REPOST: the catenary tales

[Original posting at Cocktail Party Physics is on hiatus because the writer is being forced to lounge lazily on a Mexican beach in between grueling spa treatments -- and admire the odd bit of local architecture. In honor of architectural arches everywhere, we offer this repost from March 2006. Also, feel free to head on over to Cosmic Variance to wish my shiny new Spousal Unit a happy birthday!]

Leonardo da Vinci once observed, "An arch consists of two weaknesses which, leaning one against the other, make a strength." I was reminded of this last month when I attended a press briefing on arches throughout history at the AAAS meeting in St. Louis. (Yes, I am only now getting around to blogging about it. It's been a busy few weeks, and the following required a great deal of rigorous thought.)

St. Louis is known for its Gateway Arch, a landmark structure that opened in October 1965 to commemorate Thomas Jefferson's 1803 Louisiana Purchase. The Finnish architect Eero Saarinen designed the enormous steel parabola to symbolize the gateway to the American West. There's a very specific geometric term for the arch's shape: it's essentially an inverted model of a flexible chain or rope suspended from two points. The non-inverted model's shape is known as a catenary. Its name is courtesy of the Dutch mathematician Christian Huygens, who dubbed the curve catenarius, from the Latin word for "chain."

Saarinen didn't copy the classic inverted catenary shape perfectly; he elongated it, thinning it out a bit towards the top to produce what one encyclopedia entry describes as "a subtle soaring effect." Maybe not subtle enough. In 1980, a man named Kenneth Swyers took the whole "soaring effect" a bit too literally. He tried to parachute onto the arch's span and died in the attempt, garnering a post-humous Darwin Award for his efforts.

Okay, so parachuting onto the arch is a bad idea. You can still ride a little egg-shaped tram to the top if you're so inclined, but it was bitterly cold out -- and Jen-Luc Piquant gets claustrophobic -- so we took a pass on that particular tourist attraction. The brave souls with fire in their hearts and appropriate winter outerwear who did ascend reported that the spectacular view was marred somewhat by the disconcerting sensation of swaying whenever the wind picked up. In fact, the arch is designed to sway up to 18 inches in the wind.

There are very good physics-based reasons why Saarinen chose an inverted catenary shape to build the Gateway Arch. Leonardo's codependent "leaning weaknesses" describes a delicate balance of opposing forces that gives rise to a certain degree of structural stability. A chain suspended from two points will always try to form a catenary. This happens because the chain wants to hang in a state known as "pure tension," so it will always adjust itself to find this balanced state. Only tension forces can exist in the hanging chain; inverting the shape into an arch reverses those into pure compression forces. All that compression force acts along the curve and never at right angles to it. This makes the inverted catenary very stable, particularly for spanning a horizontal distance. The 17th-century English scientist Robert Hooke phrased it best: "As hangs the flexible chain, so but inverted will stand the rigid arch."

Saarinen developed his variation on a catenary theme in consultation with an architectural engineer named Hannskarl Bandel; the slight elongation is not only pretty, but it transfers more of the structure's weight downward, rather than outward at the base. This extra stability is important because, although the inverted catenary is very stable horizontally, it is less so in the vertical direction. The higher the arch goes, the less stable it becomes vertically. The St. Louis arch rises to some 630 feet, so Saarinen had to incorporate a lot of extra material in the so-called "cross-section" to get the arch to stand up and stay up. (I'll take an uneducated stab in the dark here and surmise that it has something to do with needing more mass to reinforce the "pull" between the arch's two "leaning weaknesses" at that particular juncture.) That's also why the little tram cars to the top are so small, they're almost like five-person coffins, and why the arch is closed on very windy days.

Legend has it that in ancient Rome, whenever an arch was constructed, the architect who designed it was forced to stand underneath as the wooden supports were removed as a means of quality control. It was a terrific motivational tool: design it right, or the arch falls and crushes you. Today, architects can rely on people like MIT's John Ochsendorf, who has developed a new method for 3D modeling of the forces in a building design, using a computer graphics technique called particle spring modeling. Remember the scene in Revenge of the Sith where Yoda fights while wrapped in a cloak? That's a particle spring model. Computer animation graphics designers use it to model fabrics, because they need to understand how forces flow in different directions, in real time, in 3D, and in an interactive format -- designers need to be able to tweak the parameters and view the effect of doing so immediately. The virtual masses are connected by virtual springs that bounce around until they find an equilibrium to support the requisite loads.

Ochsendorf is the first to apply this method to the study of architectural design; he's currently modeling the physical forces at work in the complex structures of historic Gothic cathedrals. The eventual goal is to uncover more efficient ways of building modern structures. For instance, Frank Gehry's trademark leaning columns are visually striking from an aesthetic standpoint, but practically speaking, roughly 40% more material is needed to make such structures stable. Ochsendorf's 3D models can determine where the lines of force naturally want to fall, so architects like Gehry can better align columns with those lines of force.

