The Catenary - Mathematics All Around Us.

The catenary is the natural shape of a free hanging rope or chain and can be found everywhere. From the the wires of a pylon to the thread of a spider's web. From the world of bubbles to the world of architecture. What is the catenary and why is it all around us?

[Garbaz]

In case somebody is wondering what "cosh" is:
"cosh" stands for hyperbolic cosine, a function related to the cosine.
Even though their plots don't look alike at all, the mathematical notation does.


The cosine can be written as
cos(a) = ( e^(i*a) + e^(-i*a) )/2
where "e" is Euler's constant, "i" is the complex identity (i² = -1) and "a" is the angle.

The hyperbolic cosine looks quite similar:
cosh(a) = ( e^(a) + e^(-a) )/2

It's almost the same, except that the hyperbolic cosine misses the "i", the complex identity. It can also be written as
cosh(a) = cos( i * a)
since i * i = i² = -1, thereby getting rid of the "i" in the exponent and flipping the sign.

sinh, tanh and coth also exist, with similar relationships to sine, tangent and the cotangent.

Further information:

[zuzaaa]

First time I saw this video was around the time it was released. Back then I was ending my elementary school, so I didn't understand what the cosh(x) really is, but this video had surely inspired me, I thought it was one of the most beautiful videos on youtube. It had convinced me to study mathematics and that I want mathematics to be a part of my life. Since then I've watched it dozens of times along the years and I always revisit it with the same joy as when I first saw it as a young teenager. Now I am doing my masters in maths, I know what the cosh function is and how to derive the equation for hanging rope, but I still really enjoyed the video. Although the camera is a little bit shaky in some places and the quality of youtube videos has improved since 2012 I think it aged pretty well. Thank you again! I think that it wouldn't be an exaggeration if I said that this video shaped my life in terms of my interest in mathematics

[ragnkja]

I'm not sure how many times I've watched this. I just keep coming back to it; it's simply beautiful.

[MatematicaTel]

Absolutely fantastic!! I have a math channel in Brasil, but I wanna be like singbanana. Thank you teacher James Grime!!

[Idacertei]

this video deserves at least 1 million of views, beautiful, thanks doctor.

[MicheleeiRettili]

beautifully filmed and created. Great quality, mixed with the usual great way of expressing out concepts you have. A "like" is not enough

[xyz]

the way he explained nature and practical applications of the whole catenary concept left me speachless. bravo!

[SpreadTheTruthUSA]

The best and most informed mathematical video I ever watched.

[QuasarRiceMints]

I always find your explanations and videos interesting, but somehow this one is especially beautiful. Great video, Dr. Grime! :)

[olbaze]

The music makes this somehow beautiful. This is going straight to my go-to videos to show to people not into math.

[DubGoodness]

I bet, Mr. Grime, that you enjoyed doing this clip very much.
And congrats on the amazing production, hats down to Mr Bailey!

[lanreenilo]

This video on catenary is a real eye opener.  As i  further delve into the world of mathematics I gain a a greater appreciation and understanding of how mathematics works in the world.. 

[_notch]

The style and background music certainly has changed over the year, but the passion is the same as ever. Love your work.

[Mackinstyle]

Every single video you do from now on should have super profound, inspirational music in the background.

[surferdude]

James came to our school today to do a presentation on the enigma, and even brought it along! Not everyday that you get to meet your hero, haha.

I wasn't one of the ones who you needed to convince to go into maths, since GCHQ or research at Oxbridge have always been my ideal career paths, but it was a great insight into codebreaking.

That has actually made my week, so thanks for that. Also, to anyone wondering, he really is as happy and enthusiastic as he seems in his videos. :D

-Luther

[singingbanana]

You get two numberphile videos a week. We don't literally film two videos a week, that would be inefficient. Not much of a conspiracy.

[anthonymontanio]

This is really informative! Please make more videos regarding the mathematics in nature and engineering. :)
Thank You!

[wolken_bruch]

you look MAJESTIC i n this video daaamn

[experiencebg]

James, please make a video like this it's truly marvelous !

[JordanScales]

Excellent video James! I had no idea these curves existed. And the part about arches? Mind-boggling.