quote: Original post by picklejuice
could you explain the concept for me please?
I haven't learnt about the golden ratio before (i'm rather young), but I'm very interested in maths, you're help would be greatly appreciated.
Look here: (TODO: make this a clicky.)
http://library.thinkquest.org/C005449/home.html?tqskip1=1
I wrote a paper on the subject for art school recently. I'll probably get in some kind of trouble with someone over this, but here is that paper:
ON FIBONACCI
AND ON ART
by Ben Heath
INTRODUCTION
I have one pair of young rabbits, one male and one female. Next month, they'll mature, and they'll have more rabbits after that. Well, if each pair of rabbits produces one new pair of rabbits every month, and if it takes one month for a pair of rabbits to mature, how many rabbits will I have in 12 months?
Leonardo Fibonacci (also Leonardo of Pisa) posed this problem many years ago in his book, Liber Abaci (Book of the Abacus). Just reason your way through it and you will find the answer.
In the first month, I have that 1 pair.
In the second month, that 1 pair matures but they are still all that I have.
In third month, however, I have 2 pairs.
In the fourth month, the first pair produces a new one and the second pair matures, so I have 3 pairs.
In the fifth month, the first and second pairs both produce while the third matures, so I have 5 pairs.
Do you see a pattern there? 1, 1, 2, 3, 5... Each number in the sequence is the sum of the previous two. Take a look: 1 + 1 = 2; 1 + 2 = 3; 2 + 3 = 5. If you follow that pattern, then by month twelve, I have 144 pairs of rabbits.
WHAT PYTHAGORAS AND RABBITS HAVE IN COMMON
Okay, so what on earth has that got to do with art? If you're talking about the Renaissance, or ancient Greece, or 20th century Pop Art, rabbits don't come in at all. Those numbers certainly do, though.
Here's the sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... This snowballs onward forever. Now, if you were to take any two consecutive numbers from it (for instance, 89 and 144), and divide the greater by the lesser, you would get a number that comes close to... THE GOLDEN RATIO! You may (or may not) recognize the name, but the golden ratio is an infinite real number, and is said to be the “most irrational number.” Oddly enough, it's frequently found in art and science. The higher you go in the sequence, dividing a given member by the previous one, the closer you come to the golden ratio. Even still, it is irrational, so you will never pinpoint it precisely with any fraction.
It gets better. The golden ratio is synonymous with “sacred ratio,” “golden section,” “golden mean,” and “divine proportion.” What's more, it can refer to either of two different quantities. The first is phi (little “p”), which is 0.6180339887..., and the other is Phi (big “P”), which is 1.6180339887....
Now, in the ballpark of 2500 years ago, Pythagoras and his cult were using the golden ratio in their philosophy and studies. Even before that, the Egyptians, who called it the “sacred ratio” were using it everywhere. From writing to sculpture to housing, their day-to-day life was loaded with it. In the Middle Ages, various cults and guilds and societies were using the pentagon, a shape that was closely studied by the Pythagoreans for its “golden” properties, as a symbol of league and status.
In the Renaissance, use of the golden ratio was probably accelerated even further. Fillippo Brunelleschi set in motion his laws and theories of perspective, and it just so happens that the golden ratio is pleasantly found everywhere in good perspective.
Here's the greatest example of a shape that's loaded with the golden ratio: the human face. The shape of your face forms a network of lines, angles, and planes that is riddled with perspective, patterns, and the elusive golden ratio. The bottom of your nose is about two thirds down from the top of your head, leaving one third for your mouth and chin. 2 and 3 are Fibonacci numbers, aren't they?
Even today, we find the golden ratio in everyday life. The typical refridgerator has two doors, but one is around two thirds of the refridgerator's height. The standard size for an index card is 3”x5”. A credit card is a good example of a perfect golden rectangle: Its length is just about its height times Phi. It's also a general rule of thumb that when drawing a picture, an object is never put on the exact center. It's always about one or two thirds down or across.
You can find the golden ratio in film. In 1925, Sergie Eisenstein directed the silent film, “The Battleship Potemkin.” He is said to have used golden section points to divide the film up and start important scenes. What's to stop any director from doing this? Honestly, would you ever know it unless someone explained it to you?
Believe it or not, the golden ratio may even have been used in classic literature and poetry. For the Aeineid, Virgil may have consciously used the golden ratio in his poetry.
THE MOST PLEASANT? REALLY?
Of course, to say that the golden ratio produces the most pleasant shapes every time would be speculation at best. This is just an interesting mathematical property that you find in many interesting places. It's just delightful (to me) to see it. Here's the question: Are 3”x5” index cards more pleasant because they're 3”x5”, or is it because that's the standard and this is all that you're used to?
[edited by - Benjamin Heath on December 25, 2003 10:56:10 PM]
