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How to build a Giant Dome - Numberphile
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- เผยแพร่เมื่อ 3 ธ.ค. 2020
- Tom Crawford speaks about domes, curves, and catenaries.
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Unfortuantely they don't ship to India. Wish you could find sponsors who can service your worldwide audience.
I
Kiwico seems like a great sponsor. It's the first thing I've seen on TH-cam promotions that I've actually thought was worth checking out, rather than just being annoying. (No they're not paying me to say this!)
Ok
Tom, you need to learn to line up fractions!
6:42 7:20 8:02 8:18 8:52 9:43
TIL why that curve is called a catenary - catena is Latin for chain.
Wow, I had no idea!!!
Huh, that must be why concatenation is chaining!
And I now learned that the German word for chain "Kette" also comes from the Latin catena just like "chain" does.
Interestingly enough, one still says catena in Italian.
Being Italian the link between the name catenary and chain (catena in Italian) seemed so obvious that it wasn’t even worth mentioning. Forgot that English speakers have a different perspective.
If anyone builds a dome after watching this video I expect to see pictures.
Yesss
On second thoughts, just send me dome pictures. SEND DOMES.
@@TomRocksMaths YOU MUST CONSTRUCT ADDITIONAL -PYLONS- DOMES
@@oz_jones great reference.
Hey you’re the dude
"how do we make a dome without any supports?"
"well.. first make 2 domes, and then support it with a third"
I like the Roman solution better. Just do one dome using better materials than anyone could replicate until the nineteenth century. You can go into the Pantheon and see that they even made a giant hole in the middle to let in light. Brunelleschi and the Hooke/Wren team may have built bigger domes, but their domes were much more complex, and they didn't have holes in the middle.
@@Bacopa68 There is a hole in the middle of the inner dome of St Paul's, with windows near the tops of the loaded dome with gaps in the outer dome near the cupola so light can reach the inner dome, but someone inside the church can't see those windows, nor can someone outside at ground level. The gap between the domes is taller than needed to conceal the load bearing dome because Wren wanted the inside and outsides to look like a solid hemispheres and the added height closed the sight lines to the ground and lit the entire inner dome with skylights.
Well if you support a dome with another dome that itself has no support, then you succeeded in making a large dome with no support.
Can someone link a force diagram of this?
@@Bacopa68 In fact the Pantheon's dome is bigger than St Paul's, it's 43m wide. But that's not the main issue, the Pantheon's dome is very close to the ground (it's only 43 m high, you can fit a sphere in it that would touch the floor and the dome at the same time). St.Paul's dome though is quite high and that's the problem. Domes create lateral forces that need to go all the way to the ground, if domes are high they need huge pillars to get those forces to the ground. St. Peter's dome in the Vatican is 43m wide just like the Pantheon's. But comparing the 4 pillars of St. Peter to the Pantheon brings the Pantheon to shame, the Pantheon is so slim in comparison it is incredible it's still standing, and that's not even taking into account the rest of St Peter's building that also helps supporting the dome.
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.
You just explained exactly the issue, that the video should have :-D
Of course they were minimizing the bending and shear stresses in the construction and did not try to find the chain curve...
@Numberphile: I think that you guys should pin that comment. :-)
(I'm a large fan btw)
There is also the problem of the dome having double curvature
I think the video was partly about how they did it _without_ having the knowledge about the _cosh_ function.
But in explaining why the _cosh_ is the optimal dome form your comment added a lot of value. Thank you!
@@alfeberlin true! And the video is fascinating. I'm really into how they figured things out like that without the kinds of tools we have now. I just thought the explanation involving energy that they provided in the video might not work for some people, so I thought I'd add my two cents.
Doesn’t this lead to a minimal surface problem - a two-dimensional analogue of the hanging cable problem - instead of a hanging cable problem itself? I understand that it makes sense that historically, they reasoned based upon hanging cables, but wouldn’t it be cool if they realized that a “hanging sheet” would be a better model for them to use? If it turns out mathematically to be the same shape, I’d be surprised, but I’d love to see that proof using variation as calculus, or, as I guess it is usually called now, differential geometry.
