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Divine Proportion

Updated: Jan 8, 2021

As far as the laws of mathematics refer to reality, they are not certain;

as far as they are certain, they do not refer to reality.

— Albert Einstein

We humans excel at explaining things. Even if we’re utterly wrong — all of cosmology before Copernicus, for instance — we’ll come up with something that sounds good. And, once an explanation sounds good we get rather attached to it, as Giordano Bruno discovered.

You probably learned about the Golden Ratio/Section/Mean in geometry class like I did. In Euclid’s words: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.” Two lengths in this proportion are incommensurable; they have no common length or denominator.

Unless your profession took you into mathematics (science, engineering, architecture), maybe your memory faded like mine did. To review, Phi, like Pi. is an irrational number relating to certain geometrical shapes (especially pentagons) that is neither a whole number nor a fraction — so far as we know it can be calculated into infinity without resolving. The ancient Greeks discovered it, although the discovery of irrational numbers was evidently such an unpleasant shock to the Pythagoreans that the man who announced the discovery, Hippasus of Metapontum, was shunned and his tomb was built as though he had died. The Greeks eventually came around, and, we were told, incorporated the Golden Ratio into Classical architecture. Remember the picture of the Parthenon? And there was a bunch of other stuff.

If we go too far into the other stuff we’ll be here for a long time. You probably remember how a golden rectangle (the short and long sides are in Golden proportion), cut at the Golden Ratio point of the long side, creates a square and a smaller golden rectangle. Repeating this process creates ever smaller nested rectangles/squares, and an arc drawn through each square generates a logarithmic spiral, which shows up in the shell of the chambered nautilus, and in the arms of hurricanes and spiral galaxies. Numbers in the Fibonacci series (1, 1, 2, 3, 5, 8, 13 . . . ), the higher you go, come closer and closer to the Golden Ratio. One more step and you’re into fractal geometry and chaos theory. The Golden Ratio figures in arguments for mathematics as evidence of the existence of God, some referring to it as the Divine Proportion. Clearly this number is big medicine.

Returning to the human passion for explanation, exactly why some things look good and others don’t has gotten a lot of attention over the years, and one approach to aesthetics is to take something that looks good and slap it with the Golden Ratio until it fits. Very many beautiful and elegant things in nature are described by the Golden Ratio; therefore, a given thing’s degree of correspondence to the Golden Ratio can be taken as a measure of its beauty. And a thing constructed with proportions in the Golden Ratio will be inherently pleasing to the eye. Sounds good, right? Which is why we have things like the Marquardt Mask, which purports to quantify the beauty of human faces by measuring their conformity to Golden Ratio polygons. Discussing the problems with the Marquardt Mask would perhaps shed light on some of the problems with Golden Ratio aesthetics in general, but let’s get to pipes.

In the past I’ve mostly let talk of applying the Golden Ratio to pipe design go in one ear and out the other, partly because I felt that I hadn’t quite reached that level in the pipemaking hierarchy of needs. Still do, actually, just trying to make a salable pipe without getting torpedoed by a flaw takes most of my attention. That’s probably why I’m a little wary — ok, superstitious — about approaching a block with too much of a plan. As I described in The Briar Imp, the more specific my requirements for a block of briar, the more perversely it will thwart me.

Speaking of ears, as a musician you find that some people have good ears and others don’t. Either you can play in tune or you can’t, and if you can’t hear what’s in or out of tune, knowing that the third above the tonic in a major chord has to be played 14 cents lower than equal temperament in order to create just intonation is useless. And if you CAN play in tune, you do it without thinking about the number of cents. Similarly, I think that some people have a good eye, hastening to add that one may have a good eye for color, or for proportion or perspective, or for shapes, patterns, or movement. I think there are ways to make a variety of shapes balanced and pleasing and I’ve never bothered to measure for ratio, or to plan shapes accordingly. That said, I'm curious.

What brought us to this point was a YouTube video by Adam Savage (of Mythbusters fame) in which he was making a drawer divider for his drafting tools. Sounds thrilling, I know . . . what can I say, it’s been a long pandemic. One of the tools that he mentioned in passing was a dividing caliper that mechanically described a Golden Ratio. That struck me as a nifty gadget, so I asked for it for Christmas. And, the spark of curiosity having been brightened, I ordred a copy of The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number by Mario Livio.

