## Parabola v. Catenary

A separate thread for what is probably the most depressing and least consequential outcome of my Public Relations post: is this a parabola or a catenary? Discuss!

Coming next week: “Polygon” v. “A Simple, Closed, Plane Curve Composed Of Finitely Many Straight Line Segments.”

And for the record, the differences here are real. Not real to 8th graders, but mathematically real. And actually some mathematically honest version of these differences is usually interesting to a class full of eighth graders …

Dan Meyer (yeah that’s allowed):

There is a difference. The difference is important. Precision is important. Precision, for some students, is perplexing. No denying any of that. However, the attention paid that particular issue in that last thread was a dramatic instance of tree-noticing and forest-missing.

Gert:

It’s interesting that everybody seems to assume that NCTM overlooks this issue…. The companion sheet [pdf] could do with some improvements, but at least it looks at the catenary question.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Jaime

October 18, 2011 - 8:50 am -

Parabola…

A catenary is the shape a rope adopts under its own weight, which is proportional to arc length. If the load is proportional to horizontal length, the shape is a parabola.

And the big load on the above bridge is the horizontal road, so the parabola shaped arc holding it up will be axially loaded only, without bending.

October 18, 2011 - 9:27 am -

And the week after… perhaps we can tease apart the intricate differences among “minus one”, “negative one”, and “the additive inverse of one”.

3. #### David

October 18, 2011 - 9:37 am -

‘Coming next week: “Polygon” v. “A Simple, Closed, Plane Curve Composed Of Finitely Many Straight Line Segments.” ‘ seems to suggest that you think a catenary is the same as a parabola.

4. #### Christopher Danielson

October 18, 2011 - 9:39 am -

And for the record, the differences here are real. Not real to 8th graders, but mathematically real. And actually some mathematically honest version of these differences is usually interesting to a class full of eighth graders (for about five minutes, which is about what it’s worth at that level-go any longer and they’re just suckering you into skipping what you had planned for the day).

Catenaries are defined not by quadratics (nor other polynomials), but by hyperbolic trig functions. Specifically, hyperbolic cosine.

And these functions have interesting calculus properties (rather than interesting algebraic ones). The easiest of these to state is that hyperbolic sine and cosine answer the question, “What function is its own second derivative?” The natural exponential is its own first derivative; regular old sine and cosine are their own fourth derivatives. But (d^2/dx^2)(f(x))=f(x)? Hyperbolic trig functions (including catenaries) are the first interesting solutions to that baby.

5. #### Dan Meyer

October 18, 2011 - 10:15 am -

@Christopher, there is a difference. The difference is important. Precision is important. Precision, for some students, is perplexing. No denying any of that. However, the attention paid that particular issue in that last thread was a dramatic instance of tree-noticing and forest-missing.

6. #### Sean

October 18, 2011 - 10:32 am -

The last line is hilarious.

Some bigger questions, if you’re taking requests:

2. What have you learned about teacher retention/education?

7. #### DavidC

October 18, 2011 - 10:34 am -

It bothers me that some students are (already–I saw this teaching college calculus) walking away with the idea that any curve with constant concavity is a parabola.

On a more basic level… if we’re going to say ‘math is cool because this is a parabola’, doesn’t it actually have to be a parabola?

-Not the same David

8. #### Dan Meyer

October 18, 2011 - 11:04 am -

DavidC: On a more basic level… if we’re going to say ‘math is cool because this is a parabola’, doesn’t it actually have to be a parabola?

Definitely. That conversation would have been utterly on point during the creation of that poster. (Have we established that it isn’t a parabola? See Christopher’s above.)

9. #### DavidC

October 18, 2011 - 11:18 am -

Yeah. I guess maybe I’m just saying it’s unrealistic not to expect the question/discussion to happen again in the comments. In a way, the poster is a prime example of those images and videos you’re always posting to provoke questions involving math.

Just as those are designed to make it impossible not to ask (and then answer) questions, I think this image makes it impossible for your audience not to ask too.

Though I have to admit, I didn’t read the whole comment thread (signal to noise ratio was too low, so maybe the quality/respectfulness of the discussion was a bit disappointing.

