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

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

Aug 29, 2016  Use 'Golden Ratio Face' to show how beautiful or ugly someone is. Golden Ratio Face uses facial symmetry, facial structure, and the golden ratio to calculate. Putting it as simply as we can (eek!), the Golden Ratio (also known as the Golden Section, Golden Mean, Divine Proportion or Greek letter Phi) exists when a line is divided into two parts and the longer part (a) divided by the smaller part (b) is equal to the sum of (a) + (b) divided by (a), which both equal 1.618. May 21, 2019  The golden ratio grid can help save the day when it is used to determine the dimensions of a layout and where each element will go. An easy way to start a layout design is to set the dimensions to 1:1.618 from the very beginning and work with that framework going forward. The easiest approach is to use the basic 960 px width and divide it by 1.618. Licensed owners of PhiMatrix 1.618 can update now to the most recent version for free within a year of purchase. See the list of new features below. Download the latest version of the software for your PC or Mac on the Download page. Version History PhiMatrix Update Converted PhiMatrix 1.618 Professional to a new third.

The Idea Behind It

Have a try yourself (use the slider):

Beauty

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!

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:

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

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

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 calculationagain and again:

• A) divide 1 by your value (=1/value)
• B) add 1
• C) now use that value and start again at A

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

I started with 2 and got this:

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.

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

• Now 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!

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

A Quick Way to Calculate

That rectangle above shows us a simple formula for the Golden Ratio. Facetime lowers volume of other apps mac book.

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

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:

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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:

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, ..):

The Most Irrational ..

I believe the Golden Ratio is the mostirrational number. Here is why ..

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

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

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

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?

With the help of Pythagoras:

c² = a² + b²

c² = (12)² + 1²

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c² = 14 + 1

c² = 54

c = √(54)

c = √52

Solving using the Quadratic Formula

We can find the value of φ this way:

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

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

It is really cool because:

• it has Pythagoras and φ together
• the ratio of the sides is 1 : √φ : φ, making a Geometric Sequence.

The Ultimate Guide to Understanding and Using
“The Golden Ratio”.

Designers everywhere should know about the Golden Ratio. It is a mathematical ratio that creates aesthetically pleasing designs. Since the Golden Ratio exists so frequently in nature, it’s not a surprise that its results are natural-looking.

Photo by Bogomil Mihaylov on Unsplash

The Golden Ratio goes by several other names, too:

• Divine Proportion
• Golden Mean
• Golden Section
• Phi (Greek letter)

The Math Behind the Golden Ratio

I’m going to explain the Golden Ratio’s math as simply as possible and without going into the details you don’t actually need to know. If you can keep up with the math, great. But if you can’t, that’s okay – you’ll still be able to use the concept in your designs.

To understand the Golden Ratio, you have to first understand the Golden Rectangle

The Golden Rectangle is a large rectangle that has a square inside it. The sides of the square are equal to the shortest length of the rectangle:

Source: Wikipedia

The Golden Ratio is a number that’s (kind of) equal to 1.618, just like pi is approximately equal to 3.14, but not exactly.

You take a line and divide it into two parts – a long part (a) and a short part (b). The entire length (a + b) divided by (a) is equal to (a) divided by (b). And both of those numbers equal 1.618. So, (a + b) divided by (a) equals 1.618, and (a) divided by (b) also equals 1.618.

Back to the Golden Rectangle, because it’s so much easier to understand

When you place a square inside the rectangle, it creates another, smaller rectangle. Ignore the black lines and look at the red and green boxes:

The red square has four sides equal in length, and that length is equal to the shortest length of the rectangle. By sectioning off that square, you automatically create another, smaller rectangle (outlined in green). Together, they create a complete Golden Ratio layout and a base for the Golden Spiral.

You can also make a new Golden Rectangle out of the smaller rectangle, like this one I’ve outlined in blue:

A traditional Golden Ratio diagram has eight Golden Rectangles:

And here’s the smallest Golden Rectangle, #8:

If you start in the bottom left and make an arch to connect the far side of each square-and-small-rectangle cross section, you’ll get the Golden Spiral.

The Fibonacci Sequence

The Fibonacci Sequence is pretty simple to understand: you start with zero and 1, then get the next number by adding up the two numbers before it. 0 + 1 = 1, then 1 + 1 = 2, etc. The first few numbers in the sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.

If you use those numbers to create squares with those widths, you can pretty much create a Golden Spiral:

Source: Math is Fun

The Golden Circles

Sometimes, you’ll see circles drawn in the squares instead of or in addition to the spiral. If you draw perfect circles in the boxes of the Golden Ratio overlay, they’ll have the 1:1.618 ratio with one adjacent circle.

Source: Limelight Department

The Pepsi and Twitter logos use the Golden Circles:

Source: Hybrid Talks

You’ve Seen This Before, A Lot

Nature is full of the Golden Ratio. It’s in flora, shells, weather…

Source: Photo by Annie Spratt on Unsplash

Source: Photo by NASA on Unsplash

And because we see it so often, our brains prefer it. That innate attraction is why it’s such a powerful layout for designers to use.

