Is the Gateway Arch a Parabola?

IntMath.com ^ | 06-May-2010 | Murray Bourne

[grey_whiskers]

The Gateway Arch in St Loius, Mo., USA, is part of the Jefferson National Expansion Memorial.

The shape of the arch certainly looks like a parabola, but is it?

[grey_whiskers]

Huh.

[rightwingcrazy]

An upside-down catenary was the first thing I thought of, but that’s just me.

[lafroste]

Yes. Actually. it is 2 parabolas using slightly different vertices but the same focal points. It makes for a very strong structure.

[budj]

You’re right; it’s a catenary.

[Yogafist]

Mmmmmmm. Parabola.

[rlmorel]

It is a mirror image, so I think it is a parabola.

That said, I was driving on a cross country trip at 0230 in the morning, and thought I might drop over to see it, but remembered someone telling me it was an extremely unsafe place after dark, so I just looked at it as I drove.

I must say, I was very disappointed. I always had this vision of it as something far, FAR larger than it was. I know my incorrect perception was on me, but...it just seemed so small.

[Fiddlstix]

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[rightwingcrazy]

The question that comes to mind is whether the rest shape for a dangling rope implies stability for a similarly shaped structure pointing the other way. I’m too lazy to do the math.

[Tucker39]

From the highway it appears more slender than it actually is; but it is very tall.

[T. Rustin Noone]

The Gateway arch is indeed a catenary (as I learned and subsequently taught in my training as an architect). I read the article and did the math. My only objection to the article is the analog used to help the layman understand the principal and the result.

A better way of understanding a catenary is to imagine a fishing line with many small, lead weights tied to it along its length at equal distances from each other - say, a one ounce weight every 1 inch along the total length of the line. By holding the ends of the line, the (average of the) weights will distribute themselves along the line in a shape that we call a catenary.

[Olog-hai]

Agreed. Doesn’t look like the math whiz got the values for a and b quite right for the first parabola though; they are probably irrational numbers.

[Fungi]

So what! Get a life.....The country with a new president, yet this is important? Sorry. Not....

[ETL]

A parabola is a curve where any point is at an equal distance from:

a fixed point (the focus ), and
a fixed straight line (the directrix )

Get a piece of paper, draw a straight line on it, then make a big dot for the focus (not on the line!).

Now play around with some measurements until you have another dot that is exactly the same distance from the focus and the straight line.

Keep going until you have lots of little dots, then join the little dots and you will have a parabola!

https://www.mathsisfun.com/geometry/parabola.html

[ETL]

[ETL]

The 4 "Conic" (cone-based) Sections...

[zerosix]

It certainly didn’t seem “very small” when I rode up in it and looked out from the very top!

[Gene Eric]

Good grief, spare us the hyperbola.

[editor-surveyor]

.
If it isn’t a parabola, it is poorly engineered!

(wasted materials)

[SpaceBar]

The question that comes to mind is whether the rest shape for a dangling rope implies stability for a similarly shaped structure pointing the other way. I’m too lazy to do the math.

Good thought experiment. In a hanging catenary each linear element is in pure tension since a net torque would cause them to reposition themselves. Each element also has an associated weight downward due to the ~constant accelleration field g. Now we freeze the catenary, and reverse the sign of g. No elemental torques are produced since each element now has the same albeit opposite end forces which balance out with the elemental weight, so that the internal force is once again colinear with elements but instead of tension, it's in pure axial compression, the configuration for internal stability, provided the strength is enough to resist buckling. Also in a dangling rope, the forces at the supports resolve to the weight, and also in the inverted one. The magnitude of the forces are all the same but all the signs have changed. So an 'inverse' catenary in compression would suggest optimal strength. (ps - 'element' as used above is meant in the calculus infinitesimal dx sense)

[Fungi]

Don’t be so hyperbolic.....Bake some bread and use some yeast.