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: en.wikipedia.org/wiki/Hyperbolic_function
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
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
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..
So inspiring. They say poetry gives you a new outlook to life. For me, it's Mathematics. Thanks James, you showed that this glorious orchestra numbers and equations (Mathematics), is not only beautiful and elegant itself, but also has the potential to describe and express the beauty and elegance of the world around us. Thanks again.
As long as you stay curious, either maths, physics or engineering are awesome I think (and more...). There're things to learn in each field, and it'll open your doors to others. You can extend your formation rather quickly at uni, or even by yourself. I've studied civil engineering for my first year, now I'm working 2 months in a statistical environment, trust me, even this make you thirsty once you're in. Hopefully, we have internet now... With so much resources from such inspiring people.
Damn it, I just found your channel after having seen all the numberphile videos. I love you, man. You're so happy about maths and everything. Your smile is just killing me. I could never get enough of you or your videos, even if you failed me in maths. xD
James I want to say thanks for all the videos on your channel and Numberphile. You've really inspired me to try cool and interesting things with math. You always show the beauty of mathematics, and I thank you for that.
I may not always follow what you're teaching us but your enthusiasm makes the videos both enjoyable and keeps challenging me to come back for more. Thank you :-)
I'm studying physics in second semester, and we do the mathematics behind it right now. I thought it was really cool to find out the curve of a chain on my own with simple mathematics. Totally awesome. The really cool thing about these mathematics is that you can even calculate the way light goes throught various media, or how to build a slide where you can get from A to B in the shortest amount of time possible. And the maths behind it is all the same. How great is that?
You know, this newest video has probably the highest quality of all your videos I've seen. While I watched it I thought, wow, this is indistinguishable from those BBC documentaries about mathematics (eg. dangerous knowledge, the secret life of chaos etc). I don't know how difficult it was to make this video, but keeping up this kind of quality will surely open many doors for you!
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"?
Great example for great scientific journalism. Well explained, well packed, good Video quality, nice special effects, sympathic host, impressive locations, everyday applicability, eloquent anekdotes... keep on!
this is the most unique video i have seen posted by singing banana. Its actually quite soothing. *Replays* I don;t understand the math at all. but i love the music.
I just spent my entire friday night re-watching all singingbanana videos while building a post-it dodecahedron. it was amazing :) looking forward to many more videos!
i've always liked maths in my entire life, since i was in kindergarden, and when i watch these kind of videos i feel inspired to be a great mathematician like this guy :D
James, you got yourself one more subscriber. Your explanations are amazing, simple to understand and expressed with alot of passion. Greetings from Portugal and best of luck!
James, you're lovely and inspirational! I've never been too bright at maths but ever since I've begun watching your videos it's sparked an interest I never knew I had. I'd love to be there if you ever do any talks in California.
A beautiful presentation bringing to light the many wonders of Mathematics all around us. Also, as a civil engineering undergrad, i happen to have a professional interest in the catenary. Cheers! Subscribed :)
Looking at this video I have no doubt that one day, if you so choose it, you'll be to mathematics what Carl Sagan is to physics, James. All you need is some funding and a directive crew really, you've already got the rest. Keep up the good work!
I'm sure Vi Hart mentioned the non-parabolic/ellipse properties of these curves, but gave a geometric construction. If you roll a parabola along a line, its focus traces a catenary... which just adds to the elegance and confusion. Especially when you've studied the conic sections question in Technical Drawing.
You're objectifying his clothing and reducing it to sex. Whilst doing this, you've ignored his message. Shameful and sexist. Perhaps you should lose your jobs and earning potential. #metoo equality.
Although the math shown in all his videos are amazingly well displayed, I absolutely love the background music. It's beautiful and makes the video that much more pleasing. I hope he includes other pieces of classical music to enrich his videos :)
I believe because the rope bridge can change its shape (in response to any loads) as you move between its supports, whereas the suspension bridge is designed not to. This results in an increase in moment (turning forces) as you move towards the centre from either direction. The supporting cables now have to distribute that increasing 'sideways' force and the result is a slightly 'shallower' (parabolic) curve compared to the free-hanging one. HTH
A bit off topic, but quickly: they guess a word that might appear in the message and try to find where it fits. The clue is that a letter can't become itself. So if I take a German word like "wetter" it wouldn't fit where the code says "njgthk" because that means the 't' in wetter becomes 't' in the code. But maybe "wetter" fits where the code says "hgsohp" because then there are no matches. This means 'w' becomes 'h', 'e' becomes 'g' and so on. That's your starting point for breaking the code.
Great video, such high quality and clearly explained. I have my Cambridge maths interview on Tuesday - so nervous; thanks for calming me down with the beauty of maths, if only a bit.
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.
Hi James!! I just found your channel after a LONG time watching Numberphile, and I just wanted to say that your videos (here and the Numberphile ones with you in them) are totally cool, and your awesome! That's all... :)
I know your doing this channel besides your professional work and Brady is making videos for a living. But seeing this video it seems the other way round. Which shows you are truly into it, take it as a compliment ;-)
Hi James, I just came here and subscribed to your channel after watching the last numberphile video you and brady made-the hang-out. I couldn't resisnt, you are an amazing person and I love your videos.