And yeah, it does seem to be a parabola, or at least better approximated by one than by a catenary. So of course, I’m wondering why.

10. #### gasstationwithoutpumps

October 18, 2011 - 11:52 am -

A parabola would be the right shape if the load is all on the road and the arch has negligible weight. A catenary would be the right shape if the load were all in the arch and none on the road.

I suspect that the arch shape in question is neither, but an appropriate intermediate shape given the load on it. I can’t see the ends, but I suspect that this is a “bowstring-arch” bridge, with the deck in tension, which probably also changes the formulas.

Does anyone commenting on this actually design bridges? Or at least have taken a class on designing bridges?

11. #### Gert

October 18, 2011 - 4:13 pm -

It’s interesting that everybody seems to assume that NCTM overlooks this issue. The poster states (at the bottom) that more information can be found on http://www.nctm.org/more4U with access code LOV31415 (aside: I would have preferred LOV227 as the access code, but that is just my preference for fractional approximations). Following the link one finds a companion sheet to the poster which actually gives a short discussion of the parabola v catenary. The companions sheet could do with some improvements, but at least it looks at the catenary question.

12. #### Vishakha

October 18, 2011 - 8:13 pm -

It was exactly the question that came to my mind – is this a catenary or a parabola? Given that the answer is not obvious I do submit that it a good image for creating perplexity – 8th grade is maybe not the intended audience.

So I tend to agree with David C, that this was almost designed to provoke this question.

With all due respect, I can almost hear the sigh in the first line and I find the last line a tad insulting – the quest to make math relevant and perplexing should not make us have to justify a spontaneous debate on a topic of mathematical precision.

Borrowed from a friend –

“You do not study mathematics because it helps you build a bridge. You study mathematics because it is the poetry of the universe. Its beauty transcends mere things.” [Jonathan David Farley, NY Times letter, 9/1/11]

13. #### Danil

October 18, 2011 - 10:49 pm -

As long as the theme is wheel spinning, are we asking about the shape of the bridge, or the shape of the worms eye view of the bridge as it appears on the poster?

Ceci, est-ce qu’elle une pipe?

14. #### William Hall

October 19, 2011 - 6:06 am -

I know that a catenary is not a parabola, but in my algebra classes I like to introduce parabolas with the question, “What is the most famous parabola in America?” The answer I’m looking for is the St. Louis Arch, which you may know is not actually a parabola, it’s a catenary.

15. #### Fred Thulin

October 19, 2011 - 7:26 am -

I recommend the following.

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

To quote from it (retrieved 2011/10/19):

The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is incorrect.[18] It is close to a more general curve called a flattened catenary, with equation y=Acosh(Bx). (A catenary would have AB=1.) While a catenary is the ideal shape for a freestanding arch of constant thickness, the Gateway Arch is narrower near the top. According to the U.S. National Historic Landmark nomination for the arch, it is a “weighted catenary” instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form.

………………..

In a free-hanging chains the force exerted is uniform with respect to length of the chain and they follow the catenary curve. However in suspension bridge chains or cables do not hang freely since they support the weight of the bridge. In most cases the roadway is flat, so when the weight of the cable is negligible compared with the weight being supported, the force exerted is uniform with respect to horizontal distance, and the result is a parabola. If the cable is heavy then the resulting curve is between a catenary and a parabola.

16. #### Eric B.

October 19, 2011 - 7:30 am -

William

I would think the most famous parabola in America would be the “double parabola” of McDonald’s. ;)

17. #### Mark

October 19, 2011 - 12:54 pm -

@DavidC or anyone else who can answer.

It bothers me that some students are (already—I saw this teaching college calculus) walking away with the idea that any curve with constant concavity is a parabola.

I just want to see if I’m missing anything here. Off the top of my head the only curves I can think of with constant concavity are parabolas and lines or a variety of piecewise defined functions made up of line and parabola segments. Am I missing another type of curve?

If not – I don’t think that is such a horrible first impression for student to come away with. It can easily be addressed in lecture or problem sets.

18. #### Mark

October 19, 2011 - 2:52 pm -

@Chris, thanks for responding. I get that. Doing that yields what I said. I don’t see another possibility. Based on David’s reaction to students thinking you get a parabola, I wanted to make sure I wasnt missing something.

