Thursday, February 5, 2009

Are We Golden? Part III

I think that the most interesting designs combine in some relationship order and variety. If there is too much order, I think we find the design boring, and if there is too much variety, it tends to engender some sense of confusion. Some sweet spot in the tension between order and variety needs to be found - I wonder if that sweet spot might not also be a Golden Mean of sorts? That is to say, in terms of the polarities I just mentioned (just to give two) these need not be in some sort of simple 50/50 balance. The see-saw won't move unless there is some dynamism, some weight shifting.

The use of the Golden Mean in classic Western architecture was extremely numerous, and more information on that is but a couple of clicks away on a search engine. There are many sites devoted to that topic. An influential figure in recent times in terms of using the Golden Ratio was the 20th century's Modernist Architect Le Courbusier, whom I'm not a fan of otherwise - he developed, in his 1948 book, "Le Modulor", a proportioning system based on extrapolating from an idealized human (male) figure, using phi. His architecture, though employing phi as a proportioning device, is not attractive to me in the slightest, so it goes to show that using phi is no guarantee of success in design.

So, let's look at an example of how phi could be used as a design tool, exploring with our trusty compass and straightedge. Let's say we are designing a tall chest of drawers, and want to find some progression of drawer sizes - one potential application. This method produces a graduated series of boxes, nearly all of which incorporate the phi ratio of 1.6180339887... in some way.

With the chest of drawers design as an example, it makes sense to make the lower drawers larger, since bulkier and heavier things might be best kept low, and the greater mass down low gives the piece an 'anchored' feeling, some would say. So, in short, we are looking for a progression of drawer sizes, starting with the largest at the bottom.

We start with a Golden Rectangle, ABCD: This Golden Rectangle could be considered the overall form for our chest of drawers, but it need not be so. We are trying to produce a series of graduated drawer sizes, that's all. First let's divide the Golden Rectangle in half:This produces point E on the line C~D, and point F in the center. Next we will increase the 'contents' of our Golden Rectangle box from one to two squares, the second exactly the same size as the first:So, you can see that BCHI is the second square, same size as the original generating square. Now take point E as the center for the compass and swing the half circle arc, radius ED:It can be seen that this arc intersects our lines that form the edges of our two squares, at the points J and K. Next, connect J and K with a straight line, projecting it out to the edges of the Golden Rectangle:We're going to use point 'C' as a vanishing point, as the method I'm showing also has applications for creating a perspective view (another topic altogether, which I'll leave aside for the moment). The first major line from our vanishing point, moving from C to A, intersects line JK at L:From L, mark out a line parallel to AD, namely MN. This is the largest box, the lowest drawer in our series.

Now connect C to N, through point J:Line HI gives the second division, thus I~N is the height of our second drawer. Now connect C to K, projecting out to the side of the rectangle to give point P:Thus line O~P is formed. At point K, draw another parallel to line AD, giving points Q and R. Now you can see we have created our next two drawer heights, namely P~I and R~P. Simply repeat the above process a few more times, and you will generate the rest of the drawer heights:Obviously, these drawers get smaller and smaller, actually approaching what is termed a 'limit' of zero drawer height. Since there is little point in proceeding towards a drawer of zero height, and since I do not imagine needing an infinite number of drawers, I stopped the process after a few more horizontal lines were generated. Some of those smaller heights, while unsuitable for actual drawer heights, might well be useful to give sizes for components in the chest, frame rails, legs, thickness of wood dividers, etc.

As a final step in this process, I have pulled the side pair of vertical lines, AB and the JK line, out to the side to show how the boxes generated relate: I have indicated that the lower box has proportions of 1.0 to phi, and the next box but one up from that is a square, 1.0 x 1.0. The rest of the boxes all have a Golden Ratio present in some form (sometimes it is 1/ø, sometimes √ø, etc.) - I'll leave it to interested readers to discover what some of those size relationships might be.

