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Golden Ratio

The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618

It appears many times in geometry, art, architecture and other areas.

The Idea Behind It

If you divide 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

then you will have the golden ratio.

Guessing It

There is only one value that would make a/b equal to (a+b)/a. Let us try out some possibilities to see if we can discover it:

Let us try a=7 and b=3, so a+b=10:
  7/3 = 2.333..., but 10/7 = 1.429..., so that won't work
   
Let us try a=6 and b=4, so a+b=10:
  6/4 = 1.5, but 10/6 = 1.666..., closer but not there yet!
   
Let us try a=6.18 and b=3.82, so a+b=10:
  6.18/3.82 = 1.6178..., and 10/6.18 = 1.6181..., getting very close!

In fact the value is:

1.61803398874989484820... (keeps going, without any pattern)

The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number, and I will tell you more about it later.

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:

Number 1/Number Add 1
2 1/2=0.5 0.5+1=1.5
1.5 1/1.5 = 0.666... 0.666... + 1 = 1.666...
1.666... 1/1.666... = 0.6 0.6 + 1 = 1.6
1.6 1/1.6 = 0.625 0.625 + 1 = 1.625
1.625 1/1.625 = 0.6154... 0.6154... + 1 = 1.6154...
1.6154...    

It is getting closer and closer!

But it would take a long time to get there, however 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.

The Formula

Looking at the rectangle we just drew, you can see that there is a simple formula for it. If one side is 1, the other side will be:

The square root of 5 is approximately 2.236068, so The Golden Ratio is approximately (1+2.236068)/2 = 3.236068/2 = 1.618034. This is an easy way to calculate it when you need it.

Beauty

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

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

Many buildings and works of art include the Golden Ratio in them,

such as the Parthenon in Greece.

Fibonacci Sequence

And here is a surprise. If you take any two successive Fibonacci Numbers, their ratio is very close to the Golden Ratio. In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation.

Let us try a few:

A
B
  B/A
2
3
  1.5
3
5
  1.666666666...
5
8
  1.6
8
13
  1.625
...
...
  ...
144
233
  1.618055556...
233
377
  1.618025751...
...
...
  ...

The Most Irrational ...

The Golden Ratio is the most irrational number. Here is why ...

One of the special properties of the Golden Ratio is that it can be defined in terms of itself, like this:
  (In numbers: 1.61803... = 1 + 1/1.61803...)
   
That can be expanded into this fraction that goes on for ever (called a "continued fraction"):

So, it neatly slips in between simple fractions.

Whereas many other irrational numbers are reasonably close to rational numbers (for example Pi = 3.141592654... is pretty close to 22/7 = 3.1428571...)

Other Names

The Golden Ratio is also sometimes called the golden section, golden mean, golden number, divine proportion, divine section and golden proportion.