An
Elementary
Look at the
Golden Ratio

The golden ratio (symbolized by the Greek letter “phi”) is a special number approximately equal to 1.618. It appears many times in geometry, art, architecture and other areas.

The Math Behind It

If you divide a line into two parts so that:

the longer part (a)
divided by the smaller part (b)

is also equal to

the whole length (a+b)
divided by the longer part (a)

then you will have the golden ratio (1.618…)

Beauty

This rectangle below has been made using the Golden Ratio. It looks like a typical frame for a painting, does it not?

Some believe the golden ratio makes the most pleasing and beautiful shape; in human faces, as well.

Many buildings and artworks have the golden ratio incorporated within them. The Parthenon in Greece, the Taj Mahal in India  and Bottocelli’s “Birth of Venus” are some examples.

The Actual Value

The Golden Ratio is equal to: 1.61803398874989484820… (etc.)
The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number…more about this in a bit.

Calculating It

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

• Divide 1 by your number     (=1/number)
• Add 1
• That is your new number
• Start again at step A

With a calculator, just keep pressing “1/x”, “+”, “1”, “=”, around and around. If you start with 2 you will get this:


# 1/# 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…

But it takes a long time to get even close; however with a computer it can be calculated to thousands of decimal places quite quickly; with the same result.

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.
• Extend it to create a rectangle based on 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.

Fibonacci Sequence

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: if you 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. For example:

A B B/A 2 3 1.5 3 5 1.666666666... 5 8 1.6 8 13 .625 ... ... ... 144 233 1.6118055556... 233 377 1.618025751... ... ... ...

You don’t even have to start with 2 and 3; as example start: with 192 and 16 (and get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, …):

A B B/A 192 16 0.08333333... 16 208 13 208 224 1.07692308... 224 432 1.92857143... ... ... ... 7408 11984 1.61771058... 11984 19392 1.61815754... ... ... ...

The Most Irrational

The Golden Ratio may be 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 forever (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, phi ratio, divine proportion, divine section and golden proportion.