Formed by extending the diagonal of a half square creating a ratio of 1:1.618
Last Spring I gave a keynote address at the WIA (Woodworking in America) design conference in Chicago. I was a little worried about the audio visual setup and had nightmares about standing in front of two hundred people doing shadow puppets. I also dreaded the question and answer period at the end of the talk. Shooting from the hip is not my strength. I always think of the best answers about 2:00 AM back in the hotel room. The presentation went smooth and the audio visuals flawless. Everyone enjoyed my story about the time I tripped and fell headfirst down a grand staircase in the Library of Congress. Then I spied Chris Schwarz, editor of Popular Woodworking magazine waving his hand.
“George, I was curious to know why you left out any references to the Golden Rectangle in your latest video on furniture design, was there a reason?”
I was outed. Felt like an atheist in a Baptist bible college. Armed with only a pair of dividers in case an angry mob rushed the stage I laid out my reasons for putting no stock in this mystical holy grail of proportions. I’ll give you the benefit of my answers that came to me at 2:00AM that night.
Like many woodworkers I started out firmly believing the many articles and books that sang the praises of the Golden Rectangle, also referred to as the Golden mean, Golden ratio, Fibonacci series, or Phi. It’s been used to explain the design for great buildings like the Parthenon, great works of art and masterful furniture designs from the 18th century. It has some unique mathematical properties and is often tied to patterns observed in nature like the graceful volute in a nautilus shell. In simple geometric terms it’s formed by dividing a square in half and using the diagonal from the half square extended out to form a rectangle with a ratio of 1:1.618….
My own doubts about this started creeping in when I began my own investigation of how period furniture was designed in the 18th century. I studied the standard texts like Thomas Chippendales Director and the writings of Sheraton, both English furniture designers who reflected the design approach from this important era. All arrows pointed back to architecture as the wellspring and stressed the importance of mastering the classic orders. The classic orders are an ancient architectural form consisting of a column and the support structure above it that were used in Greek and Roman temple construction.
Doric classic order, drawing by author
Many of those historical design books started with a series of engravings detailing the proportions in the orders. I did something a bit odd for a 21st century woodworker. I took heed of the advice, sat down at the dining room table and began to draw and explore the classic orders. Over the ensuing weeks the sun came out, the fog lifted and my understanding of furniture design blossomed. First, the physical act of drawing the orders awakened my mind to see proportions in a new and sharp focus. Secondly, the excitement of discovery drove me deeper into period architectural literature which filled in many gaps and changed my entire way of seeing. It was exhilarating to discover a design language that is rational and allowed me to pull back the veil on the furniture I so greatly loved. Dividers became an extension of my hand as I explored the elegant and simple proportions in the engravings in design books and photo copies of furniture pieces. This approach focussed on simple whole number proportions like 2:3, 3:5 that can be easily played with and manipulated much like you can arrange musical notes on a scale. Those old engravings from the design books with odd symbols and hieroglyphics now made sense. But the more confident and excited I became about what I was learning, the more doubts began to creep in about the magical Golden Rectangle. I couldn’t find any references to it in any of the design books from the pre-industrial time period. That’s nothing short of amazing as there were over 200 architecture titles and 20 more on furniture design published in England in the 18th century. To top it off, many of these authors had an ego like Donald Trump. They vied with each other to display their knowledge like a peacock to attract the richest benefactors, yet no mention of the Golden mean. None, nada, zip. Then I started trying to reconcile using the golden ratio with the techniques I had learned using dividers to lay out a design with simple whole number ratios. I couldn’t make it work. It always felt like a train wreck or like drawing a blue print and mixing inch and metric dimensions on the same drawing. I was exploring a lot of photos with dividers so I began revisiting some of the articles where the Golden rectangle was overlaid over a form as an illustration. What I frequently saw was the rectangle applied in a way that would make no sense to a furniture builder. If you overlay the golden rectangle over any complicated piece of furniture you are bound to find something that will coincide. As a furniture maker I’m looking for proportions that align with major structural boundaries that have meaning. I always focus on the inner or outer boundaries of the cases, top of the pediment etc. I don’t care if a proportion lines up with some random drawer, that’s not how you build furniture and is meaningless.
One of these is based on the golden ratio and the other two are based on simple whole number proportions of 3:5 and 5:8
In addition there are several whole number rectangles very close to the Golden rectangle as to be almost indistinguishable. On this graphic I laid out three proportional rectangles, can you tell which one is Golden? I’ll give you a hint. One was used in the design of the “Ark of the Covenant” the most holy piece of furniture in the ancient Jewish tabernacle circa 1400BC. It’s described in the book of Exodus chapter 25 verse 10. Note: if you have a little trouble with the dimensions spelled out in cubits, the ark is a wooden chest that is 27” high by 45” long, a rectangle with a simple ratio of 3:5.
The rectangle on top is 3:5, middle is the Golden rectangle or 1:1.618, and the bottom is 5:8. With all three it’s difficult to distinguish as they are so similar.
