Art: the golden mean

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1. Abu
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## the golden mean

yeah... hello guys well, i´m new over here... well i will ask you guys to apologise my bad english

i´ve bean seeing the " da vinci´s code" i think its the right name and i saw something about the golden mean ..
that really got into my mind so i look around to find something ab about that.. i found this page
http://www.tu-harburg.de/b/kuehn/lec4.html

i´ve allready understood some basics information

if there is someone over there who can help .. i will really like that :d

thanks

2. i really dont know how this principle could apply to drawing, if its the golden mean i've been thinking about. The only golden mean i saw in the link you provided was a mathematical formula that talked about rectangles :p Isnt the golden mean a rule about visual culture or something like that?

later,
GriNGoLoco

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Yup, the golden mean IS a mathmatical equation. The 'perfect' ratio. The greeks used it to build their temples, some artists use it for painting, and amazingly enough it is (as far as I know) the most commonly reaccuring equation in nature.

IE: The bottom of most people's nose sits on the 'golden mean' of their face. Same with the wrist on the forearm, and several other things I can't recall from my classes.

One of my drawing teachers was big on the golden mean. He showed us all of this. He also used the Mean to lay out a lot of his compositions.

Hope that confused you to death!

4. Abu
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thanks the replay

i searched about the meaning of the "golden mean".. and it really is only an mathmatical equation . I´ve also read that da vinci noticed that the golden mean is found in the human body... anda that he used to make his painting....and was this treasonthat caught my attention...and i also like maths to

if some one can explane it better ...

5. hey, Hjelm, can you please explain this formula for us? it sounds interesting... For what I found, the golden mean is the following:

in the 6 perspectives for analyzing any image : the golden mean is one of the six possible ethical perspectives . Heres is the definition (taken from "The Media Through Which We See":

"Aristotles golden mean refers to finding a middle ground or a compromise between two extreme points of view or actions. Formulated in about the fourth century B.C., this philosophy of taking the middle way doesnt involve a precisely mathematical average but is an action that approximately fits that situation at that time. Compromise and negotiation are actions aimed at finding a link between the opposing viewpoints of two competing interests".

Other upossible aspects in the ethical perspective are: categorical imperative, utilitarism, hedonism, the golden rule and the veil of ignorance.

Later,
GriNGoLoCo

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Well I don't know what all that garbeldegook was that YOU found. Doesn't sound like it has anything to do with math though. The Golden Mean is also referred to as the Golden Section. I googled that term and found sites with the right information.

I'll post more about it soon, but I'm at work and I'm not really supposed to be on here anyway!

Have a good.

7. ok man, thanks for helping out in this doubt...

later,
GriNGoLoCo

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Okay... here's the shpeal. Had to do a bit of researching.

The Golden Mean (I will call it this, since I like the term better) is supposed to be the perfect ratio. Shapes build with the GM are supposed to be the most pleaseing to the eye. The ratio itself one of those hairy numbers that goes on forever. This is part of it:

±0·61803 39887 ...
or
±1·61803 39887 ...

I'm no mathemetician, so I prefer to think of the Golden Mean as the following fraction:

618/1000

It's not as precise, maybe, but it's close enough for government work.
So in order to find the Golden Mean on a line, you'd measure the line, multiply your measurement by 618, then divide that number by 1000.

Ex: I want to find the Mean on a 5" line.
5x618=3090
3090÷1000=3.09
So the golden mean of 5"(or whatever unit) would be 3.09"(units)

Take the following rectangle:

This rectangle is built using the GM. Doesn't look to special at first glance, eh? Now look at the next:

Do you follow? Dividing the rectangle's long side at the Golden Mean created on one side, a square of course. On the other side is a rectangle that is preportional to the original rectangle. The ratio for both is roughly 618/1000!

If you continue dividing these preportional rectangles, you get something like the following:

Adding quarter circles inside each square will give you the shape of a nautilus shell.

The areas lining up with the golden mean are supposed to be the strongest areas of a composition for example, pretend the following is a 6x3 canvas (units don't matter. It could be inches, feet, meters, miles, etc)

Now 6x3 doesn't follow the Golden Mean at all, but if you find the ratio on each side of a canvas and intersect the lines, you should be able to find the 'perfect' place to put focal points. (If you invert the GM lines above, bot Horizontally and Vertically, you should be able to find 4 points)

My instructor also found that using 'secondary Golden Means' (finding the Golden Mean between the edge and the primary GM lines pictured above) was a good way to place secondary focal points.

Okay so here's how the ratio applies to the human face. I haven't found any good references for bodies, arms, and legs yet, but I will try.

Here is the stunning Nicole Kidman with lines running through her face (Yey!) Hopefully you can understand what's going on here. I took a line and divided it at the golden section, then lined the top and bottom up with different landmarks on the face (chin, top of head, nose, and hairline).

Most faces fit this pretty closely as you can see below:

These are the exact same lines. All I did was swap the faces. His eyes/nose/lips line up just about the same—all on a Golden Mean between two other major points.

Not all faces will line up on on this diagram. Most of your average 'pretty people' should. If you find a picture of someone who looks a little 'off' check them against the Golden Section. Chances are their nose or something doesn't sit on the Golden Mean.

Ugh! Too much math.
If you got confused, read it again. If you're still confused after that, ask and it shall be given.