So for the past week I’ve been going around measuring things. Light switch plate? Golden rectangle. Credit card? Not, I repeat, NOT, a golden rectangle. Yes, it’s close, within about 2%, and often touted as an example of a golden rectangle, but if it was designed according to the Golden Ratio they would have added that extra millimeter to the length and made it exact. Any number of other rectangles in my life look vaguely Golden until I measure them. My HDTV, the grille on my Fender Tweed Deluxe guitar amp, various Apple products, my Craftsman machinist’s toolbox, all are in the neighborhood but not in the gold.

According to Livio, many of the examples used to advance the Golden Ratio as a universal and ancient key to harmony and beauty — including the Parthenon — are dubious. The Parthenon was (again according to Livio) built some centuries before Greeks mentioned the Golden Ratio (and they didn’t call it that). Certainly it’s possible that they knew about the Ratio and used it some time before anybody wrote it down, but it seems more likely that more recent evangelists for the Golden Ratio went looking for examples and fudged things a bit. The edges of roof are included in the rectangle but the edges of the pedestal are not, the actual dimensions vary from source to source, etc. There are Babylonian clay cuneiform tablets that come very close to a golden rectangle, and the height of the Great Pyramid’s triangular face is close to being in the Golden Ratio with half its base. And so on. But not close enough, according to Livio.

If you are selective enough in your measuring points and generous enough in your measurements, well, let’s take the front of my Craftsman tool box. Closed and latched, it measures 20.125” by 13.5”. That yields a ratio of 1.49, pleasing enough but not golden. Ok, let’s omit the lid and measure the height just to the box/lid division, that’s 20.125” by 11.5”, or 1.75. How about we include the lip of the box while open, that gives us 12.5” in height and a ratio of 1.61. Sigh. Ok, are we sure about the height? Surely it’s just a bit shy of the half inch line, a height of 12.4” yields a ratio of 1.622, that’s close enough. Another example!

Of course I put the caliper to a couple of my pipes that happened to be lying around. I’m working on a Sphinx, and the length of the shank is passably in Golden Ratio to the stem. If we ignore the fact that the stem is still unshaped and straight; finished with a slight bend it appears a little shorter.

Applying the calipers to an Acorn, if I squint I can find several measurements that are just about in the Golden Ratio. Which was not a consideration in carving it; unless the pipe is to a specific formula (like a Billiard) I will often take what the briar gives me for a stummel, leave the stem blank long, then trim until I think it looks right.

Speaking of specific formulas, let’s look at that Billiard. We know the recipe: stem length equals stummel length, shank length to bowl equals bowl height from shank to rim. The joker in the deck is the shank diameter. Above is a sketch I made to illustrate basic group 3 LB Billiard proportions for a customer. As it happens, a rectangle drawn around the stummel is pretty close to a Golden Rectangle. Although there's no bowl cant in the drawing and there is 4 degrees in a pipe. Measuring the Billiard that happens to be for sale this week, the length of the stummel is 2.8” and the height is 1.75”, which comes out to a ratio of 1.6. Pretty dang close. With such small dimensions, it would only take a couple hundredths of an inch off the top — the thickness of a minor sand pit — or a fractionally longer shank to make it exact. So, is the enduring appeal of the Billiard is due to this “Golden” relationship between its bowl height and stummel length? Might we conclude that the original designers of the shape had this in mind?

Maybe. How philosophical do we want to get about this? Or, rather, how long a blog post can you stand? I’ve undoubtedly been influenced by Livio’s book, which suggests that nobody expressly and consciously used the Golden Ratio in art or architecture until the Renaissance, and that nobody has proven experimentally that humans have an innate preference for Golden proportions. I don’t have a dog in that fight, and if I can make better-looking pipes by intentionally employing the Golden Ratio I’m down. And there may be more subtle applications of it.

There is much more that can be said about the Golden Ratio (and feel free to have at it in the "comments" below), but I’m neither a mathematician nor a philosopher and we’re talking about, well, pipes. My starting point is that tastes vary, that various factors in addition to proportion create pleasing compositions, and I guess I side with Livio that humans, in general, probably lack both the acuity to tell if something is in a Golden Ratio and a hard-wired preference for it. And you can be as precise as you want with the intended proportions of a pipe, there is still an excellent chance that the briar is going to lob a grenade at you and plans will change.

But if I stop there I can’t criticize other people for being unwilling to learn. It may be that if I am conscious of the Golden Ratio I can learn something new about shapes, or it may be that I’ve been unknowingly coming pretty close to the ratio all along. In any case, I intend to keep playing with my new toy. If I discover anything I’ll let you know.