How is the Golden Mean derived? I am told it is approximately the ratio of 1:1.6
Is this correct? How was it derived and by whom (or when - era)?
(sqrt(5) + 1) / 2 = 1.618
It's old. It certainly shows up in ancient Greek architecture. I don't remember if the ancient Egyptians used it or not...
I answered a similar question in the "Letter to the Editor" section of the upcoming issue (May/June 2004) of View Camera magazine. Here's an excerpt from that response:
The irrational number Phi (1.618033988749895...) has long held significance to both mathematicians and artists. The ancient Greeks referred to this ratio as the Golden Section. To Renaissance artists, it was the Divine Proportion. It has also been called the Golden Mean or Golden Ratio.
The value of Phi can be calculated using multiple methods. One simple method is based on a numerical sequence discovered by 12th century mathematician Leonardo Fibonacci. The Fibonacci sequence starts simple enough, with the numbers 0 and 1. The next number is computed by simply adding the previous two numbers in the sequence - which becomes:
The interesting thing about the Fibonacci sequence is that the ratio of two successive numbers in the sequence converges on the value of Phi - the Golden Mean. In other words, the further you go, the closer this ratio gets to 1.618033988749895...
3/2 = 1.500
5/3 = 1.666...
8/5 = 1.600
13/8 = 1.625
21/13 = 1.616
By the time the 40th number in the sequence is reached, the ratio of the last two numbers is the value of Phi accurate to 15 decimal places.
It's old for sure.... discovered by God (or insert the name of your favourite creator) and made manifest as part of the geometry that underlies all creation. It is also the proportion of both the 7x11 format & the coils of ocean waves as they move through time that I have been photographing of late...... Coincidentally it is also the exact proportion of the dimensions of the credit card that I have been using to purchase film for the project!
With the popularity of the book The Da Vinci Code there is a wealth of books on the subjects Dan Brown brings up in it. Amoung the subjects is Phi.
I saw a book devoted entirely to Phi in a book store in Seattle last week, but I don't remember the name. You might try searching Amazon for Phi or Da Vinci and see what turns up.
Hope this helps
As I understand it another of its properties is that if you have a rectangle with sides 1 and 1.618... and then chop off a square of side 1 by 1 you are left with another rectangle of the same proportions and so on ad infinitum or ad nauseam whichever is soonest. i.e the ratio 1/0.618.. = 1.618.. You can try it with your old credit cards:-)
The classic definition I know is as follows.
Consider the figure below:
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There's a larger rectangle which measures y by x. There's a smaller contained rectangle formed by lopping off the square measuring x by x from the larger rectangle. This smaller rectangle measures x by (y-x).
If the ratio y:x is the same as xy-x) then this is a golden rectangle. This can only happen if the ratio is the golden ratio.
We can find what this ratio must be as follows:
By our definition
y / x = x / (y - x)
If we normalize this so that x = 1, then we have
y = 1 / (y - 1)
Which we can rearrange and solve by the quadratic equation, resulting in y = (sqrt(5) + 1)/2, as mentioned by Hogarth above. (There's another solution, of course, but it's (1 - sqrt(5))/ 2, a negative value.)
There are other ways to arrive at the same value---Kerry mentions one above.
Some people assert that forms which match the golden ratio are particularly beautiful, but that's aesthetics, not mathematics.
There are forms in nature which seem to match the golden ratio, but I don't know of any fundamental reason why this should be---it's possible that out of the many forms in nature, some of them are bound to come pretty close. I'm bet that we could find plenty of things in nature that match a 3:2 ratio as well.
Cheers (and hello),
The 4:5 and 8:10 ratios are "double":two vertical golden rectangles joined together.
The ancient Greeks commonly used both single and double golden ratios.. For example, the Parthenon fits into a single golden rectangle (sideways), while the Athena Temple at Priene fits into a 4:5 ratio, with 2 rectangles joined.
The 2:5 and 4:10 ratio used by many as "panoramic" is very close to the ratio of 1: 2.236 (the Square Root of 5).
A nice book on the subject, The Power of Limits, can be found here.
There are many websites, and at least one mathematical society, dedicated to the study of this subject. For example, see GoldenNumber.net
It was just another example of bones and rattles and chicken entrals and 666 (mark of the beast) and converting lead into gold bullshit. A concept of pseudo-intellectual historic interest only.
There's an excellent book 'The Golden Ratio' by Mario Livio (ISBN 0-7472-4988-1), which explores all aspects of Phi. It's fairly readable, though, by its very nature, a little academic in places.
Incidently, he dismisses many of the ideas which suggest that the ratio appears in ancient art and man-made atifacts, simply because the ratio can be 'fitted' to anything if you take measurements from the 'right' places.