n:2n+1

[johndevlin]

does anyone out there know anything about the origin of this formula for classical proportion? I have tried to google it but got nowhere. Has it been generally discredited as having any relevance today, and if so, by whom? The Parthenon obeys it, but does this really mean anything?

[lambros]

Hi there, I am not an architect, but I have studied maths and I come from Greece, and therefore consider myself eligimate to speak!

I guess that the formula you are talking about is the "golden rule" or "chrisi tomi" in Greek (which is best translated in English as "golden section"). Pythagoras (or according to another version, Pythagoras' wife, Theano) was credited with decrystalizing in mathematical form the perfect analogy rule found in nature. A little googling on golden rule should provide you with the mathematical and geometrical derivations of the golden rule.

On a normalized case (a straght line with length=1) the golden rule points that the perfect split is to be 0.618.... This number (or 1,618...in other notations) is named phi, and in fact it can be expressed in terms of the pi (which can lead to the conclusion that the "magical" pi and phi numbers are expressions of nature's wisdom).

Parthenon, as well as other ancient Greek buildings and temples obey the golden rule in many ways, as it proves to provide harmony and beauty to the eye. Much can be said, and much have been written, but you should check the net on more info.

Nowadays, the golden rule is used by those who understand its subtle but effective properties. Take credit cards for example: my visa is 8.6 cm by 5.3, and 8.6/5.3=1.6226... which is really close to 1.618.....

Now, don't tell me that this is a coincidence, it is a way to make it pleasant to the eye..... I am not trying to build a "Da Vinci Code" story here, but that is the truth!

[kris]

I think the golden cut, as we call it in my country goes like this.

You have to cut the peace of your art in that proportion so the biger peace is in same proportion to complete peace as the smaller peace is to biger one.

Its all about proportion.

Example. If you split number one on two parts. First is 2/3 and second is 1/3.

So the biger part 2/3, is exactly 2/3 to number one, but 1/3 is exactly 1/2 to 2/3, so its not golden cut becouse they are not in same proportion.

I have headache from math so I let you find your golden cut.

[PJ]

i seem to remember that the best example of the golden ratio is the paper sizes eg A0 is 2 A1's, A1 is 2 A2 or 4 A3's ect... now i'm thinking of other proportions oooooooh my... ha ha

 

Phil

[IC]

This number (or 1,618...in other notations) is named phi, and in fact it can be expressed in terms of the pi (which can lead to the conclusion that the "magical" pi and phi numbers are expressions of nature's wisdom)......

 

Nowadays, the golden rule is used by those who understand its subtle but effective properties. Take credit cards for example: my visa is 8.6 cm by 5.3, and 8.6/5.3=1.6226... which is really close to 1.618.....

Now, don't tell me that this is a coincidence, it is a way to make it pleasant to the eye..... I am not trying to build a "Da Vinci Code" story here, but that is the truth!

 

You'll find a large chapter of the Da Vinci Code refers to this phenomenon.

90% of the readers probably skipped it as it wasn't actually relevant to the story-just the author showing off probably.

[lambros]

haha, that damn book tells so many lies... but it is so addictive! I could not put it down...Mysteries are all around us, but it seems that we people only pay attention when the are served to us in specific ways!

(by the way, my wife's name is Theano, just like Pythagoras' wife!)

[lambros]

another thing I remembered... Pythagoras and the other Greek philosophers observed the analogies used in nature. The golden section can be therefore found in the human body. Try to measure your knuckles (they are the easier to measure!) and compare their ratios. They approximate the golden section (or to be more correct, the golden section approximates the human measures!). Most features of the human body conform to the law! It is the same with plants.

[DelfoZ]

" The Golden Section is an ancient Greek concept. It is a formula developed in 600 B.C. by the Greek philosopher and mathematician Pythagoras to define the concept of natural harmony. The work began after Pythagoras decided random chance alone was not enough to account for the blending of strength and beauty found in plants, animals, and minerals. Pythagoras proved that the balance between form and function which is the basis of nature's designs is the result of precise mathematical relationships

In architecture, art, and the natural sciences, the concept has since become the foundation for achieving classic, pleasing proportions. The Greeks and Romans used the Golden Section to create architectural masterpieces, with the Parthenon as a classic example

Nearly 2,000 years later, the concept took hold in Europe where chartres and other Gothic cathedrals are highly ornate expressions of the Golden Section. During the Italian Renaissance, Leonardo Da Vinci brought the same formula to scuplture and painting. Many of his works, including the Mona Lisa, were composed with the aid of an invisible matrix set down according to the mathematical rules of the Golden Section. "

http://www.macalester.edu/~psych/whathap/UBNRP/aesthetics/preferences.html

http://www.bonsai4me.com/AdvTech/ATGolden%20Section.html

http://facultystaff.vwc.edu/~trfanney/golden-mean-WOWslides/gm4.html

and lot more in alatvista or googles etc

[AJLynn]

I used to do math, I think I remember some of this. The A0, A1, etc paper sizes are not golden rectangles, they are 1:square root of 2, approximately 1:1.4142 - put the second number in a calculator and divide by 2, you get 0.7071. 1/0.7071 = 1.4142. So you cut the paper in half and you get 2 pieces that are the same aspect ratio as the first. Start with an A0, cut it in half you have 2 A1's, cut those in half you have 4 A2's, etc. More elegant than the system we use over here.

 

The golden ratio is the ration that gives you (A:B) :: (A+B:A). n:2n+1 won't get you there, what you're thinking of is the Fibonacci sequence, which is:

a(1)=1

a(2)=1

a(n)=a(n-1)+a(n-2), for a>=3

e.g. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

For large values of n, a(n)/a(n-1) approximates the golden section.

 

n:2n+1 is just just a series that will give you all the odd whole numbers. Start with n=0, work up, and you get 1, 3, 5, 7, 9, 11, ... - useful computer programming trick.

 

Oh yeah, and I think the credit card sizes are supposed to be based on the golden ratio, mainly because they were standardized a few decades ago when people cared about these things.

 

Interesting side note:

 

Is the golden ratio really a genetic thing? For a long time people have been saying things like 'if you tell a random non-architect to draw a section of a room that looks good, he will start by drawing a golden rectangle'. Maybe this is actually because architects designed their best rooms that way for centuries, ever since the Greeks decided it was a good idea, the Romans copied them and then everybody copied the Romans. These days, likely as not, you ask somebody under 20 to do the same thing and they'll start with eith a 4:3 or 16:9 rectangle, because those are TV sizes. (It's true, people have done studies.)

[IC]

 

Maybe this is actually because architects designed their best rooms that way for centuries, ever since the Greeks decided it was a good idea, the Romans copied them and then everybody copied the Romans.

 

Ah but they weren't really architects were they?

Sorry, in-joke from another thread.

[reddy]

Sorry I know this is probably months too late but just noticed this thread there.

You might be interested in Le corbusiers modulor. ie an attempt to create a standardised measurment system which would govern his buildings. His later work such as the Unite and La tourette were built with this system and having visited them I really feel he was on to something. The proportions of spaces in these buildings is fantastic.

During the middle half of the twentieth century a lot of work was done looking at proportion and the golden mean etc.

Since the da vinci code i think interest has been re kindled.

[Brian Cassil]

Phil Straub mentions the use of the golden rule in his composition tutorial here: http://features.cgsociety.org/story_custom.php?story_id=3275&page=

[reddy]

Thats a really cool tutorial actually. Wish I'd seen it before starting my latest piece of work!!