Toodles
Robin

EDIT: fixed a few spelling errors that mighta caused confusion.

Last edited by Hjelm; June 10th, 2004 at 01:10 AM.

9. Abu
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WOW ... great incredible....

it was the best explanation that i´ve seen

thanks a lot man

i really didn´tunderstand the fourth picture very well

if..just tried touse these propotions.. and they realy fit well

see ya !

10. Woot~! Interesting stuff! Maybe god is a mathematician

11. That reminds me of the "golden mask" for perfect face proportions based on the golden mean equation. I think I have it somewhere still. Interested ?

Now as for God being a mathematician... Everything in nature seems to be orchestrated. Look at a pineapple! Or heck even any kind of tree or sprouts out there. I don't pretend to understand all of it, with my head, but with my heart, yes.

12. Hey peeps if this stuff is interesting to you watch the movie "Pi" by Darren Aronofsky. Good movie and it talks about the golden mean a lot.

http://www.imdb.com/title/tt0138704/

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...Maybe God's a nerd. J/K

Anyways... I'll try to re-explain the 'focal point' thing for Abu. This time with real measurements.

Here's a blank 'canvas' that's 5x6" large. The canvas itself was not based on the Golden Mean. 5/6 isn't preportional to 618/1000, but you can still use the GM to find what are supposed to be natural 'strong points' on the canvas.

I took the measurement of each side and found the GM on them. Anything that lays on one of these lines is supposed to naturally draw the eye, and anything that sits on an intersection of the lines is supposed to get even more attention, so if you plan a focal point on an intersection between two GM lines, it's supposed to strengthen your focal point.

Here's something to clarify what I meant by finding secondary focal point locations:

Hopefully that's clear enough. I'm afraid if I try to explain it in words, I'll just make it more confusing.

Thanks egerie and Groover for pitching in. I don't claim to be an expert on the Golden Mean, so any extra information is helpful!

I just read an article in a CG magazine about some people creating 'perfect' CG 'actors' for 3D animation. The faces are based on the Fibonacci numbers and the Golden Mean (which is somehow related to the Fibonacci numbers, though I'm not sure how). Actually I thought their 'perfect' faces looked kinda ugly, but that was probably more the fault of the skin than the model.

Eh whatever.

Have a good!
Robin

14. Hey Hjelm,

Thanks for explaning this. The diagrams were really helpful.
I was thinking that instead of using a ratio of 618/1000, you could simplify it to 61.8/100.
This way, you can always find the measurements using percentages.
The smaller side of the rectangle will always be 61.8% the length the longer side.

Gringoloco,

I think you found the philosophical idea rather than the mathematical idea.
Something like "the perfect solution that both parties can agree on and be happy about."
If only it were as easy to find this in real life.

emily

15. Abu
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wow thank Hjelm .. now igot the main ideia

Aerythes, god could be a computer

egerie, tour help would be great

see ya

16. Abu> Haha, god is the AC (Just finished re-reading Isaac Asimov's "The Last Question" )

17. A simplified formula for the proportion is this: 3:5, 5:8, 8:13, 13:21, 21:34 and so on ... I htink you get the picture. The higher the ratio is, the nearer you´re at the mathematical "Golden Cut". I´m personally working mostly with a ratio of 8:13 - higher is a bit clumsy imo.

Hope this could help.

Fipse

BTW good vizualisations of the principles, Hjelm!:chug:

18. dzu
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## Fibonacci

The Fibonacci sequence is a sequence of numbers

1,1,2,3,5,8,13,21,34....

and is calculated by adding the two previous terms to produce the third. For example
1 =1
1 =1
1+1=2
1+2=3
2+3=5
3+5=8
5+8=13
8+13=21
13+21=34
...

How is this related to the Golden Mean? The proportions of the numbers converge towards the Golden mean number of 1.618.

2/1=2
3/2=1.5
5/3=1.66666
8/5=1.6
13/8=1.625
...
377/233=1.618

As you get farther in the sequence, any number divided by the following number approximates .618 and any number divided by the previous number approximates 1.618

So how does this help your art? Well as Hjelm has already posted, you can use it for composotion. It can also be used for designwork since many things in nature and architecture follow the G.M. You can use it to help start your designs to find pleasing proportions that look also look structurally sound/realistic. There is a book called "Geometry of Design" by Kimberly Elam that describes this more in depth.

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http://www.beautyanalysis.com/index2_mba.htm
I think this webstie is the best rescource for the golden section. This is the most intriguing subject in my opinion. I mean, everything has it. The distance of branches is the same 1.680/1 ratio as well as a nautilous shell, the joints of a human, everything has that exact same ratio (even the DNA straind for god sakes). It is, to be blunt, the perfect form.

20. Abu : ok here !

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National Geographic magazine covers are in the golden mean. So are areas of the Parthenon. It's also the ratio of one finger bone to the next, on any finger.

22. Abu
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wow!

thanks guy... im really understanding it now...
great help....

23. Well, I'll try to re-read it later today. Still looks preetty complex to me though.. but in the end its all about proportions. Can this be applied only in "beautifying" things?

Oh and the stuff that I came up with are some rules about analyzing the things you see in the world. They give you a much wider perspective in how things are. They help you not to limit yourself in one or some particular aspects of that image. This particular rule (referred to as the golden mean) talks about being neutral when analyzing an image. Or so I understand. :p

later,
GriNGoLoCo

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