Ochsendorf also reached beyond Hollywood and drew on the lessons of the past for his computer modeling technique, specifically a similar design tool -- the hanging model -- used by the Spanish architect Antonio Gaudi to calculate structural balance for the arches incorporated into his building designs. Gaudi devised an elaborate system of threads that he used to represent columns, arches, walls and vaults, augmented with little sachets filled with lead shot to mimic the weight of small building components. He used the model to design the Colonia Guelli Church between 1889 and 1908, as well as the Casa Mila in Barcelona, Spain, completed in 1910.

All of this discussion proved fascinating, but frankly -- and hell is freezing over as I type these words -- I wanted to know more about the math. The St. Louis Gateway Arch is unique because it has an actual mathematical equation displayed at its base that describes -- what else? -- a catenary. Because it makes me look like I know what I'm talking about, I reproduce the relevant equation here: y = 68.8(cosh0.01-1). (Jen-Luc Piquant claims this makes perfect sense to her, but I suspect she's prevaricating just a little.) On hand to offer his expertise in this area was Paul Calter, a retired math professor from Vermont who is also a painter, sculptor and author of a mystery novel, not to mention a forthcoming book entitled Squaring the Circle: Geometry in Art and Architecture.

According to Calter, small stone arches were typically built around a curved wooden form, around which the builder would lay stones or bricks, tracing the shape with pegs and string. But modern construction materials include steel, which must be fabricated in a steelyard and then assembled onsite. As every engineer knows all too well, math can be a big help in making sure everything is sized and shaped just right to achieve the critical balance of forces. Equations for geometric figures like circles and arches weren't even available until Rene Descartes devised analytic geometry in the 17th century, thereby ensuring that millions of high schoolers would be required to take up compass and straightedge and learn about interior and exterior angle sum conjectures and the properties of trapezoids. (Personally, I enjoyed high school geometry; I would have enjoyed it even more if Michael Serra had been teaching the class. Among other pedagogical methods, Serra teaches his students how to build arches out of Chinese take-out cartons.) 

Not surprisingly, there are different equations for the various arch shapes: circular, pointed, parabolic, or elliptical. I'll spare you most of the technical details of the St. Louis arch equation, because they make my innumerate little head ache --something about giving the height y of any point on the arch at a given horizontal distance x? And what the heck is a cosh, anyway? -- but even my befuddled brain was fascinated by a little-known connection. It turns out that the equation is related to both exponential growth curves and exponential decay curves.

This is one of those strange convergences -- in this case, between math, physics, engineering, architecture, and art -- that we here at Cocktail Party Physics love so much, because it provides so many different contexts in which to discuss highly intimidating math and physics concepts. Some of us need all the context we can get.

Let's start with the exponential growth curve, which describes, among other phenomena, population growth. It turns out that the greater the population, the faster it grows. Calter used the analogy of computing compound interest. Say you invest a certain number of dollars (P) at a fixed interest rate (n); after x number of years, the interest would compound for a grand total of y. So if Jen-Luc Piquant took $500 of our hard-earned wages and invested that money at 6.5% interest for eight years, at the end of that period we would have $827.50. That's a pretty decent return; no wonder Jen-Luc handles the finances.

If I understood Calter correctly, there's nothing jaw-droppingly exponential (yet) about this example, because the interest is only compounded once a year. In fact, things don't move into the truly mind-boggling realm as long as the interest is computed in discrete intervals. Jen-Luc and I don't get a substantially greater return on our investment if the interest is compounded monthly instead of annually. The difference is something like $12.33, which barely covers the cost of a movie and one-way Metro fare to the cinema. But if we compute the interest continuously rather than discretely -- in physics terms, this would be akin to using Maxwell's wave equations for light versus the Planckian approach of quanta -- the resulting value first becomes very large, and then hits a sort of threshold, such that the value stabilizes (to something on the order of 2.7183, per Calter).

The reverse happens in exponential decay. Let's say that Jen-Luc invests our hard-earned wages unwisely, at a negative interest rate, so that we lose money instead of earning it. (Since Jen-Luc would never be so foolish, perhaps another possible example might be the interest incurred on a debt, say, our mortgage interest rate, which amounts to losing money, judging by how slowly the principal balance decreases.) The minus sign in the equation is the only thing that differentiates exponential growth from exponential decay; everything else remains the same. The value grows larger and larger, then hits a threshold and stabilizes. Calter likened it to how a cup of coffee cools in a cold room. It is the temperature difference between the two that drives the heat out of the coffee. As the coffee cools, that temperature difference decreases, and the rate at which the temperature drops decreases along with it. So the rate of change in temperature is proportional to the temperature of the coffee.

So the exponential decay curve contains the essence of entropy, a.k.a., the second law of thermodynamics -- which brings us to a bit of critical physics history. A certain well-known 19th century equation related the temperature of an object to the total amount of radiation it emits: if the temperature is doubled, the emitted radiation will increase 16-fold. In 1900, German scientists conducted experiments to verify this by measuring how much radiation came off objects at various temperatures, expecting -- per the equation -- that as the temperature rose, so would the amount of emitted radiation. Alas, that's not what happened. The equation held until the temperature rose into the ultraviolet range of the electromagnetic spectrum. Then it leveled off and began getting smaller and smaller again.