Here’s the bottom line for me. I’ll let you in on the “secret” of traditional design. Listen close. This is not rocket science. If you can count to ten on your fingers, you can learn to use simple proportions to guide your designs. Simple whole number proportions might not sound as sexy as the magical Golden rectangle, but they work without fuss to find elegant design solutions at the workbench.
George R. Walker
Design Matters 
Your selection of whole number ratios example actually give credence to the “golden” ratio, though only recently did “phi” click for me.
Look at the ratio of sequential numbers in the Fibonacci sequence (add previous 2 numbers :previous number) which is the basis for the “golden” ratio:
2:1 (rather crude approximation 2)
3:2 (somewhat close 1.5)
5:3 (really quite close approximation 1.67…)
8:5 (1.6 no real need to go beyond this approximation and just wear out your dividers)
13:8 (1.625)
21:13 (1.615…)
34:21 (1.619…)
55:34 (1.617…)
….
Several years ago I read an article that said pi=3 the reason being that for most scenarios that approximation was sufficient and much easier to use, though “heretical” for Mechanical Engineering magazine. I think that your whole numbers ratio is a similar simplification that is more than acceptable, and is what practical craftsmen and designers (rather than theoretical mathematicians) would use in their work.
Your example of the rectangles is telling. And it would be even more telling if the rectangles were not next to each other. Are the exact numerical ratios important? Will building a cabinet carcase with a 3.15 :1 or 2.9:1 ratio make it appear fundamentally different, particularly if it’s standing on it’s own.
David,
Proportions are important but I don’t hold any system up as perfect. I use this approach because it’s user freindly, time tested, and very easy to visualize after you have practiced with it for a while. I agree with you that there often is leeway from an exact proportion and still achieve the efffect you are after.
George
I absolutely agree with all of your article and have been saying the same thing myself for a long time. The golden rectangle is a bit of a myth – albeit an attractive one!
Too many articles on furniture making — praise the golden mean and repeat how it was supposedly used in Greek architecture and occurs throughout nature. Various writers and researchers have shown how misguided that approach is, and how the claims are often based on faulty methodology and assumptions (however, that’s not to say that there are not some interesting properties).
The idea of trying to use a single ratio for three dimensional objects that require considerations for function, context, surface decoration, lighting, etc — is very limiting. I think that what you’re showing in your articles and blog is such a better approach — thank you!
Terry
There is no reason whatsoever that a ‘single ratio’ is incompatible or ‘very limiting’ in three dimensional, in fact I would argue the exact opposite.
jl,
I meant that, for example using a very simplistic example, that I’d hate to have a commission for, say, a desk, where the client asked to use the golden ratio as much as possible – I would consider that an interesting challenge, but very limiting. Also that linear measurement ratios for sides of a rectangle is not the same thing as relationships between surfaces and volumes. There was once a Graham Blackburn article that mentioned the “golden solid” — a 3D version of the rectangle, and that didn’t seem very practical.
I was also thinking that I remembered a critique of Le Corbusier’s attempt to use the golden ratio and/or his Modulor, in the third dimension (that it was mostly used for facades) – the critic claiming that he failed, but I can’t readily find the reference for that. Not being an architect, I don’t know enough to argue that myself — it’s just a somewhat vague recollection.
I would find it quite odd if I were to get a request “where the client asked to use the golden ratio as much as possible”. While anything is possible I would wonder what the motivation was for such a request. If you ask me that would be a case of the ‘tail wagging the dog’. As an academic exercise I could understand such a project for students but proportioning systems are tools to assist in the designing of things. Such a request would be (almost) akin to being asked to design a piece only using metric measurements.
Perhaps your notion of a proportioning system is a bit too rigid. In practice there is quite a bit of room to maneuver with in them.
If you read the Modulor I &II, (your a glutton for punishment, lol), but Le Cobusier is actually trying to accomplish many different things (with the main goal being self-promotion). But I digress, let me end this with this image of a grid that is derived from the Modular http://www.google.com/imgres?imgurl=http://www.michael-robinett.com/isis/m-00.jpg&imgrefurl=http://www.michael-robinett.com/isis/mod-2.htm&h=365&w=361&sz=24&tbnid=xXbWVC39Py0A8M:&tbnh=121&tbnw=120&prev=/images%3Fq%3Dthe%2BModulor&usg=__2wKO1z7LN1FhXdQ2236GHUYmgMk=&ei=zQ5oS-zANI7aNZfA0IsG&sa=X&oi=image_result&resnum=6&ct=image&ved=0CBMQ9QEwBQ , now you really don’t believe that such a grid limits your ability to design a piece of furniture do you?
jl,
Of course it would be absurd to have a golden-ratio-only commission – but you said you believed that such a single-ratio system would be the exact opposite of “very limiting”. I do mean that single ratio – as shown in various articles on furniture design – whereas you might mean an extended progression of ratios like the Modulor
So maybe I’m not explaining myself properly, and am not using terms correctly. I agree a Modulor-based grid is just fine to use as the basis for a design – or one based on the Seven Orders would be just fine. My disagreement is with articles on furniture design that seem to preach that if you make a table top a golden rectangle (2D), then you’ve reached design nirvana. Also examples of trying to extend it into 3D – what Graham Blackburn called a “golden solid” in one of his articles – is of course a possibility, but somewhat limiting. Hope that makes sense…
Terry
I think I do understand the point your trying to make however I still don’t quite get your understanding of the term ‘single ratio’. If you draw the diagonal of a golden section rectangle and extend that diagonal to infinity any size rectangle you would draw that had that line as its diagonal would have the same ‘single ratio’ which would be the ratio of the golden section. The proportion of every golden section rectangle is the same ratio (single ratio) no matter what the actual size of the rectangle is.