Numberphile: * releases a video about mathematical domes *
Me, an Italian, surrounded by domes and cathedrals everywhere: 👁👄👁
The Duomo in Florence is by far the most ludicrous dome on the planet
At first I thought this video is about the Pantheon
@@janmelantu7490 It was mindblowing to see! Love Firenze!
@@janmelantu7490 Yeah, we have some insane domes. The Pantheon is the biggest dome in the world which is not made out of reinforced concrete and does not have any scaffolding. Just amazing🤩
@Nikhil RaajeMaankar where is that?
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.
This was how Gaudi designed the Sagrada familia, using string, upside down, with weights where point loads are located.
Indeed! Lookup the drawings of what they build and you see the inner "dome" is much more like a cone. This is to support the heavy lantern. A chain line is ideal to support an even load, not a point force.
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.
Gaudi used this trick for a lot of his arches. The fact that it seems so simple once you see is so nice.
30 years ago I visited the Sagrada Familia and was absolutely fascinated by the model you are talking about. Originally I planned only to spend about an hour there but stayed for the rest of the day (6hrs) after seeing the model. Not only that but went for dinner and came back in the evening only to find the place closed. Only the two end facades were completed and the core of the Basilica was still very much empty. Only the base off some of the columns were present but from the model one could visualize how it would come to completion. Would love to go back now.
Yeah, because a catenary is only the shape when it's only supporting its own weight. Add extra load and you need a different shape -- see suspension bridges, for example.
I feel like they really underplayed that part, probably because the curve would no longer fit the maths. If it's pointy I'd bet it's something to do with the added e weight of the massive stone tower at the top. Also, they did not mention the horizontal component of the force at the base of the dome is relative to how steep the sides are.
I wish this video existed when I was doing taylor series and cosh, sinh. He broke it down so well
Glad you enjoyed it!
There are also geometric and functional definitions that give a lot more insight into the parallel with trigonometric functions.
@@EebstertheGreat Yes, and they (hyperbolic & circular trig functions) are connected to each other by some neat complex relationships.
cosh(ix) = cos x . . . cos(ix) = cosh x
sinh(ix) = i sin x . . . sin(ix) = i sinh x
e^(±x) = cosh x ± sinh x
e^(±ix) = cos x ± i sin x
etc.
Fred
But... this is literally how it’s explained in every maths textbook? There’s no extra insight here?
@@ObjectsInMotion Well, sure. This stuff is a few centuries old; it contains extra insight only for those who are seeing it for the first time.
Fred
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.
supporting the outer dome... and the lantern at the top... thats a huge point load at the center of the chain.
I think Gaudi's workshop had lots of ropes, pulleys and weights for designing arches upside down.
@@raykent3211 Yes, back then, a lot of fancier structures were designed with such physical models - which also leaves them vulnerable to damage. For example, we lost a lot of Gaudi's work during the Spanish revolution.
I just go to the Dome Depot.
I sooooo hope that was a Simpsons reference, because, well... I like the Simpsons...
@@rocaza21 me 2
I only have a Hyperbolic ome Depot where I live.
nah you need doug dimmadome owner of the dimmsdale dimmadome
This man is the maths teacher your friends have and you're really jealous of
Well he teaches at oxford so yes you would be jealous
He looks like the cool uni professor that hangs out with students after class but is also amazing at teaching.
He is great at explaining and also has a lot of enthusiasm. It's a pleasure to watch him on Numberphile.
@@schifoso
Great enthusiasm.
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
The bougiest gingerbread house of all time - love it
false.
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!
What an incredible teacher. Such clarity, excellent scaffolding, and enthusiasm!
You can't really teach dome construction without excellent scaffolding!
Love learning from this boy
Tom's enthusiasm is positively contagious :)
@@TomRocksMaths Subscribed. Norwegian nautical engineer here - have had much fun with the NP content with you.
You should have mentioned the Sagrada Familia! The entire building is basically built up entirely of catenary curves and related hyperbolic and hyperboloid shapes!
You know, you sometimes have to build a giant dome in your home for a reason.
false.
This is several hundred years after St Peter's in Rome, or Brunelleschi's dome in Florence, each of which are much larger
and there is some evidence that the Egyptians knew about the catenary curve, too.
@@timbeaton5045 insert joke about dome-shaped pyramid.
