Golden Ratio

The Golden Ratio is formed by sequentially taking the ratio of two consecutive Fibonacci numbers (0,1,1,2,3,5,8,13..., each number being the sum of its preceding two) approaching 1.6180339887499... as the sequence heads towards infinity.

1 + 1 = 2

The name Golden Ratio suggests something rich, when in fact it is nothing more than the result of elementary arithmetic.

Some things are just what they are, nothing more. Yet the mysterious Golden Ratio pops up frequently in Sacred Geometry, the comfort blanket for people who cannot accept that some things are simple. Our lives are so busy and complicated that it's difficult sometimes to see that simple things are just that. Simple.

For example, 1 + 1 = 2.

It always does because that's what it does. There is nothing mysterious, magical or sacred about it.

Now let's extend this example a bit.

Upping it

If we start with the numbers 0 and 1, and then make a list in which each additional number is the sum of the previous two, we get the following:

and so on.

We now have a list of numbers, commonly called the 'Fibonacci Series', that have been created by simple additions. And as you can see, there is nothing mysterious, magical or sacred about that series.

Heavier stuff

About 800 years ago, an Italian named Leonardo Bonacci played with this series and came up with a strange number of 1.618034... which we now call phi. He wasn't, however, the first to discover that number. Phidias (490–430 BC) used it in designing some of his statues, and phi is his namesake.

To make it look even more posh and academic, it's usual to use the Greek lowercase letter for phi, which is "φ"

To get that weird looking number, and to understand why it's interesting, we take the ratio of number pairs from the above series, as follows:

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.6153846...
34/21 = 1.6190476...
55/34 = 1.6176470...
89/55 = 1.6181818...
and so on, lots and lots of times.

You will see that the quotient of each division gets closer and closer to 1.6180339887499...

This number has been created by simple divisions – elementary arithmetic that a child could do. And as you can see, there is nothing mysterious, magical or sacred about any of that.

So what is it about φ that's so special? Well, several things.

3 5

5 3 5

For example, if we draw a rectangle with side lengths of any of the above two consecutive numbers (for example, 3 & 5), that rectangle is called a 'Golden Rectangle'.

If we then use the height to create a square (5 x 5) and attach that to our rectangle to create a new rectangle, then that new rectangle is also Golden (5 x 8).

We could repeat adding squares to our rectangle over and over again, and always end up with a Golden Rectangle.

Lookin' good

It is said that the proportions of a Golden Rectangle are aesthetically pleasing. We don't know who said it, but ancient architecture has so many examples of Golden Rectangles that it's become part of human culture. There's a good chance that your PC screen is a Golden Rectangle (as are the images below).

And here's something else you might not know (and hopefully couldn't care less about): If we add 1 to φ, the result is the same as φ x φ. (In other words, φ + 1 = φ x φ.)

Why? Simply because that's what it does. There is nothing mysterious, magical or sacred about it.

And the big fuss is because...?

Typically sunflowers have 34 spirals in one direction and 55 in the other; a number pair we find in the Fibonacci Series

(Click either image to enlarge)

The internal chambers of the nautilus showing its logarithmic spiral.

The Golden Ratio is one of those odd things that pops up in nature quite a bit. And that is to be expected for things that grow and unfold in single steps.

The sunflower (Helianthus) and the Nautilus shell (Nautilus pompilius) are oft-quoted spirally examples of things that grow larger by a rate of φ.

And it is because of these natural appearances that φ has acquired mystical reverence. The ratio also appears in the pentagram; adding even more mystique. Yet when you consider that most shapes in geometry and nature are NOT based on φ, the partying is over.


In fact the nautilus shell chambers are NOT part of a Golden Spiral; its ratio averages 1:1.33

Why is it not Golden? We don't know.

Have we asked one of these little cephalopods? No.

We can only wildly speculate that the rate of growth is affected by things like sea temperature, depth, pollution, what they had for breakfast, and so on.

And the vast majority of flowers have no hint at all about the Golden Ratio.

But they are pretty anyway.

Return to the Sacred Geometry page.

OK, maybe not always. If one man and one woman produce one baby then I suppose 1 + 1 = 3. But on this page we're not talking about human reproduction, we're just talking about that basic mathematical object called 'number'.


search 🔍



privacy policy