The Mistake at the Base of the Gateway Arch
There is no dishonor in saying "I don't know"; there is great dishonor in saying "I don't care to know" and insulting those who do. What sort of standard for education is set by such arrogant ignorance?
-- Carolyn Segal, Bethlehem, Pennsylvania,, New York Times Letters to the Editor page, October 10, 2000.
Let me make one thing absolutely clear: The Gateway Arch is safe. There's a mistake at the base of the Gateway Arch, BUT it's a typo and the Gateway Arch is safe. Even though the Arch weighs 43,000 tons of stainless steel and it can sway as much as 18 inches with sufficiently strong winds, the Gateway Arch is safe. I don't want to panic residents of St. Louis, nor the National Park Service, nor the four million annual visitors to 11 North Fourth Street, St. Louis, Missouri, 63102.
It's perfectly safe. It's been safe since construction ended in 1965 on the 630-foot-(192-meter)-tall monument. Finnish-born architect Eero Saarinen knew what he was doing with his design when he won the 1947 competition for the Jefferson National Expansion Memorial.
But there's still a mistake at the base of the Gateway Arch. And Saarinen is probably rolling over in his Michigan grave about the mistake, which may well have been in place since the Arch was built.
The mistake was discovered and generously brought to my attention by my friend Matt Frank, a graduate student in the history and philosophy of mathematics when he made the discovery in September 2000.
When you enter the North Tram of the Gateway Arch, you see an exhibit on the science and engineering behind the Gateway Arch. In that exhibit, there's a white plaque which shows the mathematical equations used in designing the arch.
One of those equations is mistaken.
That's it. That's the mistake.
When Matt discovered this mistake, he found a staff person who had forms for filing complaints and mentioned this mistake. The staff person then responded, "I was never any good at algebra."
Fascinating. And I suppose that if someone discovered a misspelled word at the same plaque this same staff person would likewise plead innocent with "I was never any good at English."
Before I discuss the mistake, let me now digress into a (hopefully pain-free) math lesson to help in understanding the mistake for those with little math background.
First, a bit of algebra enters from stage left. The two key characters to focus on are x and y --the main lines that you see in any algebraic grid or Cartesian plane, where y is the up/down coordinate line and x is the left/right line (plus they're the values associated with those lines, where the intersection of x and y is zero and the values increase the further away from zero).
An important supporting character is c --a name for a number which doesn't change even though do (think c is for _c_onstant, that's good enough for me).
Vocabulary time. You might know the term "hyperbolic" (pertaining to a "hyperbola" -- a specific type of arch-shaped curve) and you might know the term "cosine" (the left-right coordinate of a curve or circle on a Cartesian plane). But you might not know the combined term "hyperbolic cosine" (abbreviated "cosh"), which refers to a type of curve expressed formally as:
cosh z = 1/2 (ez + e-z).
or, translated to words, half of the sum of some value (call it z) which serve as positive and negative exponents of e (the base of natural logarithms, roughly 2.718).
These characters--x, y, c, and cosh--combine to form the following equation:
y = c (cosh x/c)
This equation is exact formal definition of a catenary--the name for the curve drawn out by a hanging chain suspended between two points. In fact, the Latin word for "chain" is catena--the source of the very name "catenary." And an inverted catenary, as you might guess, is simply a catenary turned upside- down.
The Gateway Arch is itself an inverted catenary, whose exact formula somewhat resembles the catenary formalism above:
Y = A (cosh[X/L] C-1)
where L and A are values used in calculating parts of the Arch's base and top (diversion alert: click here for the story behind L and A). Do the algebra to find the value for X (go on, it's fun --I'll wait; when you're done you can check your answer here), and you get:
X = L/C[cosh-1 (1 + [Y/A])]
These formulas for Y and X are listed on that plaque in the North Tram--except that the X formula is listed on that plaque as:
X = -[cos h (1+ Y/A)]
The Mistake at the Base of the Gateway Arch is actually three different mistakes on this single equation for X: there's no constant L/C preceding the inverse hyperbolic cosine, there's a minus sign preceding the inverse hyperbolic cosine, and there is no inverse hyperbolic cosine (denoted by a superscript -1 above the cosh). Matt told me that the missing inverse tipped him off to the discovery.
So what is drawn out by the formula for X that is listed on that plaque? Forget about trying to draw it. It's nothing, nonsense, a misteak. I have visited the Gateway Arch myself and confirmed the error firsthand.
This could be some kind of national embarrassment, given the prominence of this mistake's locale. But its discovery may have gone totally unnoticed before Matt's visit. And I must admit that, even though I count myself as a math fan, if I hadn't known about the Mistake before I saw it, I probably wouldn't have recognized it--never mind those who see the Mistake who don't know math and are practially proud of their mathematically illteracy. So I probably shouldn't use this to springboard into a complaint about the state of mathematical literacy in the USA today.
