How to build a Giant Dome - Numberphile 

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TIL why that curve is called a catenary - catena is Latin for chain.

If anyone builds a dome after watching this video I expect to see pictures.

"how do we make a dome without any supports?" "well.. first make 2 domes, and then support it with a third"

You can add weights to the chain to change the shape of the droop to best support a structure with equivalent loads in the same spots. I suspect this is something they did to help support the weight of the spire on the top. At 5:04 you can see that the structure doesn't quite fit the catenary shape. You can imagine that if you hung a chain with a weight in the middle, you might get a pointier curve that better fits the structure drawn here. All this can of course be done and shown with calculus - which I say because I was born after calculus was invented.

Numberphile: * releases a video about mathematical domes * Me, an Italian, surrounded by domes and cathedrals everywhere: 👁👄👁

Sagrada Familia was designed with string with weights on them hanging from the ceiling. Basically an upside down model of the building that automatically optimizes the shape.

I wish this video existed when I was doing taylor series and cosh, sinh. He broke it down so well

One thing I just have to add, as an engineering student. Structurally speaking, the importance of the catenary curve shape, when it comes to arches and domes, is that it minimizes the bending moment throughout the arch. To put it a bit more intuitively, you can imagine if you picked a dome up and put it down where you wanted it to be. You wouldn't want it bending a lot from the shape you originally built it in, because if it bends too much it will break. So, when looking for a physical model, you want to use one that doesn't resist bending moments almost at all. Hence the rope: ropes, in theory, have little to no resistance to bending. If you try to bend a rope, it'll bend. So, when you get a rope and let it hang between two positions, it will make a shape that minimizes its bending moment throughout the shape. Also, strictly speaking, the catenary is just the ideal shape for an arch supporting its own weight (and then you can extrapolate that into a dome by revolution). If its supporting something else, it forms a different shape. A few examples being if a rope is supporting a point mass: say you hang a very heavy weight in the middle of a suspended rope, it will form a V shape (assuming the weight of the rope is negligible). This is why the shape of the supporting dome at St. Paul's is so steep. It supports the outer dome right at the top, before veering off in another direction. Another example would be suspension bridges: the long cables spanning between each of the towers, in theory, form a parabolic shape, rather than a catenary, just due to being loaded differently.

This man is the maths teacher your friends have and you're really jealous of

I love how he’s so passionate and so young, I love the Oldie Goldies, but some new blood is amazing too!

I will have the best domed gingerbread house at Christmas this year

Small potential correction, and a possible explanation behind the non-matching shapes at 5:00 : The catenary is only if the dome is evenly loaded (same density of material throughout). By weighting the chain according to the load that is actually experienced (e.g. supporting the outer dome), you can model it for varying loads, which creates different shapes. To support an extra load at the top like the outer dome, you would place an extra weight on the chain at the middle where the outer dome joins the structural dome, creating a sharper shape, which is exactly what we see in the actual design.

I just go to the Dome Depot.

You should have mentioned the Sagrada Familia! The entire building is basically built up entirely of catenary curves and related hyperbolic and hyperboloid shapes!

Love learning from this boy

What I like about these videos is that even if you don't fully understand everything, you do learn something and the different levels of explanation here were really useful - especially with the practical example of the dome.

I love that our understanding is always growing, and how building that cathedral's dome today would result in a stronger structure. The power of the human mind is incredible!

So when Tom Crawford doesn't show his tattoos he becomes Thomas Crawford

Tom's enthusiasm is positively contagious :)

What an incredible teacher. Such clarity, excellent scaffolding, and enthusiasm!

We now need a parody of Matt Parker's "there is only one parabola" but it's "there is only one catenary" instead

The Gateway Arch in St Louis, Missouri, USA is a 630 foot (192 m) tall "weighted" catenary curve. The weighted catenary has a subtle difference in that it takes into account the increasing weight supported at the bottom, and uses a "chain" of non-uniform thickness that is thicker at the bottom than the top. If I recall correctly, this is to keep the internal stresses uniform, while also minimizing the internal energy. (I may have that last bit confused, though. I learned about this decades ago.)

Please keep bringing Tom back! He's awesome

You know, you sometimes have to build a giant dome in your home for a reason.

This is several hundred years after St Peter's in Rome, or Brunelleschi's dome in Florence, each of which are much larger

His enthusiasm is contagious.

Don't you just love listening to people talk about sth they're passionate about? You can see how giddy he is to be explaining this, and I know that feeling and I love it a lot (shout out to anyone who's ever listened to me ramble) and I love watchin other people have it:)

So every catenary is just a piece of the hyperbolic cosine? That's handy

Great video!!! The caternary curve is also the basis for the St Louis Arch in Missouri.

I really like the videos with Tom Crawford. He explains things well and shows an excitement for each topic. Fun to watch and learn.

wonderful, math is in everything around us

Yes i have always wanted to build a giant dome now my dream can come true

matt parker: there is only one true cosh curve!

Tom's ahead of his time guys. Little do you know, that's how they write 5 in the future.

This was a great video all the way through. There are a lot of ways to approach the cosh and sinh functions and I like what he did. One of my favorite theorems is that any R -> R function can be expressed as the sum of an odd function and an even function in exactly one way.

Really interesting, but it would be nice to see a derivation of of why cosh is the solution to the chain problem using calculus of variations. More please!