@@TomLeg Indeed! Don't think they used it in their architecture, but it appears they were aware of it. Almost certainly not rigourously , mathematically, but then even Galileo thought the hanging cable curve was a parabola.
Yeah but St Peter isn't a catenary and Brunelleschi's isn't even a proper dome: the base of the outermost dome is a polygon, not a circle.
Yeah, I don't think Hooke invented using a chain to build a dome. People have been using that method to build arches and domes since the Romans. Although perhaps Hooke was the first in the West to try and describe it mathematically. (Was he the first to come up with the cubic approximation?)
So when Tom Crawford doesn't show his tattoos he becomes Thomas Crawford
Alter ego. Thomas likes drinking tea and long walks in the forest.
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.)
Actually, for an arch/dome it _maximizes_ the potential energy of the building blocks. Any other curve of the same length would have less energy. For a chain it obviously _minimizes_ the energy of course. That's the reason in the first place the chain takes that shape.
@@alfeberlin Yes, that makes sense about the potential energy. I don't think of gravitational potential as "internal."
When I said it minimizes the internal energy, I meant the energy in the bending moments and internal strain. (I think. Again, I learned this in undergrad over a decade ago.)
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
Yes, and from someone else in the comment section I learned that catenary is Italian for chain and now you know why :)
@@stylis666
Caternary is one of those words that invariably you pronounce incorrect until someone informs you.
Originally I said cater nairy. Doh.
@@michaelslee4336 His pronunciation is British. The American pronunciation is "CAT-uh-nary"
@@jacemandt
I’m an Aussie and I say kuh tin er ree
@@michaelslee4336 That's how I've heard it forever. CATuhnary. This is the first time I've heard it said a different way... and it's very different...
Same with "kosh" and "shine"....
Please keep bringing Tom back! He's awesome
Indeed!
His enthusiasm is contagious.
Great video!!! The caternary curve is also the basis for the St Louis Arch in Missouri.
Yes! Came here to say this. It's quite a famous structure, and it's a shame that a picture of it didn't make it into the video.
@@jacemandt That's because it's bigger than 34 meters!
matt parker:
there is only one true cosh curve!
*coshoid
Gloria In X-squaris!
Just what I was thinking
Illuminati is run by Numberphile confirmed?
Matt would've done it better than Tom
Yes i have always wanted to build a giant dome
now my dream can come true
Really 🤔🤔
You and me both.
Shine crew for the win.
He pronounces sinh as "shine"? That makes no sense but I like it.
How is it normally pronounced? I know these functions exist but I don’t know what they are so I wouldn’t know. I feel like it’s just simply a way to make it verbally distinguishable
@@redsalmon9966 when I learned it my maths teacher called it hyperbolic sine
@@frenchyf4327
Yeah surely that’s one way to say it but that’s just standard
@@redsalmon9966 "sintch"
I've only seen it called hyperbolic sine or sinch
We now need a parody of Matt Parker's "there is only one parabola" but it's "there is only one catenary" instead
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.
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?
He said "one of", which doesn't mean the most famous. Also, the Pantheon snickers contemptuously at Il Duomo
I would argue that Hagia Sophia
is the most famous one, and also oldest.
That being said, the perception certainly completely depends on where you live, and it does not matter much.
The Capitol building: Am I a joke to you?
@@leodarkk I believe the Pantheon in Rome, is at least four hundred years older than Hagia Sophia.
Hagia Sophia in Istanbul...
Tom's ahead of his time guys. Little do you know, that's how they write 5 in the future.
This^^
I really like the videos with Tom Crawford. He explains things well and shows an excitement for each topic. Fun to watch and learn.
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!
That sounds like a numberphile2 video waiting to happen.
How about a proof that one must use calculus of variations to prove that it is the only way to solve the problem? In fact, what even does that mean?
I love that he just pronounces “cosh” as it’s written, and sinh as “shine”. Would that make tanh “than”?
Why, yes. Yes, it does. One of my lecturers (30 years ago) used "cosh", "shine" and "than" for cosh, sinh, tanh.
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).
Now i understood *how* cosh and sinh are defined by e^x and e^-x, great video!
wonderful, math is in everything around us
Yes,of course
What kind of math is in a potato then..?
especially in your phone
@@Pakuna So much, where do I begin. XD
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!
