The golden ratio (symbol is the Greek letter "phi" displayed at left) is a distinct number around equal to 1.618

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

You are watching: (1-sqrt(5))/2

The Idea Behind It

We uncover the golden ratio as soon as we divide a line into two parts so that:


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Have a try yourself (use the slider):

Beauty

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This rectangle has actually been made making use of the golden Ratio, Looks prefer a typical frame for a painting, doesn"t it?

Some artists and architects believe the gold Ratio renders the many pleasing and beautiful shape.


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Many buildings and also artworks have the gold Ratio in them, such together the Parthenon in Greece, however it is no really recognized if it to be designed that way.

The really Value

The golden Ratio is equal to:

The digits just keep on going, through no pattern. In truth the golden Ratio is recognized to it is in an Irrational Number, and I will tell you more about it later.

Formula

We saw over that the gold Ratio has actually this property:

ab = a + ba

We can separation the right-hand portion like this:

ab = aa + ba

ab is the gold Ratio φ, aa=1 and ba=1φ, which it s okay us:


φ = 1 + 1φ


So the gold Ratio deserve to be defined in regards to itself!

Let united state test that using just a few digits of accuracy:


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

With an ext digits we would certainly be more accurate.

Calculating It

You can use the formula to try and calculation φ yourself.

First guess the value, then carry out this calculation again and also again:

A) division 1 by your worth (=1/value)B) add 1C) now use that value and start again in ~ A

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

I started with 2 and got this:


value1/value1/value + 1
21/2 = 0.5 0.5 + 1 = 1.5
1.51/1.5 = 0.666...0.666... + 1 = 1.666...
1.666...1/1.666... = 0.60.6 + 1 = 1.6
1.61/1.6 = 0.6250.625 + 1 = 1.625
1.6251/1.625 = 0.6153...0.6154... + 1 = 1.6153...
1.6153...

It gets closer and also closer come φ the much more we go.

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

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Drawing It

Here is one means to draw a rectangle v the golden Ratio:

Draw a square of dimension "1"Place a dot fifty percent way follow me one sideDraw a line from that suggest to an opposite corner

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Now revolve that line so the it runs follow me the square"s sideThen you can prolong the square to it is in a rectangle v the golden Ratio!

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

A Quick way to Calculate

That rectangle above shows us a straightforward formula because that the golden Ratio.

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


φ = 12 + √52


The square root of 5 is around 2.236068, so the gold Ratio is about 0.5 + 2.236068/2 = 1.618034. This is one easy means to calculate it once you need it.


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


Fibonacci Sequence

There is a one-of-a-kind relationship in between the golden Ratio and also the Fibonacci Sequence:

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

(The next number is discovered by including up the 2 numbers prior to it.)

And here is a surprise: as soon as we take any two succeeding (one ~ the other) Fibonacci Numbers, their proportion is really close come the gold Ratio.

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Allow us shot 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 need to start through 2 and also 3, below I randomly decided 192 and also 16 (and obtained the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):


A
B
B / A
192
16
16
208
13

208
224
224
432
1.92857143...

...
...
...

7408
11984
1.61771058...

11984
19392
1.61815754...

...
...
...

The many Irrational ...

I think the gold Ratio is the most irrational number. Below is why ...

We saw prior to that the golden Ratio deserve to be identified in terms of itself, like this:
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(In numbers: 1.61803... = 1 + 1/1.61803...)
That deserve to be expanded into this fraction that go on for ever before (called a "continued fraction"):
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So, it nicely slips in between straightforward fractions.

Note: many other irrational numbers room close come rational numbers (such as Pi = 3.141592654... Is quite close to 22/7 = 3.1428571...)

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Pentagram

No, no witchcraft! The pentagram is much more famous together a wonder or divine symbol.And it has the golden Ratio in it:

a/b = 1.618...b/c = 1.618...c/d = 1.618...

Read an ext at Pentagram.

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Other Names

The gold Ratio is also sometimes referred to as the gold section, golden mean, golden number, divine proportion, divine section and golden proportion.

Footnotes because that the Keen


* where did √5/2 come from?

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With the assist 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 uncover the worth of φ this way:


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

Which is a Quadratic Equation and we have the right to use the Quadratic Formula:

φ = −b ± √(b2 − 4ac) 2a

Using a=1, b=−1 and also c=−1 us get:

φ = 1 ± √(1 + 4) 2

And the hopeful solution simplifies to:

φ = 12 + √52

Ta da!


Kepler Triangle


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

That influenced a man called Johannes Kepler to produce this triangle:

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It is yes, really cool because:

it has actually Pythagoras and φ together
Nature and The gold Ratio Fibonacci sequence Pentagram Geometry table of contents