# Golden ratio



## tnimble (22 Nov 2007)

After some little debate on using the golden ratio in the calculator thread I decided to make a little tutorial about designing a piece of furniture.

The idea is to design a table with a table top of 25mm thickness about 160 by 200cm and 80cm high.

The critical visuals measurements would be the overall size of the table, the legs thickness, skirt height and inset of the legs and skirts.

Background
The golden ratio is based on the relationship of measurements found in nature. In nature nothing happens at random or by coincidence (although most of the time it would look like all is random due the shear number of factors involved). Mathematics is a abstract method to describe and work with things in nature (physics) and boyond.

Things like the cells that make up stuff and larger structures like shells can easily be studied. They exist of regular repeating patterns and recurring sizes. A spiral shaped shell for instance can be represented by a equation. Taking any size of shell the proportions between each and every element remains roughly the same.

The algebraic representation is called a set or sequence. One of them is the Fibonacci sequence. Each and every number in this set is relative to its neighbour. When taking an infinate large number in the set it is 1.618034 larger then the previous number.

There are quite a few special numbers around. Below are a couple of common ones. Each of these has a relation to a sequence. For each calculating an infinite amount you will obtain the special number.

2.7182818284 e (Euler 1/0! + 1/1! + 1/2! + 1/3! + 1/4! +1/5! + ......)
3,1415926535 Pi (Leibniz 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ......)
1.6180339887 Phi (Euclid 1 + 1/(1 + 1/(1 + 1/(1 + 1/(.....)))) )

The last of these numbers is the golden ratio. Very simply said Euler took a line and cut it into a infinite number of small line segments each time only dividing a previously undivided segment. When reaching a infinite number of infinite small line segments the relation between these lines is 1.6180339887..... (all these numbers have an infinite number of decimals)

Instead of looking at the whole of these infinite number of line segments we could look while obtaining these segments. When breaking the line in half we have obtained 2 smaller lines (2/1 = 2). From these 2 segments we'll get 3 (3/2 = 1.5). Then 5 (5/3= 1.66667).
8/5 = 1.6,
13/8 = 1.625
21/13 = 1.6154
When we go on like this for infinity we get Phi 1.6180339887...







The number of line segments can also be calculated instead of breaking sticks in half or drawling lines. Fibonacci saw the recurrence and developed a way to calculate any of the numbers. Take the sum of the previous two amounts of segments. This set of numbers is now known as the Fibonacci set or sequence.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ....

This number set can be used to describe populations (like number of bees, number of cells, growth rate of organisms, etc). When the set can be used to describe a population of something it also describes the size of object that is composed of groups of cells (a population) when the size of such a cell is known.

For instance the spirally shell of a nautilus. The center would be structure of 1 large, spiralling outwards one of 1, 2, 3, 5, 8, ... large.

The design
For the design to work there has to be a smallest unit defined, like the cell of an organism. I could have defined this a 1 wood fibre, or even the size of an carbon atom. However since the table is a magnitude larger, and those things can't even be seen when looking at the table it would make no sense to do so.

I picked the thickness of the table top as the smallest visible measurement. The table top is 25mm so our 'cell' are cuboid of 25x25x25mm.

A table build from 25mm cuboids could have larger structures (cuboid populations) of: (all in centimetres)
2.5, 5.0, 7.5, 12.5, 20, 32.5, 52.5, 85, 137.5, 222.5, 360, 582.5, 942.5, 1525, 2467.5, ....

So we start picking our important sizes from this list.
Table top: 222.5 x 137.5 x 2.5
Table height: 85
Leg thickness: 7.5 tapered to 5, taper starting 2.5 below skirt
Skirt height: 12.5
Table top overhang: 7.5
Skirt inset: 2.5


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## TonyW (23 Nov 2007)

tnimble, interesting stuff but have to say that I do wonder if worrying too much about mathematical progression could stifle the creative process. I am not trying to suggest that the golden ratio is not relevant, merely that slavishly adhering to the principles may not always give the best results.

Proof of the pudding - how does the design look. In the case of your table the proportions look very pleasing IMO. Some designs, objects etc. just look right to the majority of people. When investigated I suspect that size and shape will probably be found to follow some mathematical progression.

If anyone is interested this web site offers a lot of information on Phi with many examples Golden Number

To add balance have a look at Fibonacci Flim-Flam
Brief extract
A search of the internet, or your local library, will convince you that the Fibonacci series has attracted the lunatic fringe who look for mysticism in numbers. You will find fantastic claims:

* The "golden rectangle" is the "most beautiful" rectangle, and was deliberately used by artists in arranging picture elements within their paintings. (You'd think that they'd always use golden rectangle frames, but they didn't.)
* The patterns based on the Fibonacci numbers, the golden ratio and the golden rectangle are those most pleasing to human perception.
* Mozart used f in composing music. (He liked number games, but there's no evidence he ever deliberately used f in a composition.)
* The Fibonacci sequence is seen in nature, in the arrangement of leaves on a stem of plants, in the pattern of sunflower seeds, spirals of snail's shells, in the number of petals of flowers, in the periods of planets of the solar system, and in stock market cycles. So pervasive is the sequence in nature (according to these folks) that one begins to suspect that the series has the remarkable ability to be "fit" to most anything!
* Nature's processes are "governed" by the golden ratio. Some sources even say that nature's processes are "explained" by this ratio. 

Of course much of this is patently nonsense. Mathematics doesn't "explain" anything in nature, but mathematical models are very powerful for describing patterns and laws found in nature. I think it's safe to say that the Fibonacci sequence, golden mean, and golden rectangle have never, not even once, directly led to the discovery of a fundamental law of nature. When we see a neat numeric or geometric pattern in nature, we realize we must dig deeper to find the underlying reason why these patterns arise.

Other links
Patterns for Fibonacci guage HERE

Video of Guage in use HERE

Hope this may be of interest to someone

Cheers  
Tony


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## tnimble (23 Nov 2007)

That is so true Tony. The golden ratio is the most common known system but there are many many others around such as my faforate the plastic number. But you should only use it if you WANT to do so. If you find it limiting your ways your doing something wrong. Since the thinge you do should never be limiting you.


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## AdrianM (23 Nov 2007)

I might try incorporating some of that into my Sketchup Ruby scripts. I've written several to semi-automatically design most of the common units I make.

Could be interesting....


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