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.
Thanks for the suggestion! A very interesting piece to read. It's not that I have a problem with mathematics in particular, I actually am a physicist myself and I like mathematics very much. But I agree, that mostly people are scared of mathematics is because it is taught as some very abstract thing without any connection to a real world.
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.
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:
en.wikipedia.org/wiki/Hyperbolic_function
thanks very helpful
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
This is wonderful. I'm glad it did so much for you!
I'm not sure how many times I've watched this. I just keep coming back to it; it's simply beautiful.
You are on your way to being a leader and ambassador for science and mathematics, and I think everyone will know you name someday!
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
Your coat is awesome.
The best and most informed mathematical video I ever watched.
That’s why everyone loves James!
The music makes this somehow beautiful. This is going straight to my go-to videos to show to people not into math.
the way he explained nature and practical applications of the whole catenary concept left me speachless. bravo!
I love how all comments are about this video being much more professional than your previous ones. - And I love the video itself, great job.
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..
I bet, Mr. Grime, that you enjoyed doing this clip very much.
And congrats on the amazing production, hats down to Mr Bailey!
this video deserves at least 1 million of views , beautiful , thanks doctor.
So inspiring. They say poetry gives you a new outlook to life. For me, it's Mathematics. Thanks James, you showed that this glorious orchestra numbers and equations (Mathematics), is not only beautiful and elegant itself, but also has the potential to describe and express the beauty and elegance of the world around us. Thanks again.
Wow! I hope you enjoy it!
As long as you stay curious, either maths, physics or engineering are awesome I think (and more...). There're things to learn in each field, and it'll open your doors to others. You can extend your formation rather quickly at uni, or even by yourself.
I've studied civil engineering for my first year, now I'm working 2 months in a statistical environment, trust me, even this make you thirsty once you're in. Hopefully, we have internet now... With so much resources from such inspiring people.
Absolutely fantastic!! I have a math channel in Brasil, but I wanna be like singbanana. Thank you teacher James Grime!!
The style and background music certainly has changed over the year, but the passion is the same as ever. Love your work.
Damn it, I just found your channel after having seen all the numberphile videos. I love you, man. You're so happy about maths and everything. Your smile is just killing me. I could never get enough of you or your videos, even if you failed me in maths. xD
Every single video you do from now on should have super profound, inspirational music in the background.
Wow, this video has amazing production value! At first I thought it was an announcement for a full length documentary about math at first...
James I want to say thanks for all the videos on your channel and Numberphile. You've really inspired me to try cool and interesting things with math. You always show the beauty of mathematics, and I thank you for that.
Really liked this video, loving the way it was shot as well. Seemed really professional as if the BBC had put it together or something.
I may not always follow what you're teaching us but your enthusiasm makes the videos both enjoyable and keeps challenging me to come back for more.
Thank you :-)
I'm studying physics in second semester, and we do the mathematics behind it right now. I thought it was really cool to find out the curve of a chain on my own with simple mathematics.
Totally awesome.
The really cool thing about these mathematics is that you can even calculate the way light goes throught various media, or how to build a slide where you can get from A to B in the shortest amount of time possible. And the maths behind it is all the same. How great is that?
You know, this newest video has probably the highest quality of all your videos I've seen. While I watched it I thought, wow, this is indistinguishable from those BBC documentaries about mathematics (eg. dangerous knowledge, the secret life of chaos etc).
I don't know how difficult it was to make this video, but keeping up this kind of quality will surely open many doors for you!
Ok, this might be one of the best videos on youtube!
I always find your explanations and videos interesting, but somehow this one is especially beautiful. Great video, Dr. Grime! :)
I believe that's cause of the music in the background! Really good choice of music here!
James, please make a video like this it's truly marvelous !
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"?
wow, this is really professionally edited and presented, and fascinating as usual of course! great work!
beautifully filmed and created. Great quality, mixed with the usual great way of expressing out concepts you have. A "like" is not enough
really liking the professionalism! Especially the audio quality and the graphics.
Bah, who needs TV anyway. We have RUclips.
Excellent video James! I had no idea these curves existed. And the part about arches? Mind-boggling.
Great example for great scientific journalism.
Well explained, well packed, good Video quality, nice special effects, sympathic host, impressive locations, everyday applicability, eloquent anekdotes...
keep on!
this is the most unique video i have seen posted by singing banana. Its actually quite soothing. *Replays* I don;t understand the math at all. but i love the music.
I just spent my entire friday night re-watching all singingbanana videos while building a post-it dodecahedron. it was amazing :)
looking forward to many more videos!
i've always liked maths in my entire life, since i was in kindergarden, and when i watch these kind of videos i feel inspired to be a great mathematician like this guy :D
the production work in this video is phenominal!
James, you got yourself one more subscriber.
Your explanations are amazing, simple to understand and expressed with alot of passion.
Greetings from Portugal and best of luck!