19. #### DavidC

October 19, 2011 - 3:04 pm -

Oh, I think I see the confusion. When I said “constant concavity”, the values I had in mind for concavity were ‘concave’ and ‘convex’. Sorry if that was confusing.

I didn’t mean constant second derivative. (Or constant curvature, which is also different.)

So the students I have in mind would think that any function which is (globally) concave, or (globally) convex is a parabola.

Better?

20. #### Mark

October 19, 2011 - 3:50 pm -

Thanks David. I see what you mean now. That makes complete sense and I agree with you about that being a big deal. Sorry to get so off topic Dan.

21. #### Mr Macx

October 20, 2011 - 9:56 am -

So, I know these are different curves, but can’t we curve fit a parabola onto this arch within the finite space of the poster? The curves behave in different ways, but that is at larger limits than what i’m looking at. At least, that’s always been my impression. In a relatively small finite amount of space, I can create a catenary or a parabola that look identical to the naked eye.

22. #### Brendan Murphy

October 20, 2011 - 6:54 pm -

Thanks for the quote Vishakha, I think I will use it in my email signature.

As to the parabola or catenary, flattened or otherwise. I think it is well suited for 8th grade high school algebra. We can take this picture put it on a graph and discover that it isn’t actually a parabola.

23. #### Mark

October 21, 2011 - 6:46 am -

I like the idea of putting the picture on graph paper, but I think it would be nice in a more advanced class to pick three points, solve the resulting system of equations and get an equation for a parabola. I think this bridge is a catenary, so the initial equation might look quite close, but not exactly the same. If the students had already done the same activity with a suspension bridge that actually formed a parabola, you could talk about why this wasn’t an exact fit. If the class had discussed hyperbolic cosine functions previously then you could lead them in that direction. For a more elementary class, you could have students compete to find the best fit with a parabola by changing the parameters. At the end, you reward the students that got the closest, and if you want to talk about physics tell them what the catenary is.

24. #### Chris Sears

October 21, 2011 - 6:37 pm -

Eric B.:
I would think the most famous parabola in America would be the “double parabola” of McDonald’s.

Mark:
I like the idea of putting the picture on graph paper, but I think it would be nice in a more advanced class to pick three points, solve the resulting system of equations and get an equation for a parabola.

Well, it just so happened that I was at McDonald’s tonight for a mass play date with my children’s friends, and I am teaching quadratic modeling in my College Algebra class. So, inspired by Eric, I grabbed a place mat, scanned the Golden Arches ® into the computer, and dumped the picture into GeoGebra.

It turns out that the Golden Arches ® is not a parabola at all. It turns out that they are best described by ellipses. I’m working out a few pictures now.

25. #### Tony

October 21, 2011 - 7:23 pm -

I inserted the picture into my TI-Nspire Software, plotted about 7 points and then performed a quadratic regression. My R^2 value was 99% and it looked pretty darn good…much better than the catenaries I’ve used in similar investigations with my class. Based on my previous studies into catenaries vs. parabolas I thought for sure this was a parabola but all this discussion has me confused. Hey…the calculator says it’s right so then it must be…

26. #### Caleb

October 24, 2011 - 11:27 am -

I happened across this blog while looking for pictures of falling painos (linked from http://radar.oreilly.com/2009/06/when-do-your-beliefs-become-kn.html). And I started reading Dan’s blurg on infographics. Then I saw Dan spin off this thread on can-thingys and parab-thingys (on purpose using ‘thingy’).
I rarely comment, if ever, on blogs but I have to say just 3 things:
1) My context – loved math my whole life, minored in it in college, wouldn’t now be able to reproduce even half of whatever it was a learned back then

3) I still don’t really understand what you guys are taking about: Catenary vs. Parabola other than there is some difference mathematically that may or may not be relevant to you depending on if you’re a match major, architect, song writter, or poet. So if I find the topic interesting enough to read the whole thing ,but still complex enough not to really understand it – how is a 12 year old supposed to understand the discussion?

Anyway, thanks for an enjoyable 30 minute lunch break!

PS really far away from the thread…most useful math concept in my life: A AND B = NOT A OR NOT B