# The Catenary – Mathematics All Around Us.

[music plays] You may be aware that mathematics is all around us. It’s in the petals of a flower, it’s in the honeycomb of a bee hive, it’s in the spirals of a sea shell. So it’s all around us in the natural world, but what about closer to home? Here I have a piece of rope. If I hold it between my hands, it forms a curve. This is called a catenary curve, and it is not a parabola. It is its own type of curve, with its own type of mathematical description. [music plays] The word ‘catenary’ is derived from the Latin for ‘chain,’ because it’s the shape formed from any chain or rope hanging freely under its own weight. And you will see this simple and beautiful piece of mathematics everywhere, from the wires in an overhead power line, to the strands of a spider’s web. But why do we see this curve appearing again and again in the natural world? To answer that question, we’re going to turn to the world of bubbles. To minimize their surface tension, bubbles form perfect spheres. If we try to form a bubble between two rings, it cannot make a sphere, so instead makes a curved shape. This time, to minimize its surface tension, it makes the catenary. So a chain, like in the world of bubbles, wants to minimize its tension. So it forms the catenary, and it is not a parabola. The parabola is the arc of a ball flying through the air. [chalk board noise] Its mathematical description is y equals (ax) squared plus bx plus c. [y=ax^2+bx+c]. By eye alone, it looks like a catenary. Galileo realized that the curve of a free-hanging chain was very similar to a parabola, but something was amiss. The mathematical description of a catenary is actually quite different. It is y equals a cosh x over a. [y=a cosh(x/a)] That’s equal to a over 2 outside of e to the power x over a plus e to the power minus x over a.

[y=a/2(e^x/a + e^-x/a) The curve changes as the distance between the two ends and the length of the chain changes. But they can be described with the same formula, and are never parabolas. Man uses this natural form in the worlds of engineering and architecture. For example, the wires of a suspension bridge are parabolic due to the weight of the bridge it carries. But without this weight, it is more like a simple rope bridge, carrying the same weight along its length, and will again be a catenary curve. But if the catenary is the shape of a rope or a chain that minimizes forces in tension, then by turning this upside down, we form an arch, which minimizes the forces in compression. And it’s actually the strongest kind of arch– strong enough to support its own weight. For example, That’s what the master builders of the Middle Ages did to construct this, the ceiling of King’s College Chapel in Cambridge. So to make the strongest kind of arch, you can simply mark out the shape of a free-hanging chain, and turn that shape upside down. [music plays] So now that you know what a catenary curve is, you’ll be able to spot it everywhere. So the next time you see an archway, a bridge, or a spider’s web, remember: you’re seeing mathematics! [music plays]

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

Yes!

I've always heard it with a short middle syllable as well. Oh well, tomaeto, tomahto.

Every point along the chain is pulled by gravity while the chain resists being pulled apart. The chain links weigh the same, but those nearer the ends are carrying the weight of those below them which makes the curve straighter near the ends. I've not seen proof that the gothic cathedral designers modeled their structures on catenaries, but Spanish architect Antonio Gaudi did. There're photos of the inverted models he made using chains w/ weights to simulate point loads.

definitely

Fantastic explanation. I dont have any of the equations memorized or anything but they are nonetheless fascinating to me. (:

Awesome!!!Beautifully made!!

The professor Brian Cox of mathematics. 😀

Still doing these? Would you consider doing a video on cycloids?

i think your cool jacket or no jacket

What is the difference b/w a parabola and a catenary?

now you've made my maths assignment so much more complicated 🙁

2:35 i am lost.

I can see now why hes students calls him the master^^

Jesus Christ I just saw the most hardcore intro on Youtube

This is really informative! Please make more videos regarding the mathematics in nature and engineering. 🙂

Thank You!

You should be the next doctor.

I seriously can't be the only person who was distracted by how sexy his outfits were. I mean, dat coat was amazing.

An excellent video, keep up the good work

great video. I'm embarrassed to admit that I've never heard of cosh.

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

Wow

Now I'm certain that I'm not smarter than a Singing Banana.

In classical mechanics we had to calculate what the shape of a free hanging chain would be. It's like a 2 page problem involving tensors and crap like that. Yeah, I didn't understand then, and I still don't understand. So I have a dislike for the catenary now.

Thanks

If the catenary is the shape of something that wants to minimize tension or compression, does the bottom of a water droplet form a catenary curve? (my mum asked me that and i didn't know how to answer, so I am relaying the question to you.)

Fun With Flags

I wanna learn to graph a catenary more then I want my next breath.

I went to my local public library and just picked up a maths textbox, and that was where I learnt the shape of the cartenary.

Your coat is awesome.

Not only is the catenary related to the hyperbolic cosine function, but its length is related to the hyperbolic sine function. The length L of the catenary y = a cosh(x/a) suspended from two points of the same height that are 2x units apart is L = a sinh (x/a).

Interesting stuff! And I may have been thrown for a second there when I realised I was watching the video from the very block of flats in this video :O

This looks like the trailer for something bigger….

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

Love the way he says Bubbles

Wow, this video was fantastic! It was so beautiful!

It was almost like art! Math is also art.

Great video!

Thanks this was inspiring. I'm a somewhat jaded mech engr and currently studying for the California Professional Engr's license. I just happen to be currently working for a amusement park/construction design company but before that I was always involved in mass production design processes where science is seldom used.

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

Brilliant video james!

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

This was hugely helpful. Am teaching e to a group of gifted 5th graders using Joy Hakim's series. This was great. Got anything similar on phi?

This was wonderful, well done.

Epic video 😀

when you come towards the camera you look like one of those christian channel guys.

