Golden Ratio

by Jody Carlisle

Euclid describes the golden ratio as "to cut a line segment in extreme and mean ratio." (Proposition VI,30) The golden ratio is the ratio of the length to the width of what is believed to be one of the most optical pleasing rectangular shapes. This rectangle is called a golden rectangle. The golden ratio can be represented by:

where point B is said to divide the segment AC into the golden ratio. That is, the ratio of the shorter segment to the longer is equal to the ratio of the longer to the entire line segment or AB/BC = BC/AC. This can also be written as a/b = b/(a+b), with a< b. Let's say that b/a is equal to x therefore we have x^2-x-1=0. When factoring this equation you get (1+ or - sqrt 5)/ 2 and the positive root gives 1.618.

Euclid worked mainly with geometry. In Proposition II, 11 he says "to cut a given segment so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment" The construction is as follows:
First construct the square ABCD

Next find the midpoint of AC label it E. Then draw a segment from E to B.

Now using E as the center and EB as the radius find the point where the circle intersect the line CA after it is extended. Call this point F.

The only thing left is to complete the square using the segment FA, and where the square ABCD intersect the small square this is the golden ratio, call it H.


The golden ratio also comes up in the Fibonacci Sequence. This sequence is where any term after the second is the sum of the two preceding terms. The golden ratio is the ratio of any two successive terms.


In column A is the Fibonacci Sequence, and in column B is the ratio of successive terms which approaches the limit of the Golden Ratio.

The golden ratio can be found in may things such as index cards, Greek Parthenon, other architecture, tree growth, snails, star fish, art, and proportions of both human and animal bodies. An example of these human body proportions is (the measure from head to toe) / (measure from belly button to toe) is equal to (the measure of the head to the belly button) / (the measure of the head to the under arm) which is equal to 1.618. There are many other facial proportions that also gives the golden ratio.
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