Regarding Graham Blackburn I found the article you must have been referring to (with the ‘golden solid’)in the FWW archives. I didn’t read the whole thing nor do I wish to advocate for the term he seems to have coined but I did notice something that would indicate he isn’t all that hardcore about proportions
“A word of caution before applying the golden ratio as a design paradigm: Remember that form must follow function. Even the most sublimely proportioned piece of furniture can be a failure if it does not function because it is too small or too large or otherwise unable to be used comfortably. Practical considerations, therefore, must come first. In fact, most furniture designs require that you start with some given dimensions: A table must be a certain height, a cabinet may have to fit a particular space, or a bookcase may require a fixed number of shelves. But almost certainly you will be left with many other decisions regarding dimensions to which you can apply this proportion. It will be worth the effort to see whether the golden ratio might work for these other elements. Deciding on dimensions by eye alone—or worse, on the basis of the lumber that is conveniently at hand—is a less certain way of achieving a well balanced, nicely proportioned piece.”
I find this to be rather sound advice.
JL,
I agree that the Blackburn text you quote is good advice. But elsewhere in the article are these two little gems ( in “A Guide to Good Design” in Fine Woodworking; Jan/Feb 2004):
“Chief among the many paradigms that designers have used — and continue to use — to ensure balance and good proportions in furniture design is the golden ratio.” …and…
“But even the attempt to approach perfection (which may be defined as measurements that correspond precisely to a system like the golden ratio) is virtually guaranteed to produce a better result than designing with no regard for any such paradigm. Even if you are close to perfect proportions, the eye is inclined to accommodate slight imperfections and fill in the gaps. Don’t think that everyhing has to fit the formula exactly.”
I fully agree that most can profit by using some paradigm, but I disagree with his emphasis on the golden ratio being perfection. You might interpret this as his saying that adherence to whatever system you choose is defined as perfection, not the actual system itself — but the less-experienced would likely get the impression that the golden ratio is considered perfection.
I understand that a single ratio (e.g. 1 : 1.618) – can be scaled to any size. The article also shows a method for a progression of drawer heights using the golden ratio. Certainly that is one solution, but I believe it is rather limited, and rarely if ever used in most dresser design. However, it is interesting because it shows the use of a system that determines dimension progression without imposing the golden ratio for the drawer fronts themselves.
well now after all that we are in agreement (more or less) if you ask me… lol, funny how things work out (sometimes)…
Another awesome entry George.
Isn’t it funny that the only poster supporting the Golden Rectangle theory felt the need to hide his identity?
Just to throw another wrench into the works, but I’d be interested in hearing your opinion in regards to comparing the golden rectangle to the silver rectangle? From what I understand, the difference is that instead of using the diagonal of H x 1/2W, you use the full diagonal as the radius to draw the arc down. The neat thing is that with these rectangles, when cut in half, the proportions remain the same. A-series paper sizes follow this. Take a piece of A4 paper, cut it in half, and you have a piece of A5, and the proportions remain the same. That’s another one that’s fun to play with.
Dave
I am not familiar with the term ‘silver rectangle’ when it comes to using the arc of the diagonal of a square. Where I come from it is referred to the “square root of 2” rectangle (which, if you remember your Pythagoras theorem, describes the diagonal mathematically). Do you remember where you first heard the term?
Dave,
That’s an interesting point you raise. I have not heard of this refered to as a silver rectangle. A rectangle formed from the diagonal of a square, unlike the golden rectangle does have some strong references in historical design literature. Palladio listed it as one of a handfull of shapes favored for sizing a room along with a square and other simple whole number rectangles. It’s significant as Palladio was the major source for English design in the 18th and early 19th century and so impacted American designers also. I have to admit I have not investigated how it might have been used by furniture makers. I’d be interested to hear from anyone who has explored this.
George
George
I assume you have certain desired results (at least generally) regarding these posts, I would be interested in hearing your take on the responses you have gotten from this post. Feel free to email them if you feel its more appropriate.
I first heard the reference to “silver rectangle” when I lived in Europe for awhile. One of the first things I had to adjust to was using A4 sized paper. A little investigation into the format (out of my insatiable curiosity) turned up the term. I was initially fascinated (OK, I’ll admit, I still am) by the fact that the proportions remain the same when you cut it in half. That’s just cool.