sign. You should find it produces a series that looks like this:
1, 1.618, 2.618, 4.236, 6.854, 11.09 . . . .
Can you find any other number that works?
How about 1?
10, 11, 12, 13, 14, 15. . . .
Clearly, 1 does not work because 1 raised to any power is 1. (Remember, any number raised to the 0 power is 1.)
Does the number 2 work? How about 3? Can you find any number other than the golden ratio that works?
If you have seen the Parthenon, built in Athens in the fifth century B.C., or pictures of it, you will find that it was built with the golden ratio in mind. Make some measurements of pictures of the Parthenon and other buildings or of buildings themselves. Can you find the golden ratio in the structure of any of these buildings? Do you find buildings with the golden ratio to be more attractive than square buildings or buildings with rectangles other than the golden ratio?
Le Corbusier, a twentieth-century architect, whose real name was Charles-Édouard Jeanneret (1887–1965), found the golden ratio in the structure of the human body. You can look for the ratio yourself. Begin by using a measuring tape to find the following lengths in a number of different people. Be sure to use the same units (centimeters or inches) in all your measurements. Measurements that require subtracting one value from another are indicated by the minus sign (–).
Height
Distance from floor to navel
Distance from navel to top of head
Distance from floor to tip of fingers of arm raised straight upward
Length of upper arm
Distance from tip of nose to tip of fingers when arm is full outstretched to side
Distance between tips of left-hand and right-hand fingers when both arms are outstretched
Span (distance between tips of thumb and index finger of one outstretched hand)
Distance from top of head to tip of fingers of arm raised straight upward
Inseam (length of inside of leg)
Cubit (elbow to tip of middle finger)
Length of lower leg (knee to heel)
Height – Inseam
Height – Distance from floor to navel
Distance from floor to tip of fingers of arm raised straight upward – Distance from floor to navel
Distance from floor to navel – Inseam
Look at the measurements you have made. Which ones can be paired to form ratios equal to, or very nearly equal to (± 0.2), the golden ratio. If the ratio of a pair of measurements is equal to the golden ratio for one person, is it equal or nearly equal for other people as well?
Reprinted with permission from Science Projects About Math by Robert Gardner. © 1999 by Enslow Publishers ( www.enslow.com ).
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