THE GOLDEN RATIO

The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is often symbolized using phi, after the 21st letter of the Greek alphabet. In an equation form, it looks like this:

a/b = (a+b)/a = 1.6180339887498948420 …

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The golden ratio symbol is the Greek letter "phi" shown below

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This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it?

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Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

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Calculating It


You can calculate it yourself by starting with any number and following these steps:

  • A) divide 1 by your number (=1/number)
  • B) add 1
  • C) that is your new number, start again at A

With a calculator, just keep pressing "1/x", "+", "1", "=", around and around. I started with 2 and got this:

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But it takes a long time to get even close, but there are better ways and it can be calculated to thousands of decimal places quite quickly.

Drawing It

Here is one way to draw a rectangle with the Golden Ratio:

  • Draw a square (of size "1")
  • Place a dot half way along one side
  • Draw a line from that point to an opposite corner (it will be √5/2 in length)
  • Turn that line so that it runs along the square's side

Then you can extend the square to be a rectangle with the Golden Ratio.

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There is a special relationship between the Golden Ratio and the Fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

(The next number is found by adding up the two numbers before it.)

And here is a surprise: when we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio.

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation.

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Pentagram

No, not witchcraft! The pentagram is more famous as a magical or holy symbol. And it has the Golden Ratio in it:

  • a/b = 1.618...
  • b/c = 1.618...
  • c/d = 1.618...
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Da Vinci himself used the Golden ratio to define all of the proportions in his Last Supper, including the dimensions of the table and the proportions of the walls and backgrounds. The Golden ratio also appears in da Vinci's Vitruvian Man and the Mona Lisa.