Golden Ratio

φ

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

We find the golden ratio when we divide a line into two parts so that:

the long part divided by the short part
is also equal to
the whole length divided by the long part

golden ratio (a+b)/a = a/b = 1.618...

Have a try yourself (use the slider):

Beauty

golden rectangle

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

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

Do you think it is the "most pleasing rectangle"?

Maybe you do or don't, that is up to you!

parthenon golden ratio

 

Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way.

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, and I will tell you more about it later.

Formula

We saw above that the Golden Ratio has this property:

ab = a + ba

We can split the right-hand fraction like this:

ab = aa + ba

ab is the Golden Ratio φ, aa=1 and ba=1φ, which gets us:

φ = 1 + 1φ

So the Golden Ratio can be defined in terms of itself!

Let us test it using just a few digits of accuracy:

φ  =1 + 11.618
=1 + 0.61805...
=1.61805...

With more digits we would be more accurate.

Calculating It

You can use that formula to try and calculate φ yourself.

First guess its value, then do this calculation again and again:

With a calculator, just keep pressing "1/x", "+", "1", "=", around and around.

I started with 2 and got this:

value 1/value 1/value + 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 = 0.6153... 0.6154... + 1 = 1.6153...
1.6153...    

It gets closer and closer to φ the more we go.

But there are better ways to calculate it to thousands of decimal places quite quickly.

golden ratio construction step 1

Drawing It

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

golden ratio construction

(Where did √52 come from? See footnote*)

A Quick Way to Calculate

That rectangle above shows us a simple formula for the Golden Ratio.

When the short side is 1, the long side is 12+√52, so:

φ = 12 + √52

The square root of 5 is approximately 2.236068, so the Golden Ratio is approximately 0.5 + 2.236068/2 = 1.618034. This is an easy way to calculate it when you need it.

Interesting fact: the Golden Ratio is also equal to 2 × sin(54°), get your calculator and check!

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: 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. 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...
...
...
  ...

We don't have to start with 2 and 3, here I randomly chose 192 and 16 (and got 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 ...

I believe the Golden Ratio is the most irrational number. Here is why ...

We saw before that the Golden Ratio can be defined in terms of itself,
like this:
right arrow phi = 1 + 1/phi
  (In numbers: 1.61803... = 1 + 1/1.61803...)
   
That can be expanded into this fraction that
goes on for ever (called a "continued fraction"):
right arrow phi continued fraction: phi = 1+1/(1+1/(1+1/(1+1/...

So, it neatly slips in between simple fractions.

Note: many other irrational numbers are close to rational numbers (such as Pi = 3.141592654... is pretty close to 22/7 = 3.1428571...)

 

pentagram lengths

Pentagram

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

Read more at Pentagram.

Other Names

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

Footnotes for the Keen

* Where did √5/2 come from?

golden ratio square root of 5 over 2

With the help of Pythagoras:

c2 = a2 + b2

c2 = (12)2 + 12

c2 = 14 + 1

c2 = 54

c = √(54)

c = √52

 

Solving using the Quadratic Formula

We can find the value of φ this way:

Start with:φ = 1 + 1φ
Multiply both sides by φ:φ2 = φ + 1
Rearrange to:φ2 − φ − 1 = 0

Which is a Quadratic Equation and we can use the Quadratic Formula:

φ = −b ± √(b2 − 4ac) 2a

Using a=1, b=−1 and c=−1 we get:

φ = 1 ± √(1 + 4) 2

And the positive solution simplifies to:

φ = 12 + √52

Ta da!

 

Kepler Triangle

We saw above that:φ2 = φ + 1
And Pythagoras says a right-angled triangle has:c2 = a2 + b2

That inspired a man called Johannes Kepler to create this triangle:

Kepler triangle

It is really cool because: