Got Phi?

[Jim Koepke]

No, that is not a typo of Pie on this Thanksgiving Day.

Phi is the 'Golden Ratio' of approximately 1:1.618.

Lee-Valley now has rulers to do the calculations for us.

http://www.leevalley.com/US/wood/page.aspx?p=57625&cat=1,43513

There is also a relation ship with Fibonacci numbers. As the numbers get larger, the ratio of two numbers in sequence is also the 'Golden Ratio'.

168261

jtk

[Andrew Arndts]

Save money, Here is a how to vid to make your own cheaply

http://www.youtube.com/watch?v=yLTja49yKFk

[Andres Sender]

It's worth noting that the golden ratio was not the only proportion considered beautiful by those who, through the course of history, have made beautiful things. If one has the time to wade through them, these two articles will put it in perspective:

http://en.wikipedia.org/wiki/Golden_ratio

http://en.wikipedia.org/wiki/Proportion_%28architecture%29

[Frank Drew]

The couple of times I tried to use the Golden Mean when designing upright pieces, I didn't care for the look -- the proportions weren't as pleasing as I assumed they might be.

[glenn bradley]

I use the ratio as a general guide and get some nice results, I have done "what if" drawings using the strict ratios and agree that in certain scale and in certain relationships the feeling is awkward. In general, phi along with the Hambridge Progression supply a great relational dimensioning point to start designing from.

[John Shuk]

I have an app for that on my Ipod. It can help when thinking about proportion.

[Jon van der Linden]

The couple of times I tried to use the Golden Mean when designing upright pieces, I didn't care for the look -- the proportions weren't as pleasing as I assumed they might be.

This is the main problem with using numbers to design things. Numbers can have all sorts of interesting relationships without being able to produce anything visually pleasing from them.

Developing an eye takes years of constant practice. Numbers can be used as guides, but really it's the eye and the mind that make things work. If you're using numbers, they have to be for the same item, style, and character. You can't make something different and expect proportions to apply in the same way.

Take a look at Greek Doric temples as an example. The talk about the Golden Mean comes from the facade of the Parthenon on the Acropolis in Athens. However there is a huge range of ratios used in the Doric style. Just look at some of the Greek temples in Sicily for example, the proportions and character are totally different and they still work.

[Frank Drew]

Numbers can have all sorts of interesting relationships without being able to produce anything visually pleasing from them.



Quite true; in my case, when the Golden Mean didn't work during the drawing stage, I decided that was enough experimentation and went back to proportions I thought looked good.

[Andres Sender]

The talk about the Golden Mean comes from the facade of the Acropolis.

Even the claims about the Acropolis may not be true, see the above-linked Wikipedia article.

[Jon van der Linden]

Even the claims about the Acropolis may not be true, see the above-linked Wikipedia article.

Those are just claims. The wiki article doesn't tell you much if anything.

What is a fact and known is that the Parthenon uses very sophisticated design to compensate for optical illusions. Which is to say that the columns do not have equal spacing in some places precisely to make the spacing appear equal. The base, angle of columns, the roof, virtually every part is curved or angled to make it appear straight and perpendicular.

The idea that you can put a rectangle over the facade and say that is how they came up with the shape is a bit of a joke really when you consider the level of sophistication that is known to have gone into the design. It's a theory that appeals to the ignorant because they can understand it, but as you noted there's not one shred of evidence for it whatsoever.

The idea of using proportions and that they are somehow fixed as rules comes from classical architecture. It is important to understand that all of these proportions had their critics. There was considerable debate among great architects as to the correctness of various ratios proposed by Palladio and other equally gifted talents. The Greeks gave themselves considerably more leeway and produced a larger range of character with the same elements.

All of this doesn't mean that certain ratios aren't useful, or that progressions like that found in a nautilus shell aren't beautiful. It's what you do with that knowledge that counts.

[Federico Mena Quintero]

I'll point to this for anyone interested in proportioning and design:
http://people.gnome.org/~federico/news-2010-06.html#28

(Each part links to a PDF for that chapter.) The parts on proportions, universal scaling, patterns, etc. are tremendously interesting.

Nikos Salingaros is one of the mathematicians/architects that, along with Christopher Alexander, has been redefining the foundations of design, mainly for achitecture and urbanism. But their work applies to any manufactured object; proportioning and design considerations apply to everything we make, not just buildings.

[Federico Mena Quintero]

Among those lectures, numbers 1, 2, and 3 talk about proportioning in general.

Lecture 9 talks about moldings and their structural / ornamental function.

Frank Drew says that the Golden Mean for upright pieces didn't look right for him. I think the first lectures provide an answer to him - it's not the overall proportions of the piece, but rather the sequence of scales in the vertical and horizontal axes, separately. You wouldn't build a grandfather clock with the Golden Mean (that would be a pretty fat clock), but instead you would ensure that the vertical and horizontal dimensions of components have a progression of scales.

[Tony Zaffuto]

Back to Jim's original post, I bought one of the Phi rules, and for the couple of bucks it cost, I would definitely buy again. Quick and easy to use and my use is more for measuring/studying existing pieces/drawers than designing anything I might build. Got the rule a week or so again, but whenever I see a piece that is pleasing to my eye, I like to measure and try to understand what it is that makes it pleasing. The rule makes it quick to do a check.

Along this line, a book that I have found indispensable over the past decade (at least to me) is Taunton Press's "Practical Design, Solutions & Strategies".

Finally, I'm also somewhat either at an advantage or major disadvantage in that my oldest daughter is an architect, having studied the "artsy craftsy" part of architecture (in Rome nonetheless). I get criticism whether I ask for it or not. She now lives a hundred odd miles away, but it was amusing have dinner conversations of Fibonacci or Hambridge (sp.) progressions, etc. In terms of furniture likes/dislikes, though we agree--Shaker and then Craftsmen style. Sorry for going a bit off topic!

[James Taglienti]

some folks get pretty daffy with it, applying it to the stock market and stuff.

[mike holden]

Try "The Golden Ratio" by Mario Livio, for a complete discussion of the golden ratio and its relationship to plant and animal life, and it's necessity for building the pyramids!
The book is quite readable and I am not a math-head.

FWIW, I believe that craftsmen were taught to use a 5:8 ratio in place of the golden section as it was easier to remember and mark out.

Mike

[Pam Niedermayer]

You may want to read/watch George Walker on this issue before tooling up and basing everything on this golden ratio.

Pam

PS He says, basically, it's all silly and that everything works better with whole number ratios.

[mike holden]

You may want to read/watch George Walker on this issue before tooling up and basing everything on this golden ratio.

Pam

PS He says, basically, it's all silly and that everything works better with whole number ratios.

True, but that is *HIS* opinion, the guys that built the furniture he based that feeling on, *DID* believe in the golden ratio. See any of the 18th century builders guides.

Mike

[Pam Niedermayer]

True, but that is *HIS* opinion, the guys that built the furniture he based that feeling on, *DID* believe in the golden ratio. See any of the 18th century builders guides.

I wouldn't characterize Walker's work as based on "feelings." Plus, I love the look of his furniture, as well as his methodology, golden ratios aside. Also a good idea to remember that we can be wrong, or not exactly right, for thousands of years; so yes, there's a lot written about the golden ratio. Another good watch is the Nova show on rebuilding the Parthenon, in which they reexamine how it was built.

Pam

[Rick Markham]

Hi gang, it's been awhile... I've been hiding in the turner's forum. The golden ratio, is seen in many "beautiful" natural forms, some chili peppers/fruits etc. Nature doesn't follow it completely but in some forms it does. It works really well with natural objects, and particularly roundish things. Not too many things in nature are truly flat and square. I have no doubt that it's been taken into account in some very beautiful masterpieces of furniture. Heck nature doesn't follow it in every case, so every piece of furniture isn't going to look right strictly following it. Plus our ideas of aesthetics and form/function are always changing. Furniture likes to follow certain proportions, that for whatever, reason's have evolved to suit us, when something gets out of those proportions we expect, it tends to look awkward or strange.

Wikipedia, is really an absolutely horrible resource to do research from. (it is an unsubstantiated resource) An advanced search on google with sites ending in .edu usually guarantees that there has to be well documented resource (with bibliography) to "back up" any information presented in the article. ;)

[Don McConnell]

True, but that is *HIS* opinion, the guys that built the furniture he based that feeling on, *DID* believe in the golden ratio. See any of the 18th century builders guides.

Mike

Mike,

I would like you to point out a single example of an 18th century architect or published furniture designer which mentions the "golden ratio." I've looked at a lot of them and haven't found a single mention yet. Lots of discussion of the classical orders (which, claims to the contrary, are not based on the golden ratio), as well as squares, and various whole number ratios -- but not the golden meaningless.

Even most honest proponents of the golden ratio acknowledge that there is no solid evidence for any serious attempt to use it until well into the 19th century. And while many claims are made about the antiquity of "phi" representing this concept, the truth is that this usage wasn't established until the early part of the 20th century. I don't have the reference on this available to me at present, but it is traceable to a particular person in a specific publication.

Don McConnell
Eureka Springs, AR
(temporarily in Athens OH)

[Adam Cherubini]

Mike,

I would like you to point out a single example of an 18th century architect or published furniture designer which mentions the "golden ratio." I've looked at a lot of them and haven't found a single mention yet.

Don, I wish you'd soften your stance on this issue. There's quite a bit we believe to be true that isn't documented. And your statement may lead people to believe there are great documentary sources for furniture building from the 18th century. There are not.

In my opinion, one would need to look at individual pieces of furniture or architecture to see if the golden mean was used or not. The problems with ths approach are many:

1) How does one account for slight variations or errors? golden section is .618 (or it's inverse 1.618 - I know cool, right?) 5/8 is .625. So if you measure something at .62, is it golden section or 5/8? What was the builders intention?

2) You have to measure things the builder would have measured. I sincerely doubt cabinetmakers accurately measured moldings such that their addition was accounted for in some proportion rule. More likely from what I see is that they may have divided a case side with a proportion rule- for a drawer divider say, without considering the effect of a molding. And this makes sense because he's likely laying out the guts long before he cuts the moldings.

Ditto if he's doing a length and width sort of thing. You'd find that in a single element of the design, but when the moldings are added, maybe it wouldn't "fit" the proportion rule.

3) Lastly, I have seen things that led me to believe they were squeezing designs into the wood they had. I've seen little bits of live edges etc. So again, maybe his intention was for 14", but his board was only 13-7/8" and he had to square up some edges which cost him a lttle more etc etc.

When Herculaneum was discovered, folks went out of their way to incorporate what they thought was "classical". They got some stuff wrong. But I wouldn't be surpirsed if they thought golden section, which is arguably easier to use than whole number ratios, wasn't used. It would e fun to look in Pompeii to see if there's ny evidence there. There are borders around frescoes on walls that wouldn't have been limited to the size of the room. Ditto for mosaic floor patterns.

Personally, I'm not convinced they did use it, Don. But I don't think the lack of a mention of it in scare 18th c texts mean it wasn't used. We just don't know for sure.

100% honest- I laid proportion rule templates over existing museum pieces. Some of them fit into the golden section-ISH proprotions. I thought using a template would be helpful to woodworkers and I still think so. But I also realize that with these sorts of things, you can find them where you look for them. I couldn't look anyone in the eye and say I know 18th c woodworkers used golden section. Maybe they did and maybe they didn't. But that shouldn't stop modern woodworkers form using it. Period woodworkers didn't use white glue either but most guys use that without explanation.

Adam

[Don McConnell]

Adam,

I'm not quite sure what you would like for me to soften. I was responding to an implied claim that the golden ratio was commonly mentioned in 18th century builder's texts, and I was requesting that evidence be provided for that claim. I've looked at a number of those (including some from the 17th century), as well as texts on architecture and furniture design and have yet to find any such mention. I don't think it's an unreasonable request, and I've made it before in similar situations. In each case, my request has been met with a deafening silence.

Also note, that despite what you imply, I did not claim that the lack of such textual evidence "proves" that they didn't use it. What I do think, though, is that this lack of textual evidence makes it very difficult to "authoritatively" assert that they did.

Why? Well, largely because of the difficulties you pointed out, yourself, with inferring such intent/usage from period furniture or architectural examples. My experience with published claims of finding such usage in period works is that they are often based on arbitrary or meaningless reference points. In addition, the "golden" ratio is typically used in such an elastic manner as to undercut the very sense of precision implied by carrying it out to three decimal places as well as any sense that it's some kind of "special" ratio.

In fact, since you've decided to join this discussion, I'd like to illustrate this by referring to the drawings of cornice mouldings (from Moxon, etc.) in which you illustrated finding the golden ratio in one of your articles. Some time back, I studied them in some detail. In several of the drawings, the reference points you used had little, if anything, to do with the way in which such mouldings are designed. In other words, they didn't correspond with meaningful transitions between elements of the mouldings. Secondly, the orientation and placement of each rectangle/ratio varied noticeably from drawing to drawing. Finally, upon checking the actual ratio of each finding, it became very evident that this, too, varied from drawing to drawing. Based on all this, I find it very difficult to infer that the authors of those drawings intended to use the golden ratio in their creation.

Likely, you will disagree with this assessment. Which only helps to underscore the difficulty of inferring the intent of using the golden ratio from existing period objects or drawings.To my mind, the only situation in which we can authoritatively assert such intent is if there is textual evidence indicating it to be so. Which leads to the issue of the quantity of period sources. While I agree that there is very little 18th century textual detailed descriptions of trade practices, there is a little more textual information regarding furniture design and much more regarding architectural design and drawing. In fact, there is so much of the latter type of textual evidence, that it's misleading, in my estimation, to imply there is a paucity. And, I don't think it's unreasonable to assume that if the the usage of the golden ratio in design was common at the time that the concept would explicitly appear, with some regularity, in such works. I haven't found this to be the case, yet.

When it comes to whole number ratios, the texts, including at least one on furniture design, often include methods of dividing a given line into any number of equal parts through the use of evenly spaced parallel lines and intersecting sloping lines. These were typically produced through the use of dividers, parallel rules, squares and straight edges, and it is easy to see how such constructions would easily lend themselves to the creation of whole number ratios using a pair of dividers. While it may not be likely that workmen at the bench used such constructions and tools. it isn't much of a stretch from this to the workshop method of dividing a given line into any number of equal segment through the use of a rule (canted) and a square. This plus a pair of dividers would also lend itself readily to creating whole number ratios. Not to mention the sector, mentioned many times in design/architectural texts, which included the double "line of lines" which was elegantly suitable for dividing a line into any number of equal segments. It would be a very simple and direct process to create any whole number ratio using the sector and a pair of dividers.

I'm not claiming that workmen at the bench used sectors or parallel rules, but I find it puzzling to think the instruments being used by architects and designers make provision for all kinds of computations and operations, but none for the golden ratio, while being asked to assume that workmen of the same period were making common use of the concept.

Incidentally, I would be interested in an explanation of how the golden ratio would be easier to use than whole number ratios.

Don McConnell
Eureka Springs, AR
(temporarily in Athens, OH)

[Pam Niedermayer]

I think it would be cool to closely examine story sticks from the past, if available. They should tell us a lot. Also, whole numbers are much easier. For example, if I use a small enough number it's a piece of cake to prove that everyone used whole numbers. That's not my intent, just saying.

Pam

[Rick Markham]

Ok, there is a common misconception about Phi, that it is exactly 1.618, it is not... it is a continuous non repeating decimal. Euclid is the first one in mathematics to describe Phi...

If you don't believe me... here are the mathematic courtesy of Wolfram... with references ;)

http://mathworld.wolfram.com/GoldenRatio.html' (Weisstein, Eric W. "Golden Ratio." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GoldenRatio.html)

There are some great points there, if you take the time to read through it. :D

[Adam Cherubini]

I'm not quite sure what you would like for me to soften.

I know you are skeptical about the "golden meaningless" as you put it. All I'm saying is, they may or may not have used it.

Otherwise, I agree with everything you've written. Reference points are important, I tried to say the same thing earlier.

Here's my take on easier:
Imagine we are dividing a carcass side to put a drawer divider centered on golden section (hereafter GS). To do this you simply set your dividers for ANY integer increment of the carcass side.

Suppose you have a carcass side that's 41" tall. You set your crappy iron dividers to roughly 2" and come up with 21 divider movements perfectly (so the dividers are 1.95").

To find Golden section, you make a triangle with its horizontal leg 1 movement and it's vertical, 1/2. Swing the 1/2 to the hypotenuse. The part of the hypotenuse left over is GS.

Set your dividers to that and walk off 21 movements and you have GS.

==========
With any whole number proportion, you'd really need to:

1) set your dividers to some whole multiplier of the proportion (so that the fraction would be a whole number)
2) or you'd need to cross multiple and divide (which they were trained to do), but then you'd be left with a fraction a movement. In this case, the use of divders would be a pointless waste of time since you'd be better off simply measuring, then applying the "Rule of threes" (cross mulitplying and dividing).

For the example above, if you were after 5/8, it would SO much easier if your dividers were set to 16, 24, or 32 movements, so that the drawer divider position could be established in 10, 15, or 20 movements. That can be frustratingly time consuming to divide a long space with that many movements to some specific number.

Now what may have happened historically (and what happens to me every time I do this) is I loose patience setting my dividers perfectly so I bump them a little and say "That's good enough". That tiny error is enough to throw off the proportion (easily confusing 5/8 with GS).

I'm not sure they used GS or did what I described above. It would be fun to find a little triangle inside some carcass side somewhere. I wouldn't be surprised to find one. Some pieces I recently examined suggested to me they simply eyeballed the design. No proportions or measurements made any sense with what I saw.

Adam

[Jim Koepke]

Incidentally, I would be interested in an explanation of how the golden ratio would be easier to use than whole number ratios.

This question seems to imply that the golden ratio and whole numbers are not or can not be the same thing.

In one response a poster mentioned a preference to use whole numbers or a 5/8 ratio.

8/5 = 1.6

As was mentioned in my post that started this thread, the golden ratio is related to the Fibonacci sequence.

Starting here with zero: 0-1-1-2-3-5-8-13-21-34-55...

As the numbers get larger, the ratio of any two consecutive numbers is close to 1.6.

Here is a site discussing the Fibonacci sequence and the golden ratio including how it can even work with random numbers:

http://www.mathsisfun.com/numbers/fibonacci-sequence.html

jtk

[Terry O'Donnell]

Great topic! I'll dare to list a few counter arguments (to the notion that the Golden Section is the key to all things) by various writers:

I agree that the golden section may have been used in buildings, even if we don't have a record of it -- but Marcus Frings' article in the Nexus Journal surveys all known architectural texts, and claimed that the golden section (GS) wasn't mentioned until about 1850 (by Zeising and Fechner). The Greeks discussed the mathematical concept of phi -- but we don't have a record of them mentioning it in relation to architecture.

George Markovsky wrote an article in the College Mathematics Journal (1992), "Misconceptions about the Golden Section" - wherein he claims that recent research has shown that phi was not used in many places it has often been claimed to be, e.g. the Parthenon, the Great Pyramid, etc. (I agree that Livio's book contains great examples of where it does occur).

Robin Evans, in his book "The Projective Cast", claims that no architect has successfully demonstrated how to use the GS in 3 dimensions, and that those whom we know tried (like Le Corbusier, who used it mainly for facades) -- failed.

Stanley Abercrombie, in his book Architecture as Art, examined the four main categories of analogies to support the GS (e.g. music, arithmetic, nature, etc. ), and wrote that it failed all four. However he concluded that intelligent use of any proportioning system -- is much to be preferred over using none at all.

All that's just to say that some go overboard in making claims. I don't claim to know that the above writers are any more correct than the completely pro-GS writers, but they do present an interesting perspective.

Terry

[Don McConnell]

Adam,

Sorry I've been so slow responding to the issue of the relative ease of use between the golden mean and whole number ratios. The pause has been necessitated, partially, by needing to drive back to Arkansas. But, also because I suspect further discussion may not be very helpful to (m)any people and is necessarily quite tedious. But, in the final analysis, I'm concerned that leaving the discussion as it stands may leave people with an unhelpful and inaccurate impression regarding the use of whole number ratios.

Before getting into the use of whole number ratios, though, I'd like to address a couple of your descriptions concerning the layout of the golden mean. Yes, it can be done by randomly setting your dividers to any "integer increment" of the carcass side, but you seem to gloss over the fact that it may, very likely, take several tries to get your dividers set so that the number of movements comes out completely even. Further, you seem to be indifferent to the additive error which will come into play while walking off the golden mean over so many movements with your "crappy" and, apparently, quite small dividers. This becomes especially important if you decide to be somewhat indifferent to accurately setting your dividers for this operation. Also, you glide past finding half of the base line as if that doesn't take a separate operation.

As to your description of the difficulties with using whole number ratios, they are based on approaches which are far from sensible. Specifically, randomly setting dividers and "walking" them down the full length of the starting dimension is not where I would start. So the fact that, when using this approach, I would further have to randomly work out the number of movements of my dividers to some multiple of 5 or 8 (depending on whether I was using 3:5 or 5:8 for example) is a complete red herring. The same goes for the assumption that I would begin by using some form of cross multiplication or division - especially if it results in some fraction of a movement. The latter, especially, is a fairly ludicrous assumption.

Having said that, though, it is good to be aware when arithmetical operations can be useful. For example, if one is wanting to work with a 5:8 ratio, any length which is a whole inch dimension can be easily arrived at in one's head. In the instance, for example, of the 41 inches you posit, one eighth of this can be easily arrived at because eight will go into 41 five times, with a remainder of one inch. An eighth of one inch is ... well, 1/8". So, one-eighth of 41 is 5 1/8 inches, to which you can easily set your dividers using a rule, and you can then very directly establish the 5:8 ratio by walking your dividers down the length five times. Can't imagine it being much simpler.

Of course, though, this doesn't work so well if using a 3:5 ratio or if the given dimension is something like 41 7/16 inches. In which case, geometric operations are usually the better way to go. If using the 5:8 ratio, the given dimension can be divided into eight equal segments by repeatedly bisecting to arrive at 1/2, then 1/4 and, finally, 1/8 of the whole. This can be done in a number of ways, but the use of trammels (if working full size) and/or compasses to do this was, I believe, very common practice in the 18th century. And, actually, the number of separate operations could be minimized by bisecting the whole, then only bisecting one of the halves and then bisecting only one of the resulting fourths (next to the original half bisection) to arrive at the 5:8 ratio. No need of further operations to accomplish it.

If working with other whole number ratios, 3:5 for example, one needs to find other ways to divide any line of a given length into any number of equal parts. For this portion of my post, I've decided to stay with the 5:8 ratio for consistency, but the same approach works for dividing a line into 3, 5, 7, 9 &c. parts just as well. There are a number of ways to approach this task, some of them quite intricate, but I've decided to focus on only a couple of them which are relatively straightforward and lend themselves to use in the shop or at the drawing table.

For the first method, I've adapted a drawing and description from _A complete Treatise of Practical Mathematics_ by John MacGregor, published in 1792. The section is entitled "To divide a line AB into any number of equal parts":

http://planemaker.com/photos/5to8ratio-1.jpg

The line AB is the starting dimension, and the line BC is drawn from B at any angle. Then, using a pair of dividers, one establishes a parallel line from A to D (a parallel rule could be used if one had it). Finally, one sets the dividers at any given length and walks off 7 divisions on BC and AD (starting from B and A respectively) and connects these points to intersect the original line, creating eight equal segments. The fifth one will give the 5:8 ratio and one's dividers can be set to that if working to scale (no need if working full size). In the example I've drawn up, that divider setting would be used for eight movements (1/8 scale, originally) to find the full-sized 5:8 ratio. No partial movements. No messy math.

Another approach depends on the use of a rule and square, and is one I've used in the shop many times:

http://planemaker.com/photos/5to8ratio-2.jpg

The horizontal line is 41 inches drawn to 1/8 scale (originally), and I've indicated using a rule, at 3/4" increments to divide the resulting distance into 8 equal parts. Actually, one would only need to drop the line from the 3 3/4" gradation on the rule in order to arrive at the 5:8 ratio, which would then need to be walked off eight times to arrive at the full-sized ratio value.

I've done these at scale in order to, hopefully, be consistent with the forum format, but these operations could also be carried out full-sized with normal rules, straight-edges (which you'll have if drawing full-size) as well as trammels and average sized dividers. Setting dividers can be a bit tedious, but with these approaches, when critical, they are being set to an already established point. No need to randomly adjust them and use them over 20 plus movements in order to find out if they're set accurately.

I know this has been tedious, but hope it has been somewhat helpful to at least one or two people.

Don McConnell
Eureka Springs, AR

[Adam Cherubini]

Thanks Don. I'll give this a try and hope others do the same.

Adam