The use of phi in design, it should be noted is more than some sort of 'plug-and-play' procedure. It is but one tool to use in considering the size relationships, both of the piece as a whole, and of the components within the piece. I think any approach to design, as mentioned at the start of this post, should have some 'tension between opposites' inherent - between order and variety, between rectilinear and curvilinear, between formal and playful, stable and lively, etc. The given parameters of the situation, such as setting, client tastes, the tools one has to work with, all bear upon the process, and may tip the balance one way or another. Tipping it too far to one extreme though, rarely gives an entirely pleasing result.

Getting back to the design of Client C's tansu, I chose to use the phi ratio in a number of ways, one of which was as a simple progression of stair tread lengths. Since this tansu was not intended as a functional staircase, where predictable stair rise and run lengths are of paramount importance, I felt that I could play with the arrangement a bit. More so than that, I was looking for a way to work from the maximum height available, and down to the base of the window, so that all the boxes would fit and I would achieve the storage requirements the client was after. A phi progression turned out to give me the results I seeking - I made the uppermost 'tread' around 6" long, and each subsequent tread, measured from the left side of the cabinet, was a multiple of ø times longer, vis:

6" - 9.7"- 15.7" - 25.4" - 41.125" - 66.5" - 107.625".

You can see that these last two numbers, if you subtract the smaller from the larger, 107.625" - 66.5", gives 41.125", the length of the lowest shelf, which is the top of the third box in the overall assembly of three boxes. As far as phi is concerned, if the overall length of the tansu is 1.0, then the length of the tans from the left side to the window frame (left side) is 0.618 of that length.
In the above picture of client C's dining room, you can see that there are a few more pieces in the suite of furniture, which I made prior to the kandan dansu. I will talk about those separately in upcoming posts. I thought I'd show a couple more images of the tansu to end here - there are other places where phi might have played a role in this piece.

I used Maple for the drawer sides, as I recall:

I'm not quite done - not nearly done - with the topic of phi, so I'm planning a part IV and possibly V in this series. Hang in there! Hopefully I haven't put anyone to sleep yet.


  1. Hi Chris,
    Great thread here. Just to let you know, I'm not seeing any of the graphics in this latest post (Golden Rule III). Guys like me need the pictures!


  2. Oh - thanks for letting my know. I posted this first this morning and it ended up previous to the post before this, so I deleted it and well, I guess I need to repost completely.

    Thanks! This will be corrected soon.


  3. Well, I reloaded the images - can everyone see them now?

    I'm a newbie blogger with a dial-up connection, so I am learning as I go - please bear with me. I have now learned that I can't start writing a blog entry on a particular day, set it to one side and do another separate entry a day later - and then try and post the original entry - the software posts it by the date it was begun on.

  4. Yes, all visible now, thanks. Computers can be sooo stupid, eh?

  5. Chris,
    I don't think I'd worry about Le Corbusier - by most accounts he kept changing his Modulor - he failed to apply it very well, relied on others to supply some solutions, and drew buildings and then tried to superimpose Modular measurements after the fact. Also, there is no evidence that he ever figured out how to use it in three dimensions - his examples were mostly for 2D facades. I can't claim to confirm all that -- it's just what I've read!

    What I like about what you've done is that you've tried to show how to actually apply a geometric construction for a real project. Some designers are solely intuitive, and some are purely analytical. It would be interesting if each could learn a bit from the other end of the spectrum.

    By the way - what works for tansu steps might not work for a more subtle progression of, say, dresser drawers. I once found that the Hambidge rectangles (easily found in various books and websites) gave a more subtle progression (of several possible alternatives). Just a thought! Thanks for sharing your thoughts on this!

  6. Great comment Terry. I am familiar with the Hambridge rectangle progression, and yes, it's but one of several alternatives for proportioning parts. There's no way I have enough time or space to elucidate every sort of possibility - drawing a pentagon, for instance can be done by at least half a dozen methods. Also, I didn't use the method I showed for obtaining a progression of drawer heights on my tansu - I wanted to show an application, one possibility of many, for phi as a design tool. On the tansu, I simply used a phi multiplier to get a series of lengths, and the height of the steps remained constant.



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