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.
Please do a bonus video on this where the full calculus of variations treatment is shown.
Is it bad that all I can think is "I've never heard anyone pronounce hyperbolic cosine like that."
I was resting easy til he got to sinh and then he blew my mind :P
Much more used to hearing “sinch”
In my university we always said the whole thing or "sine h" and "cosine h"
"cosh" I'm fine with, but.. "shine"? That's not where the h is! But it's hardly the weirdest pronunciation in maths.
@@SimonBuchanNz Yeah, I didn't notice he was saying shine until after I wrote the comment
I and my schoolmates called them cosh, shine and than (starting part of thanks). Maybe because our teacher used those.
Ok. This one blew my mind. A very simple but brilliant idea.
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
(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!
Literally every group project I was in school. Feelsbadman
Love you numberphile!!!❤️❤️❤️❤️
As an engineer, here’s my take on catenary curves…
When a chain or rope is held from two points under uniform gravity, there’s pure tension in the structure (and no local bending, which could cause a crack).
When the shape is flipped, there’s pure compression (and no local bending), which is super-efficient. The Sagrada Familia in Barcelona is all about this.
In the St Paul’s case, it looks like the support curve has a ‘point load’ at the upper dome, which is non-uniform loading, and therefore not a pure catenary shape.
I can feel a simulation model coming on…!
I have been waiting for a numberphile on hyperbolic curves, please add a part 2!
Gaudí used the same method of strings and weights to plan his buildings, especially the columns.
omg that upside-down Sagrada Familia model is mesmerizing on its own
Yeahhhh, straight from the top of my dome,
as I watch, watch, watch, watch Numberphile at home.
Brilliant
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.
Dr Crawford you legend! Always enjoy your videos
Wren: Sir, can you build me a cubic support chain?
17th century building contractor: ?
First time in life i understand the cosh(x) and sinh(x)
same
But they get greater than 3 pretty quick...
Happy to help!
Lucky you, so glad I stopped learning maths when I was able to count to ten. Live is perfect without these nerdy bits
Building a dome looks like so much fun! The power of math is endless!
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
If the gravitational constant had a different value, then would it still be a cosh curve?
Like the force is stronger now, right?
as long as it's constant, yes, because the potential energy would just be scaled, and you'd get the same curve to minimize it
It would still be a cosh curve, the shape distributes the force evenly, no matter the actual value of that force.
The amazing answer is : Yes, it would!
As long as your dome is not so humongous that the gravitation could not be considered as a constant (and I mean a mountain-sized humongousness, here), the shape will always be a cosh, disregard the value of the constant.
Of course, there would imply a change in some parameter inside the function, but still a cosh.
Oh I get it now, mass would still be distributed in the most efficient manner.
Cosh curves are awesome
@@aplanosgc6963 yeah! But I wonder what the parameter would be, like there's only one variable x
8:04 and I thought my 5's were lazy...
Thank cosh I wasn't the only one who noticed that monstrosity 🤣
Not my best work...
Took me a bit to work out why there was a square root sign there. 🙂
Well, if you're gonna say "shine", you might as well make your 5's look like √'s. Tom seems a bit of an odd function.😁
An intuitive explanation
A chain can only transmit pulling forces, no moments. When the chain is hanging under its own weight, each link aligns in a way to only transmit longitudinal forces, with no bending. When you flip the whole thing, now the structure is in pure compression, again with no bending. That’s why it's so strong.
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.
I want to be as happy as the guy talking about domes lol.
Find something you love and keep doing it
It’s hard to find something to love :’)))) everything are just everywhere
I've heard them pronounced as
cosh -> cosh
sinh -> sinch
tanh -> tanch
And I'm more used to cat-en-ary, not ca-tin-ery.
sinch doesn't shine as much though
it takes a second to get used to pronouncing it as shine tbh when i first learned it i kept writing shin x and having to rewrite it
Math and history and history of science. Thanks for this amazing video 😊
I learned so much about St Paul's Cathedral that I didn't know. 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…
I had no idea what sinh or cosh was. And why it was related to e^x. Thanks Tom for explaining it so well!
love that the angelic sphere appeared right at @3:14
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.)