And yet it exists and is real. And it doesn't help in the matter that a key mathematical mistake is enshrined in the tallest national monument in the United States. John Allen Paulos calls the unwillingness or fear of dealing with math "innumeracy," and in his book on the topic (also called Innumeracy) he writes:
The same people who can understand the subtlest emotional nuances in conversation, the most convoluted plots in literature, and the most intricate aspects of a legal case can't seem to grasp the most basic elements of a mathematical demonstration.
They seem to have no mathematical frame of reference and no basic understandings on which to build. They're afraid. They've been intimidated by officious and sometimes sexist teachers and others who may themselves suffer from math anxiety. The infamous word problems terrify them, and they're convinced that they're dumb. They feel that there are mathematical minds and nonmathematical minds, and that the former always come up with answers instantaneously whereas the latter are helpless and hopeless.
But having a mathematical mind (whatever that means) isn't the point. The point, I think, is to invoke a sense of discovery and to cultivate some appreciation and understanding of a subject, even if you're not "good" at it. After all, you don't have to have a major league fastball to appreciate a good baseball game.
Should we fix The Mistake at the Base of the Gateway Arch? Perhaps--if people would know enough math to be shamed and embarrassed about the Mistake. But then again, maybe we should let it be. This Mistake at the Base of the Gateway Arch just might be the catalyst to encourage people to learn more math. I hope you learned a little reading this essay.
Either way, more people would know more math. And a good thing too. To quote John Allen Paulos one final time: "As has been said about many subjects, mathematics is too important to be left to the mathematicians."
BIBLIOGRAPHY
- Clapham, Christopher. The Concise Oxford Dictionary of Mathematics, 2nd Ed. New York: Oxford University Press. 1996.
- Equation for the Catenary Curve of the Centroid of Arch Cross-Section. National Park Service/Jefferson National Expansion Memorial handout, United States Department of the Interior. (I got my copy in April 2001.)
- Explore St. Louis: The 2001 Official St. Louis Visitors Guide. St. Louis: Publishing Concepts, 2001.
- Frank, Matt. Personal Communication by e-mail, March 26, 2002.
- Gateway Today: The Official Visitor's Guide to the Gateway Arch and Old Courthouse. (I got my copy in April 2001.)
- Maor, Eli. e: the story of a number. Princeton, NJ: Princeton University Press. 1994.
- Paulos, John Allen. Innumeracy: Mathematical Illiteracy and Its Consequences. New York: Hill and Wang, 1998.
- "Saarinen, Eero." Britannica 2001 Deluxe Edition CD-ROM. Chicago: Encyclopaedia Britannica Inc, 2001.
- Segal, Carolyn. Letter to editor, The New York Times, Oct. 11, 2000, pg. A30.
APPENDIX A: So what are L and A anyway?
L is half of the value of the centroid of the arch base. (A centroid is the point of convergence of the medians of the lines extending from the lines of a triangle). L equals 299.2239.
A equals:
fc/((Qb/Qt) - 1)
where
fc = the maximum height of the centroid = 635.0925
Qb = the maximum X-section at the arch base = 1262.6651
Qt = the minimum X-section at the arch top = 125.1406
C in equations of the Gateway Arch, by the way, is the inverse hyperbolic cosine of the quantity Qb divided by Qt, which equals 3.0022.
APPENDIX B: How to go from Y to X?
Y = A (cosh[X/L] C-1)
The correct equation for X in the North Tram.
Y/A = (cosh[X/L] C-1)
Divide both sides by A.
Y/A + 1 = (cosh[X/L] C)
Add one to both sides.
cosh-1(Y/A + 1) = [X/L] C
Take the inverse hyperbolic cosine from both sides.
1/C [cosh-1(Y/A + 1)] = [X/L]
Divide C from both sides.
L/C [cosh-1(Y/A + 1)] = X
Multiply L to both sides.
X = L/C [cosh-1(Y/A + 1)]
Reorder the sides, and you get the equation for X
that should be in the North Tram.
EPILOGUE
Date: Fri, 27 Jun 2003 14:05:52 -0500
From: [Bob Moore]
To: msszczep@midway.uchicago.edu
Subject: Gateway Arch Formula
Dear Mr. Szczepanczyk:
I work as the historian at the Gateway Arch in St. Louis. I wanted to let you know that we discovered your website recently about the inaccurate mathematical formula on one of our exhibit labels in the north load zone beneath the Arch. We ran this formula by our park engineer, who agreed that the plaque was incorrect and that your website was right. We are now in the process of fabricating a new plaque with the correct formula on it. It is our intention to try to present accurate information to the visiting public, and we thank you for drawing this to our attention. Had you contacted us directly we would have corrected the problem even sooner.
Sincerely,
Bob Moore
Historian