I'm sure somebody can point you to a nice web resource, but I would recommend a book: "Geometry in Design" by Kimberly Elam. It's a "light" work, but contains a nice treatment of various geometric "ideals" in design, and Nature. Of course it has a section on the Golden Section.

What bugs me about the Golden Section is this: since it deals with proportions of length, when you use it for surfaces (like in type design), do you square-root it? I've never seen a convincing discussion of that.

BTW, I also have an article from an old issue of PEI (back when they encouraged thinking, instead of being a source for Qwik Trix with Fotopix type ••••) that discussed the Golden Section with great insight. But please don't ask me to find it! :-/

hhp

hmmmm. The golden section is quite a big subject.
It's not that difficult. (I wrote a paper about it ). Maybe first a link to start: the pages of mr Knott, which have a big deal to say about the golden section, and helped me alot. Non-designed, but filled with information & examples on the subject. (I always liked the part about Fibonnacci and his rabbits )

In fact, it is just a proportion of 1 to 1,618.... Some people believe this is the "divine proportion" because it appears in nature so often (this has rather economical reasons, see Professor Knott's pages). I think it is rather difficult to apply to design, because of it's irrational nature. Le Corbusier used it to design his buildings (he made a system out of it, called 'le Modulor'); Wiedemann used it in his Corporate ACE-typefaces (and the Mercedes-star). Many bookcovers have this proportion also. There is alot written about applying this to bookdesign (see Bringhurts "Elements of Typography" for instance), and I think Tschichold studied it and distilled a method to apply it to the margins of a page...

hope this gets you going...

Hrant, you're to quick for me! Yes 'Geometry in Design' is a indeed good book for showing some examples of applying the GS. I forgot about it ...

There are many proportion-systems, Palladio used the proportions that are based on the musical harmonies. There is a magnificent book on that (only architecture tough) by Rudolf Wittkower: Architectural Principles in the Age of Humanism (1949).
I think one should just find a system (or make up his own) that suits him (and the job) and apply that to his design, so you have a certain consistency in your design.

Hrant:
What bugs me about the Golden Section is this: since it
deals with proportions of length, when you use it for
surfaces (like in type design), do you square-root it?

No! If you have a rectangle and sections the longer side
according to the golden ratio the areas you get will still
be, of course, in the (A+B/A = A/B) relationship.
This is a misleading question. The "visual assessment" of
the relationships of surfaces was not very much the point.
The two more important applications of the golden ratio
were the golden section (=cut) which deals with the
positioning of an element inside a certain width, and the
golden rectangle, which is a shape. Though the ratio is the
same, these are quite unrelated perceptual experiences.
Naturally, if the rectangle you sectioned above was a
golden rectangle to begin with, you get a square and a
new golden rectangle, so you begin to tie the loose ends.

Now, the mathematical validity of the golden ratio is
universal, it turns up in many natural phenomena (*),
etc., but the question is: does it have any aesthetic
validity outside the area of Greek cultural influence
(the so-called "West")? For instance does it appear in
Islamic (**) architecture and decoration somehow? (As
you know, the Muslims were the heirs and "keepers" of
Greek culture through the middle ages.) In China? In
Japan?

For those interested in the golden ratio, I recommend a nice
little book called The Divine Proportion, by H.E. Huntley.
It's edited by Dover, and not hard to find, I think.

(*) http://www.guardian.co.uk/online/science/story/0,12450,875198,00.html

(**) The golden ratio is directly related to the pentagon,
and the Muslims have a much greater interest in the octagon,
AFAIK.

Interesting.

> does it have any aesthetic validity outside the area of Greek cultural influence

It must, for a very simple reason: we're animals too. Culture is a side-effect.

hhp

"Is the golden section useful in type design?
Not really..."
Gunnlaugur SE Briem DIXIT.

Please, check:
http://briem.ismennt.is/2/2.3.3a/2.3.3.03.golden.htm

http://www.amazon.com/exec/obidos/tg/detail/-/0881791164/qid=1043855642/sr=1-1/ref=sr_1_1/002-2435857-0064834?v=glance&s=books
This book puts in in great perspective. It really helped me get a grasp on it. Highly recomended.

Also this might help. http://world.std.com/~wij/glad/tschichold.html
But no promisees on this one. Get the book if you can.

cheers.

Thanks for all the links. I'm currently reading The elements of Typographic Style... almost to this section, hopefull that will help me to get a greater grasp on the usefullness of the golden section.




hildebrant.

Somebody already mentioned _The Divine proportion_ by Huntley

Another useful book: _The Geometry of Art and Life_ by Ghyka (http://www.amazon.com/exec/obidos/search-handle-form/103-7707676-6859048)

The Geometry of Art looks like a pretty good read.

The book I mention below is helpful visualizing a lot of the principles in geometry. Examples include the Barcelona Chair, the new VW Bug and Der Berufsphotographer poster by Tschichold among others.

Elam, Kimberly. Geometry of Design. Princeton Architectural Press. 2001.

You can see a flash animation demonstrating how to construct a rectangle conforming to a Golden Section at Textism <a href="http://www.textism.com/bucket/fibsquare.html"> here. </a>
The ratio of a golden section can be represented by a line bisected in such a way that the ratio of the smaller "piece" to the larger piece is equal to the ratio of the larger piece to the whole.

Sort of like this:
----------|------
(but don't count those hypens, it can't be shown with whole units - it's roughly 1:1.6, but really it's an irrational.

Thought this would be of interest to those of you following this thread... Helaman Ferguson is a renowned sculptor and friend of the family.

http://www.maa.org/mathland/mathtrek_10_21_02.html