Absolutely brilliant, You are very inspiring James
So classy and professional.
Love the video! Great music too.
Can't wait for the next one!
Great editing Dr. Grimes!!
I like the high production values of this video, James.
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.
Excellent video, Dr. Grime. I can't believe Ben never told me about this.
you came to my school today and i just wanted to say that you did so well! you actually made me like maths a bit more :D
not to lose the focus on the incredible math that was going on this video but your outfit was amazing
WoW that was mind-blowing!
Awesome!
James, you're lovely and inspirational! I've never been too bright at maths but ever since I've begun watching your videos it's sparked an interest I never knew I had. I'd love to be there if you ever do any talks in California.
A beautiful presentation bringing to light the many wonders of Mathematics all around us.
Also, as a civil engineering undergrad, i happen to have a professional interest in the catenary. Cheers! Subscribed :)
Awesome!!!Beautifully made!!
I could listen to you all day! :D Thank you for bringing me some understanding to what I once thought was impossible for me to do so!
Looking at this video I have no doubt that one day, if you so choose it, you'll be to mathematics what Carl Sagan is to physics, James. All you need is some funding and a directive crew really, you've already got the rest. Keep up the good work!
I'm sure Vi Hart mentioned the non-parabolic/ellipse properties of these curves, but gave a geometric construction. If you roll a parabola along a line, its focus traces a catenary... which just adds to the elegance and confusion. Especially when you've studied the conic sections question in Technical Drawing.
I seriously can't be the only person who was distracted by how sexy his outfits were. I mean, dat coat was amazing.
no shit right. where did he get it??!
You're objectifying his clothing and reducing it to sex. Whilst doing this, you've ignored his message. Shameful and sexist. Perhaps you should lose your jobs and earning potential. #metoo equality.
@@LitoGeorge man shut up lol
I like your style, Dr. Grime.
An excellent video, keep up the good work
I admire your passion, sir.
I really like this new format! Hope to see more of it in the near future!
This is really informative! Please make more videos regarding the mathematics in nature and engineering. :)
Thank You!
Although the math shown in all his videos are amazingly well displayed, I absolutely love the background music. It's beautiful and makes the video that much more pleasing. I hope he includes other pieces of classical music to enrich his videos :)
The beginning of this video reminded me of a BBC documentation. Well done!
GREAT video! Music in the background was amazing
This is like a television public service announcement...really well made!
I believe because the rope bridge can change its shape (in response to any loads) as you move between its supports, whereas the suspension bridge is designed not to. This results in an increase in moment (turning forces) as you move towards the centre from either direction. The supporting cables now have to distribute that increasing 'sideways' force and the result is a slightly 'shallower' (parabolic) curve compared to the free-hanging one.
HTH
A bit off topic, but quickly: they guess a word that might appear in the message and try to find where it fits. The clue is that a letter can't become itself. So if I take a German word like "wetter" it wouldn't fit where the code says "njgthk" because that means the 't' in wetter becomes 't' in the code. But maybe "wetter" fits where the code says "hgsohp" because then there are no matches. This means 'w' becomes 'h', 'e' becomes 'g' and so on. That's your starting point for breaking the code.
Awesome!!! Love your work.
This was wonderful, well done.
you look MAJESTIC i n this video daaamn
wolkenbruch It's in the editing.
Great video, such high quality and clearly explained. I have my Cambridge maths interview on Tuesday - so nervous; thanks for calming me down with the beauty of maths, if only a bit.
Cleaned up extremely well
very enlightening. very inspirational
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.
This man need his own television show, to be seen on the likes of bbc2 maybe? :) keep up the great work, sir.
Love the way he says Bubbles
This is very like a mini-documentary about math(s)! I would love to see more!
Very nice. And production values are up, too. Good job all around.
Agreed. This guy ought to be on TV!
good job man this is like an epic movie.
Hi James!! I just found your channel after a LONG time watching Numberphile, and I just wanted to say that your videos (here and the Numberphile ones with you in them) are totally cool, and your awesome! That's all... :)
That was beautiful!
I know your doing this channel besides your professional work and Brady is making videos for a living. But seeing this video it seems the other way round. Which shows you are truly into it, take it as a compliment ;-)
Excellent, very Cool James!
This was rather AWESOME
such a simple and lovely presentation
what a beautiful video, great work,
Amazing production!!! Thank you so much James!!! That suit looks great on you! ;)
I mostly watch this video because I love how he says "bubbles."
Yes! Very cool.
nature is so beautiful, *wipes away tears*
Hi James, I just came here and subscribed to your channel after watching the last numberphile video you and brady made-the hang-out. I couldn't resisnt, you are an amazing person and I love your videos.
Brilliant presentation 👍 thanks a lot
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.
You should get a show on tv! I would certainly watch it!
Thanks for the suggestion! A very interesting piece to read.
It's not that I have a problem with mathematics in particular, I actually am a physicist myself and I like mathematics very much. But I agree, that mostly people are scared of mathematics is because it is taught as some very abstract thing without any connection to a real world.
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.
oh man, you nailed the tv documentary style of walking around and talking.