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The Golden Mean

I have always been fascinated by ancient mathematical rules and how they have been applied in design. The "golden rectangle" or "golden mean" is one such rule if you like, that has often sneaked its way into my design work - sometimes planned but more often than not it just seems to happen.

Let me explain. The Golden Mean, just like PI (3.14) is another of those strange numbers that we seldom question and very often take for granted. This number is represented by the Greek letter PHI, but dissimilar to PI, the golden mean goes very much unnoticed in our everyday life in such things as buildings, plants and even in living creatures - yet we find these things strangely pleasing on the eye. This is the magical number 1.618.

So how is this number found? An ancient mathematician by the name of Fibonacci discovered that if you start with the numbers 0 and 1 then add them together you get a new number - in this case 1. Easy enough but what if you add the last number and the new number together? You get another new number, 2(See figure below). Keep doing this and you will end up with a very very long list of unique numbers.

This is known as the Fibonacci Series.

0,1 --> add them together gives new number 1
0,1,1 --> add the last two number together and new number is now 2
0,1,1,2 --> add last two numbers together and new number is now 3
0,1,1,2,3 --> add last two numbers together and new number is now 5

The series eventually grows like below into a series of unique numbers

0,1,1,2,3,5,8,13,21,34,55,89,144, 233,377 to infinity and beyond!

So what I hear you ask? Well, starting from zero and if you take any two SEQUENTIAL numbers and calculate the ratio between them then a very interesting pattern emerges below.

1,0   Ratio = 1 to 0 = 0
1,1   Ratio = 1 to 1 = 1
2,1   Ratio = 2 to 1 = 2
3,2   Ratio = 3 to 2 = 1.5
5,3   Ratio = 5 to 3 = 1.6666
8,5   Ratio = 8 to 5 = 1.6
13,8   Ratio = 13 to 8 = 1.625
21,13   Ratio = 21 to13 = 1.61538
34,21   Ratio = 34 to 21 = 1.61538
55,34   Ratio = 55 to 34 = 1.61764
89,55   Ratio = 89 to 55 = 1.6181
144,89  Ratio = 144 to 89 = 1.6179

If you keep going you will see that the decimal figure will revolve around the magic number 1.618. OK, I here you ask, but what is the point? Well lets look at the example of how the golden mean occurs in nature. Take a look at the diagram below. Notice that it is made up solely of squares, yet the overall image is a rectangle. This rectangle, if you measure it, has the magic ratio of 1.618. Also if you look at the curved lines within each of the squares you will notice that these are infact quarter circles, but, as a whole you would be forgiven for thinking that they look like the cross section of a sea shell.
And you'd be right, for this is the same as the growth rate of the beautiful Nautilus Sea Shell - i.e. 1.618.

Another interesting phenomena of nature is the sunflower. If you count the spirals you will see that there are 55 with either 34 or 89 on either side going in an anti-clockwise direction.Check it and see.

(Photo - Jeremy Merrifield at

The Mean Screen.

The golden mean can obviously be of huge benefit to designers when presenting new treatments to your clients. As we all know the client is always right and we have to go with their final say although sometimes we'd all like to think that they would accept some of our more illustrious designs.

Consider however, the following...Whilst I was was watching TV a few years ago (I can't remember the programme but I do remember the subject), they were discussing an experiment to obtain the most pleasing size of TV screen. They made a number of various rectangular shaped TV screens with differing ratios and asked a large sample of people to state the best looking TV. The results were staggering and almost all of them preferred the TV with the ratio of 1.618.

About a year ago, I bought a wide screen TV so I measured this and it has a viewing area with a ratio of 1.618 - check yours for size. Remembering the TV programme I thought I would put some design treatments to the test with my clients. I presented half of them with what they wanted and the other half with designs conforming to some form of the golden rectangle ratio. Each of the designs conforming to the ratio was accepted almost straight away and the client was pleased with the results.The others, well you know the story, 'can you change this', ' I don't like that' etc., etc. I thought there must be something in this so I reintroduced some of my less mainstream designs to clients who had rejected them but tweaked them to have ratios around the golden mean and some of them were preferred to the existing design. Yes the samples were small but weird stuff indeed.Whilst, I am not saying that this is the be all and end all of design, I am saying that it is worth giving a try. You never know, it may just save you time and effort.

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