Golden Ratio

Golden Ratio



φ=1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614…


…if you need more of the digits of the Golden Number, you can visit the GoldenRatio first 10,000,000 Digits Website

You might have heard of the Golden Ratio called as:

  • golden section (Latin: sectio aurea);
  • divine proportion;
  • divine section (Latin: sectio divina);
  • golden proportion;
  • golden cut;
  • golden number;

And most likely you might have been wondering:

Where does the Golden Ratio come from?

Why is it so widely known and used in Graphic Design?

First things first.

The Golden Number it’s usually approximated to it’s first three decimal digit as 1.618… and widely known as φ or ‘phi’.
φ it’s defined as an irrational number, it means Phi is an endless string of digits that never repeat and loops.
The Algebraic way to express this irrational number is define as:

\varphi=\frac{1+\sqrt5}{2}\text{ =1.618... }

But all of this it’s just making you more confused i guess…
And won’t really explain you why the golden proportion it’s used in Graphic Design!
So if you are curious about the math-part of φ, i suggest you go and read more at the Wikipedia Page about Golden Ratio, which it’s quite complete and understandable.

To relate this irrational number to design, sculpture, architecture and other fields we need to understand how we can visually represent the Golden Ratio [φ]. And where can we actually see it.

Golden Ratio, The Golden Rectangle's ratios

The method to create a Golden Rectangle it’s very easy.
As in the image above we started from a square with side length ‘a’, and found the exact half of one of the sides [in the image the bottom one].
With center the half of that side, and radius the distance from that point to one of the corner we can design and arc [light blue in the image above] that allow us to find the size ‘b’.
This rectangle will so have two sides:

  • ‘a’ from the initial square;
  • ‘a+b’ that we defined with the arc.

The two sides have ratio:

\frac{a+b}{a}=\varphi=\text{1.618...}

So you now have a Golden Rectangle, which it’s often use to define the ratio of a frame we can use to design a graphic, to crop a picture, to start building a logo in…
and that some famous device producers use to define the ratio of the sides of the display [that are no longer 16:9 or 4:3].

So if you see that rectangle in any of the logo i designed, you’ll know that one it’s a Golden Ratio based Logo!

With an even simpler method, we can define a series of squares, each in golden ratio [actually an approximation of it] with the previous.
This thanks to the Fibonacci Series.
Fibonacci developed a sequence in which each number is the sum of the previous two, as example:

\text{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...}

One of the many features of this sequence is that as the values are growing, by dividing a number of the series with its previous, we will obtain approximations of the Golden Number, an example:

if we try it with 102334155 and 165580141 which are two sequent numbers of the Fibonacci Series, we will obtain a number quite close to:
φ=1.618033988749894848…

\frac{165580141}{102334155}=\text{1,618033988749895}

So creating a sequence of squares that have as side length the values of the Fibonacci Series we will obtain, as the values are growing, a rectangle always closer to the Golden Rectangle we saw before.

Golden Ratio, Series Of Fibonacci's Squares

Using the same structure of squares, we can create a spiral, matching quarters of circumferences with radius of the same length of the side of each square.

Golden Ratio, Fibonacci's Squares and Spiral

The spiral we obtained it’s known as Fibonacci Spiral, not to be mistaken with the Golden Spiral, as it’s just an approximation of this last one.

Golden Ratio, Golden Spiral

The Fibonacci Spiral, as well as the Golden Spiral are claimed to be everywhere in nature, an therefore widely used to create the proportion while illustrating the body-parts of a human being, to place the focal center of a painting or of a photo [and of a logo too!].

But Golden Ratio it’s only one of the many kind of proportions we can use in graphic design, we also can have also a modular grid, π [Pi≈3.14…], √2 [≈1.414…], ecc…
If you’re curious to see it used in the graphic design, you can see some of the logo based on the Golden Ratio I designed.