It's actually our language, the yanks messed it up.
4:42 Euler again
The answer to pretty much every maths exam question that has ever existed.
@@TomRocksMaths WAIT YOU'RE THE GUY
@@TomRocksMaths didnt know u had a channel
Subbing right away
"...because Euler does what Euler does!"
-- James Grime
Wow, cool: The outer dome is a hemisphere and the inner dome is a hemisphere so that most people will feel like that's just a hemisphere, allowing you to hide the stable shape inside an inefficient shape without anyone noticing! Pretty clever!
Great video. Now I can finally put that dome on my garage!
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.
Brunelleschi: Am I a joke to you?
Well, his dome's a classical gothic pointed arch. And there are no three stacked domes, just one dome with ribs. And chains. Now I have to know which pointed arch proportions best approximate the catenary curve.
@@LordEvrey wasn't it two stacked domes?
Also, as far as I know the catenary method was known and used to model most gothic cathedrals' domes, supporting arches etc, but it was a well-kept secret of the architects/Stone masons guilds
I think when he was answering Brady's question about a small dome on his house (around 10 minutes), Tom misspoke slightly, saying that the smaller house would necessitate a squatter dome. He later partially (but very vaguely) corrects this mistake by saying there are two variables: the distance between the ends and the length of string/amount of stone available. Basically there are three variables, two of which are free and the third can be calculated given the first two. I'm fairly sure this gives you the freedom to choose the height and width of your dome, provided there's no restrictions in terms of material available.
Amazing video!
8:15 no kinkshaming please
7:56 He said SINCH in a very strange way........
Wonder how he says tanh
@@pyraliron Probably something like "George"
@@pyraliron If we use his existing algorithm, he'd pronounce it "thane". (Or maybe "than".)
Sin-cha-cha-real-smooth
@@pyraliron »Than« I bet.
I dont know how cosh and sinh are conventionally pronounced in english, but calling it "cosh" and "shine" really got me aggrevated.
I'm from germany and here I usually just heard sin-h and cos-h, maybe its that I am not used to it, but damn I flinched everytime he talked about coshs and shines...
I've always said them as the English words "cosh" and "cinch".
In German I always used _kosinus-hyperbolikus_ etc. Coming from that, _cosh_ is a great improvement.
Precisely the information I was looking for a couple of weeks ago. I ended up doing the chain suspended from two points over a graph.
"That's doing the business" 😆
Sir Christopher Wren
Said, "I'm out dining with some men.
If anybody calls,
Say I'm designing St Paul's"
HOW they build the dome(s) - and all the stuff filling the space in between - is it's own amazing tale.
Now this explained me why cosh and sinh are used in architecture... thanks!
they got DanTDM on numberphile this is crazy
That man has a Wii on his table, and I want to be his friend
First illustration I saw of this in a structure was the Gateway Arch in St. Louis, Missouri, about 150 miles from where I grew up, definitely based on a catenary... completed when I was about 4 years old.
I noticed if you take two endpoints at even height and do one period of a cosine wave between them and then break that into a bunch of end to tail vectors along the curve, you can then sort these vectors by increasing y-value and they arrange end to tail in this new order into what I think is a catenary. This reminded me I was trying to prove that a while back and forgot.
Have you ever heard about Hagia Sophia :D ?
Knew there was the inner and outer dome, didn't know there was the third support dome in between.
Not once did I feel like I didn't understand what was going on. Excellent explanation!
When talking about domes one should mention the largest unreinforced solid concrete dome in the world, the Pantheon in Rome build under the emperor Hadrian in around 120 A.D. With its 43.4 metres in diameter it was the largest dome for over 1300 years. It has a perfectly spherical inner shape and it still stands after nearly twenty centuries and numerous nasty earthquakes. A real marvel.
like what? sinh is pronounced as "shine"? I've always pronounced it as "sinch"...
I've heard: hyperbolic sine, sinus hyperbolicus, sinsh (similar to cosh) and sin-h, but I've never heard it pronounced shine which is my new favorite.
You're welcome :)
I know that for the same distance between my fingers, if the string is long enough, I can get it to droop down as mush as I want.
From this it follows that for any width, you can build a dome of arbitrary height. Cool!
i appreciate the explanation