The Golden Ratio

What is the Golden Ratio?

Most people are familiar with the number Pi because it can be found in so many different math problems and equations. There is, however, another irrational number like Pi. This number isnÐ'ÐŽÐ'¦t as well known as Pi however. This number is called Phi. This number is also called the golden ratio. The golden ratio is equal to the square root of five plus one, divided by two. If you work this out it comes out as 1.618033988749895. This is also the only number that if squared, is equal to itself plus one. Mathematically speaking, Phi^2 = Phi + 1. Also if you find the reciprocal of Phi, it is equal to itself minus one, Phi^-1 = Phi Ð'ÐŽV 1.

The Golden Ratio is the basis for many things in nature. Even ones fingers use the Golden Ratio. First measure the length of the longest finger bone. Then measure the shorter one next to it. Finally if you divide the longer one by the shorter one, you should get a number that is close to 1.168 which is really close to the Golden Ratio. Most parts of the human body are proportional to the Golden Ratio.

The Golden Ratio can even be traced back into the times of the Romans and Pyramids. For example, the Great Pyramid of Giza, which was built in 2560 BC, is one of the earliest ways the Golden Ratio was used. The length of each side of the base is 756 feet while the height of the Pyramid when build was 481 feet. If you divide 756 by 481, you would get 1.5717 which is very close to the Golden Ratio.

Another good example of the Golden Ratio is in Athens Greece. The Parthenon, which was build during 440 BC, uses the Golden Ratio also. The spaces in between the columns are proportional to the Golden Ratio. This shows that the golden ratio has been used for a very long time.

Since the Golden Ratio is an irrational number, it cannot be written as a regular fraction. You could however, get a very close estimate. One of the easiest ways is using the Fibonacci numbers. The Fibonacci numbers is a sequence of numbers where the numbers are the sum of the two numbers before it. For example, the first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, and 21. You can use this sequence for finding out the Golden Ratio by dividing two consecutive Fibonacci numbers. If you do this, you would get a number very close to Phi. For example, if you divided 21 by 13, you would get 1.615384615. This number is just off by 0.002649373. If we continued the pattern and used the numbers 144 and 89, we would get 1.617977528 which is only off by 0.000056461.

Another easy way to find the Golden Ratio is by taking the square root of a number, add one to the answer, and take the square root of that. If you kept doing that, you would eventually get to the Golden Ratio.

Constructing the Golden Ratio

A Golden Section

Ð'ÐŽÐ'§Given a line AB, construct a point E on AB so that AE/EB=The Golden Ratio.

Step 1. At point B, draw a perpendicular line BC, BC being 1/2 of AB.

Step 2. Draw a line through AC.

Step 3. Draw a circle at point C with radius CB. Circle C intersects AC at D.

Step 4. At point A, draw a circle with radius AD. Circle A intersects AB at E, the Golden Section. AE/EB=Phi=1.618....Ð'ÐŽÐ'Ё

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The Golden Rectangle

Ð'ÐŽÐ'§Given a square CBGD, construct the Golden Rectangle.

The sides of the rectangle will be in the proportion of Phi.

Step 1. Find the midpoint A of DG.

Step 2. Draw a circle centered at A and with radius AB.

Step 3. Extend line DG until it intersects circle A at E. The new side DE is the length of the rectangle. DE/DC=Phi.

In the figure, DE divided GE equals Phi. BG/BF also equals Phi. Rectangle BFGE is exactly