The Golden Ratio in Art and Design

Sometimes the Golden Ratio super easy to recognize:

Source: staceysdetailinginc.com

Sometimes you go, “I have no idea what you’re talking about… oh wait. Now I see it. I think.”

Source: Marketing Insiders

Other times you could go crazy looking at it…

Source: Widewalls

…but if you zero in on the main Golden Rectangle, it becomes a little more clear:

Let’s take a look at a commonly-referenced example: the Parthenon

Source: Creative Bloq

At first, you might see this and go, “That just looks symmetrical to me. How does what I’m looking at fit into that Golden Rectangle Spiral thing?”

The Golden Ratio isn’t about how each part of a design fits completely and only into the specific sections. If that was the point, the right side of the Parthenon would be one big block and the left side would be sectioned into smaller blocks.

Instead, the ratio is used to create harmony and proportion, and that can be interpreted in a few different ways.

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While the Golden Ratio is grounded in math, it can be adapted in creative ways. In the case of the Parthenon, the Golden Ratio determines the height and placement of design components. Plus, there are a number of ways to lay Golden Ratio diagrams over it:

Source: Archinect

Source: Esther Sugihto on Medium

Source: GoldenNumber.net

The Golden Ratio and Website Design

Whether you’re into math or your head’s about to explode, the Golden Ratio is a bit easier to understand in terms of design. You’ve done the heavy lifting. Now it’s time to take the basic overlay and make your web components perfectly pleasing.

The Golden Ratio and Layout

If you want a perfect Golden Ratio layout, set the dimensions to 1:1.618. For example, you can set the width to 960 pixels and the height to 594 pixels. The Golden Rectangle is 594 pixels on each side and the rectangle takes up the rest of the layout (594 x 366).

Calculator Soup has a helpful Golden Ratio calculator where you can set any term (A, B or A + B) to find the correct Golden Ratio values.

Or, you can simply use this type of two-column layout, where one column is quite a bit wider than the other column. It’s organized and clearly shows hierarchy.

Source: National Geographic

The Golden Ratio and Spacing

The Golden Ratio can help you determine where to place elements of your design, the proportions to use and where to leave negative space. Here’s a simple example, and you can almost see the Golden Ratio overlay without even having to put it on top:

Source: Digiarts 2011

Here’s what it looks like when I apply the Golden Spiral in Photoshop:

Again, the Golden Ratio is grounded in math, but when it comes to applying it to design, it’s not perfect. That design isn’t created on a Golden Rectangle, so the Golden Spiral is out of normal proportions. However, you can see how it can guide a designer to choose where to put the largest element of the design, as well as the smallest elements and negative space.

You can also layer the Golden Ratio overlay to apply it to different elements of the same design:

Source: Branding by Lemongraphic. Example from Canva.

The Golden Ratio and Content

When you think about the Golden Ratio’s layout and spacing together, you can start deciding where to place content on your website.

Let’s look at the National Geographic website again, this time with Canva’s Golden Ratio overlay on it:

The layout is split so that content lines up along the spiral’s center line. To the left, there’s a large block of content. To the right, the content becomes denser and there’s a lot more negative space. Toward the center curlicue of the spiral, you’ll see a second National Geographic logo – there’s no better way to drive home branding than to place it where the eye naturally goes.

Here’s a great example of how the Golden Spiral can lead your eye through a design, even past its main component. This is useful if you have a lot of content to squeeze onto one page. You’ll also notice that even with such a packed and detailed design, there’s still negative space in there.

Source: Design by Helms Workshop. Example from Canva.

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Honorable Mention: The Golden Ratio and Images

The Golden Ratio is also used in photography composition. Instead of creating a Golden Spiral, the Golden Ratio splits the image into six blocks. The same Golden Ratio is used in this type of grid: the widths and heights of the sections are either 1 or 0.618.

Source: Canva

You then use the intersections to compose the shot. The goal is to put a subject or main part of a subject on one of the intersecting lines – the subject shouldn’t be centered, and some blocks should be left empty (in most cases, at least – macro photography and close-up portraits will fill almost all of the frame). By doing this, you create a more interesting portrait than if the subject were centered.

A much simpler and more accessible way to follow this rule is to use the Rule of Thirds grid, which you probably have on your phone’s built-in camera or your DSLR.

Here’s a picture I took of my cousin’s son. I’ve laid the Rule of Thirds grid over it to show you where the subject does, and does not, fill the frame.

Also, look how the Golden Spiral almost-perfectly wraps around the subject:

The Golden Ratio differs from the Rule of Thirds because the Rule of Thirds grid has sections with equal lengths and widths. However, it’s so close – and so much easier – that this is what photographers commonly use when composing or editing a photo.

Wrapping Up

The Golden Ratio can be used as-is or adapted to your purposes and tweaked for size – math may have hard-and-fast rules, but creativity doesn’t. While you can use the Golden Ratio from the get-go to guide your design, you can also use it after you’ve started designing to make tweaks and improvements. The goal is to have the ratio guide you, not to force fit a design into it.