4:02 It is so pointy because, in addition to hold it's own weight, it had to hold the concentrated load of the lantern (the structure on top of the dome). Architect Antoni Gaudì extensively used this "trick" to design the Sagrada Familia: he used to create a model of the church made out of chains hanging from an upside-down board on which he drew the plan of the building. Where those "arches" had to carry a load, such as a pinnacle or something, he hung a little sandbag which weight was proportional to the load to be carried. The resulting shapes of his chains were much more V-shaped than without the sandbags. Since chains can freely bend but do not extend, the shape they took was the one that made them subject to tensile stress only: no sheer nor bending moment. Reversing the direction of gravity, traction becomes compression, but sheer and moment remain none. Stone and bricks are great at withstanding compression but very poor with shear and bending moment. The Gateway Arch in St. Louis by Eero Saarinen is also a great example of a catenary.

Now i understood *how* cosh and sinh are defined by e^x and e^-x, great video!

Ok. This one blew my mind. A very simple but brilliant idea.

Please do a bonus video on this where the full calculus of variations treatment is shown.

I have been waiting for a numberphile on hyperbolic curves, please add a part 2!

Building a dome looks like so much fun! The power of math is endless!

Love you numberphile!!!❤️❤️❤️❤️

I love that he just pronounces “cosh” as it’s written, and sinh as “shine”. Would that make tanh “than”?

Math and history and history of science. Thanks for this amazing video 😊

Intriguing! I believe this is also the way the Sagrada Familia in Barcelona was designed by Gaudí, just hang the design upside down to find the right shapes

He pronounces sinh as "shine"? That makes no sense but I like it.

Is it bad that all I can think is "I've never heard anyone pronounce hyperbolic cosine like that."

Amazing video!

Really interesting video ... well done 👍

We're going to talk about one of the most famous domes in the world - St Paul's in London Il duomo in Florence: am I a joke to you?

First time in life i understand the cosh(x) and sinh(x)

This makes it so clear what the sinh and cosh are. I honestly never really knew what they were except for their formulas because I needed those for statistical physics. This makes it much easier to remember!

I love this channel, thank you so much for the knowledge!

Wren: Sir, can you build me a cubic support chain? 17th century building contractor: ?

Gaudí used the same method of strings and weights to plan his buildings, especially the columns.

Dr Crawford you legend! Always enjoy your videos

I learned so much about St Paul's Cathedral that I didn't know. Thank you!

Yeahhhh, straight from the top of my dome, as I watch, watch, watch, watch Numberphile at home.

I want to be as happy as the guy talking about domes lol.

While it wasn't mentioned in the video, catenary curves are very closely approximated by parabolas for (x/a) < 1. This is useful for creating parabolic reflector troughs for concentrated solar applications simply by hanging flexible sheets between two guide wires and letting them cure/harden into that shape. The catenary curve given by y=a⋅cosh(x/a) is closely approximated by the parabola given by y=a+x^2/(2a).

Super cool. Thank you

I noticed Tom has a Pokeball tattoo. A man of culture I see... 5:11

Explaining the physical reasons why a suspended chain takes that form may be interesting…

Thats amazing, guys! I would like to see more math´s content on architecture.

The bit about the architects not having the mathematics to get really precise makes this a great pairing with Grimes’ video last week, cause both have to do with the inherent limitations (in the form of approximation) of determining something like this from it’s output, rather than concrete and predictive mathematical models.

(3:40) lower hemisphere: look at me! i'm pretty! higher hemisphere: look at me! i'm majestic! catenary: look at me, damn it! i do all the f*cking work!

Brunelleschi: Am I a joke to you?

Gaudi also did some of the same things for the Sagrada Familia, except he also included weights on his string to model the uneven loads

The nice things about cosh is that it's just a normal cosine, but without the complex numbers. cos(x) = (e^ix+e^-ix)/2 and cosh(x) = (e^x+e^-x)/2. It's amazing to see how connected trygonometric, hyperbolic and exponential functions really are.

8:04 and I thought my 5's were lazy...

Very interesting, thank you. Missed opportunity to also mention the tanh function though. tanh is a very nice sigmoid that's very useful as e.g. a waveshaper function in audio processing.

love that the angelic sphere appeared right at @3:14

i appreciate the explanation

4:42 Euler again

That man has a Wii on his table, and I want to be his friend

Great video. Now I can finally put that dome on my garage!

Wonderful!

"That's doing the business" 😆

8:15 no kinkshaming please

Awesome vid. Love the enthusiasm. V pleased with myself deducing first term of e^x as x^0 / 0! and then googling to confirm 0! = 1. Hah, love maths.

Now this explained me why cosh and sinh are used in architecture... thanks!

*MINIMIZE THE ENERGY*

Have you ever heard about Hagia Sophia :D ?

Really nice

Beautiful interpretation of the hyperbolic functions

they got DanTDM on numberphile this is crazy

Knew there was the inner and outer dome, didn't know there was the third support dome in between.

I had no idea what sinh or cosh was. And why it was related to e^x. Thanks Tom for explaining it so well!

Numberphile really is the best!

Sir Christopher Wren Said, "I'm out dining with some men. If anybody calls, Say I'm designing St Paul's"

I've heard them pronounced as cosh -> cosh sinh -> sinch tanh -> tanch

Where was this at the beginning of my calculus 2 semester? Lol Thank you for the video.

I like it that they never go too practical about what seem really practical

7:56 He said SINCH in a very strange way........

I'm digging that Pokeball tattoo!

HOW they build the dome(s) - and all the stuff filling the space in between - is it's own amazing tale.

fantastic

ka-TEEN-er-ee (for "catenary") and SHINE (for "sinh") are driving me up the wall. But I guess Brits pronounce things differently. (Doesn't excuse randomly moving an "h" around, though.)