This became known as the "ultraviolet catastrophe," which might strike non-scientists as over-reacting. But any time theory doesn't agree with experiment, it constitutes a scientific disaster of sorts.  And in this instance, the disaster led to a revolution: in an attempt to devise a theory to explain these experimental results, Planck came up with the notion of discrete quanta, inadvertently giving birth to the field of quantum mechanics.

"Okay, fine," Jen-Luc sighs impatiently. "But what the hell does any of this have to do with catenaries and arches?" Well, in the case of entropy and black-body radiation, I'm a little fuzzy on the connection myself. I think it has something to do with how calculating interest in discrete (quantum) intervals doesn't give us a (catastrophic?) exponential curve, while calculating the value continuously (like a wave) does. But Calter pointed out that if you put the exponential decay curve together with the exponential growth curve, you get something that looks like the figure at left.

Stripped of all the technical jargon, what Calter was saying is that in the classic catenary shape, the descending portion of the curve behaves like exponential decay, while the rising portion of the curve exhibits the characteristics of exponential growth. Combined, they form a classic catenary which, when inverted, in turn forms an arch.

I am, as I made clear in a prior post, functionally innumerate. The above represents my own rambling thoughts as I try to grope my way towards a better understanding of the complex web of underlying interconnections at work in something as gloriously simple (on its surface) as an architectural arch. I'm particularly intrigued by the unexpected cameo appearance of the second law of thermodynamics, perhaps because I struggle so much with the entropy of my own ignorance on a daily basis. (For anyone who cares to weigh in with corrections, clarifications, or further insights to help me combat this intellectual entropy, feel free to post your comments.)

In medieval Europe, arches weren't just structural, but were also a spiritual device, intended to evoke a transcendental quality in the building. We've lost that symbolic view to some extent, embracing a far more pragmatic approach to architectural design. But even in the midst of pragmatism, I discovered that it's possible to derive inspiration from the arch, and not just from its aesthetics. Even the underlying math and physics inspires.

Comments

[inwit]

I apologize in advance for nitpicking, but the catenary is technically not a parabola, as is stated in the second paragraph. Galileo Galilei, who tried to solve the catenary problem and failed, also thought the catenary was a parabola, so you're in excellent company.

Another way to approach the catenary problem, the way it is usually taught to physics students, is to begin with the observation that the catenary's shape is one which minimizes the total gravitational potential energy of the chain, or, equivalently, minimizes the height of the center of gravity of the chain. Any shape other than that of the hyperbolic cosine (cosh), such as Galileo's parabola or, say, a vee shape, would have a higher potential energy or center of gravity, and thus would not be favored. ( For more info see http://mathworld.wolfram.com/CalculusofVariations.html or http://en.wikipedia.org/wiki/Calculus_of_variations ) Incidentally, many physics problems become much simpler when reformulated in terms of energy (a scalar quantity) rather than in terms of forces, "pure tensions," or shear stresses (which are vectors or 2nd-rank tensors).

The hyperbolic cosine function is just the average of two exponential functions, an exponentially growing one [exp(x)] and an exponentially decaying one [exp(–x)]. In equation form, cosh(x) = [exp(x) + exp(–x)]/2 .

>"Legend has it that in ancient Rome, whenever an arch was constructed, the architect who designed it was forced to stand underneath as the wooden supports were removed as a means of quality control. It was a terrific motivational tool: design it right, or the arch falls and crushes you."

Did the architect who designed the Gateway Arch have to undergo (I use that verb advisedly) a similar ordeal? If so, is this how the expression "to be under the cosh" meaning "to be under pressure [to succeed]" came about? No, but it might make for a good apocryphal anecdote for your next cocktail party.

[inwit]

By the way, no one goes to the APS March Meeting anymore. It's too crowded.

For a somewhat different example of the intersection between physics and architecture, you might be interested in this brief snippet from a lecture by Richard Feynman:

http://www.youtube.com/watch?v=VtDQ-jPtkac

[JenLucPiquant]

Thanks for both these helpful comments... now I know what a cosh is, and I plan to spread the unofficial "legend" of the architect of the Gateway Arch being forced to stand under it while being constructed. With any luck, it will show up on the Urban Legends site one day... :)

[inwit]

You're welcome. Actually, the story I was hoping to have you spread was the bit about the origin of the expression "under the cosh". But perhaps that would be too much of a betrayal of your English-major roots.

(By the way, that equation for the gateway arch, "y = 68.8(cosh0.01-1)", seems to be missing the variable x in the right-hand side. Thought you'd want to know.)

[Kenny Jaworski]

Interesting post - Kenneth Swyers actually succeeded in his attempt to land on top of the arch. But the next step was to parachute off the top of the arch, and the wind caught his chute and he slid down one of the legs and died.

[Chris]

To be pedantic: exponential growth is still exponential growth, even if measured at discrete intervals. Your bank account that compounds yearly will, after n years, have (initial balance)*(1 + interest)^n dollars in it, which could also be written (initial balance)*(exp(n * log (1+interest)). Continuous compounding can be treated as changing the effective rate parameter; there are a bunch of formulas here to translate between equivalent interest rates at different compounding rates in this wikipedia article: http://en.wikipedia.org/wiki/Compound_interest