Math is Beautiful

Thank you proffesor!

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..

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

Why aren't all your videos shot like this?

Great editing Dr. Grimes!!

Great video! But I have just one question, it says that the curve changes depending on the length of the chain, and the distance between the two points of support, which makes perfect sense, but in the equation, which variable is which? Is a the distance and x chain length, or vice versa?

https://www.youtube.com/watch?v=GFFHh4nZKwE

Here's a simple video I made a while back. In the clip I superpose the live footage of a hanging chain over the mathematical graph of the Catenary, and they match very nicely. Notice that it would be impossible to obtain the same result with another curve, say a y = x^2 or something similar as pointed out in the present presentation by +singingbanana.

I love hyperbolic trig!

When he mentioned arcways he connected the dots in my mind so well

Projectiles follow ellipses; the parabola is just an approximation that assumes gravity is always pointing in the same direction.

How does catenary minimizes force and tension?

And how is surface tension related to catenary?

Is it the only such video, or there is a whole series of videos "Mathematics All Around Us" ?

Ask a physics professor, they see math in everything. The light from a lamp, the sound of a distant train, a rocket, architecture….formulas everywhere

How do you manage to look that good all the time?

😉

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:

https://en.wikipedia.org/wiki/Hyperbolic_function

I mostly watch this video because I love how he says "bubbles."

What is the difference between a catenary curve and a parabola??? I want to do an expirement on which can hold the most weight for a 9th grade science fair, but I need to understand it first. Pleeaaase help me out.

great video man, i think it'd be great if you made more like this one 😀

Looking Handsome!

Was I the inly one who watched about 5 of these vids then realized that singing banana isn't copying somebody from Numberphile and that's there actually channel?????

AWESOME !!! XD

You didn't mention the St. Louis Arch!

3:50 AM and I'm watching math videos on youtube

What have I become

While I appreciate your description in this video you posted 3 and a half-ish years ago, I also appreciate your sense of style. You look quite dashing.

you look MAJESTIC i n this video daaamn

but but, what is a?!??!

not to lose the focus on the incredible math that was going on this video but your outfit was amazing

He's ready for BBC2

Where did you get the music?!!?! It's amazing!!!!!!!!

http://www.msri.org/people/staff/osserman/papers/NNJ_v12n2_Osserman_pp167-189.pdf

https://www.youtube.com/watch?v=i5axtPj2GyU

Does any hanging cable or chain always describe a visible catenary over a long enough distance, however tight it's pulled between the posts? Long power cables between big pylons look as if you can never get rid of that sag. Can the formula y=cosh(x/a) describe a straight line at finite non-zero values for x and a? Could the line connecting the face of a half moon with the sun when both are visible in the daytime sky be described as a catenary rather than an illusion (the so-called "moon terminator illusion") since light rays "sag" over long distances?

does the bubble also follow caternary before breaking off

sir, you're amazing

make part 2

I'm sorry but him calling it cuh- TEE-nuh-ree is driving me nuts it's CAT-in-air-y

Why would the chain want to minimize its internal tension? It will try to minimize the potential energy and stay in equilibrium

Much informative! Thanks!

love this guy and love maths keeping do it!

Im thinking gravity??

Super

I feel like crying now!!

wow great video!

I love mathematics and this video increased that love even more.. ?

top bien g adoré ct fentastik

it's a shame this idea never took off

https://www.youtube.com/watch?v=ToMk1Tqx3hE&t=8s

I like your enthusiasm for the subject, tx! If like to learn more about the strength of the catenary in architecture. So freken interesting.

Тот случай, когда приятно не только слушать, но и смотреть…

Thanks for the proper pronunciation. Of note this 'Cate' prefix of the strongest Arch seems to apply to those of us who build linguistic structures with fonts. And for those in the Atomic professions… Catenation deals with the linking of particles.

hello can you help me

i'm looking to creat a maths channel too and i like the way which you make your videos

actually i'm not well on editing video

please share your experience with us

i'm waiting your positive response

such a simple and lovely presentation <3

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

Cateenary? Are stressing that vowel so it sounds similar to canteen? That’s ok I guess.

The best and most informed mathematical video I ever watched.

This video popped up for me again, and I love it! I once showed this to an engineering calculus class to motivate why they should care about the hyperbolic functions we were seeing (the curriculum only covered the definitions and derivatives of sinh and cosh, but nothing else).

Unrelated, seeing this video pop up again, I had a thought about my own area of study, commutative algebra. There is a type of ring called "catenary ring". Seeing this video gave me an "aha!" moment about why these rings might be called catenary rings.

In commutative algebra, one often defines invariants based on chains of prime ideals. For example, the height of a prime ideal P in a ring R is the supremum of the lengths of chains of prime ideals in R which end at P (though we count the "links", not the ideals themselves; in other words, a single prime P is a "chain of length 0"). Pondering this definition for a while, one can notice that you only really need to consider "saturated chains", i.e., chains in which you cannot insert an additional prime ideal. A very surprising thing is that, given two prime ideals P and Q with P contained in Q, you could have saturated chains from P to Q of different lengths, though it's quite challenging to come up with "natural" examples of such a situation.

Catenary rings are those rings in which every saturated chain of prime ideals (from any fixed prime P to any fixed prime Q containing P) has the same length. In some sense, you can think of it as having the property that all chains of prime ideals are "hanging freely". If you fix prime ideals P contained in Q, then you can think of P and Q as the posts, and imagine all saturated chains from P to Q. Shorter chains would have more tension in them than longer chains. So since all the saturated chains are the same length, you can think of them as all having the same tension. And why not consider that to be "hanging freely"?