Brilliant. It makes you realise just how good at maths you have to be to get into places like Oxford and Cambridge. It also highlights really well the difference between being able to 'do' a topic in maths and really understanding it, and once you've got the understanding how you go about applying it. It's also great to see how much fun you can have doiing maths.
With good education brought up in developed nations, you should be quite geared towards those academics, especially math, which is largely logic. Th rest of math, you can explore on your own when you're interested (dopamine rush?), but do take note that most mathematicians who are too into their games might at times, lost touch with the current world...
Well kinda not I mean those questions surely are quite hard to someone who hasn’t looked into math. However for someone who enjoys doing math (in their spare time)at least the first question should have been easy. I’d also like to note
This is such an awesome collaboration between two of my favourite maths UA-camrs. Unfortunately I was rejected before interview this year which sucks especially considering I was rejected after interview last year haha, but that was so much fun to watch!
I'm glad you enjoyed it, but sorry to hear about your application. Keep your head up, there are SO many other brilliant universities out there that would love to have you :)
@@TomRocksMaths Thanks for the kind words! I wasn't expecting a response haha. At the end of the day, there is still the opportunity for a masters degree at Oxford should I choose to pursue it, and I will be sure to make the most of wherever I go. One things for sure, I will stick with watching your videos for as long as I can!
Same thing just happened to me now , do you reckon its worth it to take a gap year and re apply , or should i go to warwick and apply for masters. Are you doing a masters now or next year?
wow, this was so amazing! thank you! I kinda always thought all professors at Oxford were really old and closed-minded, thinking only they are right, keeping minimal interaction with "normal" students or others, this has been really eye-opening for me to see how such fun ppl are at Oxford uni. Also, it was great to see these interviews for real and the things that go on in them!
This was so awesome to watch! 😁 Sad I've only just seen it now. I had 6 interviews for Chemistry at Oxford but unfortunately didn't get in. I cherish that experience because it really taught me about what interesting questions actually are. I've thought a lot about the answers I gave and the likely answer expectations, but my GCSE grades were meh compared to the average applicant and there were no entrance exams for Chemistry when I applied about 12-13 years ago. It's always been a dream to study at Oxford. I have a Chemistry degree. I'm 2/3 through a maths degree and doing well and loving it. One day maybe I could come back to Oxford and ace the interviews. Well done Steve for making that look easy! In the normal case there would typically be a lot more give and take between interviewer and interviewee I presume... it depends very much on the experience of the interviewee (and interviewer to a degree). Plus the passion in the way Steve communicated, written and verbally, would surely have out-shone anything incorrect he may have written accidentally on the day. Then you have me who blurted out an incorrect answer to the first question I was given in my first interview. 🤣 I'll never forget the bond angle of CO2 ever again though! 🤣 Luckily idiocy is often a very useful quality! 🙃
I'm forwarding so many of these to my youngest. He's a budding STEM major in his freshman year of high school. Hopefully will give him the confidence to consider some of the top schools seeing what the interview questions involve.
It's analogue to a model of the universe, geodesics on the surface (light traces) travel for ever without ever coming back to the same point so we say the Universe is infinite but the circumscribed volume is finite so therefore there's finite mass in the Universe.
If the paint is infinitely thin it would actually be possible to cover the whole surface with LESS paint than pi units. The only reason Gabriel’s horn seems paradoxical is because we instinctivily apply physical properties, eg. like paint having thickness, to a mathematical construct which is impossible to build in the real world. But it is actually possible to imagine paint with finite thickness being used to cover the surface. Create two horns, one bigger than the other, fill up the big one with paint and then insert the smaller horn inside it. Remove the bigger horn and you are left with a horn that is both filled AND covered by the same amount of finite paint :)
Once again we witness that people are cool, when as feel confident about something. Not being able to solve a problem makes us all nervous. Great video and wow, what a knowledge man.
For more special guest appearances check out the 'Interviews' playlist including videos with 3blue1brown, Hannah Fry and several famous mathematicians: ua-cam.com/video/UsRfECCPsCY/v-deo.html
If all Oxford mathematics interviews are like this, I'll definitely encourage my son to apply. Maths should be fun! Do you ever teach computer science students?
When you start to study it is not than entertaining most of the time because you have to learn the basics of higher mathematics. It is great when you start to see the patterns and when you can apply the rules easily to solve different problems.
The paradox isn't really a paradox. If you paint a surface in the real world, you would need to apply a layer of paint with some thickness, and below a certain thickness you wouldn't consider it to be properly painted. But the horn gets narrower and narrower, so no matter what thickness you pick for the paint layer, the horn will be thinner than that at some point, meaning a finite ammount of paint wouldn't "properly" paint the surface of the horn in the real world. Any volume can be split into infinitely many 2d surfaces however, giving a volume an infinite ammount of surface area. So mathematically, a volume of paint can cover an infinite surface area. It is kind of like how you can travel a finite distance by halving the distance to your destination an infinite number of times. So long as the time steps also become infinitely small, you can do this in a finite time, but if you needed a set time per halving, you could never finish.
Thank you very much for this video. Before I always treated dx like part of notation and just did integration as though it was algebra but now this video actually explained what dx is.
Thanks for the insightful problem - I've heard of Gabriel's horn and did these calculations a couple years ago, and now looking at problems to hopefully have some questions for an interesting school math competition, and just to practice my own maths, this is pretty amazing to watch. Right now my high school is new [only around 2 years old right now], and I really don't have much "history" to go off of, so these are the types of problems I hope to add... problems that require no more than a basic intuition of derivatives, integration, and a bit of cleverness since at the end of the day, cleverness is what separates the great from the best ya know. If I'm smart enough to apply - it would be pretty fun to have you or a similar interviewer, since games are a lot more fun than a 80 year old staring you down lmao. tldr; tysm. i needed this, and it was pretty fun to follow along
I have only just discovered this channel. I absolutely love that you are an Oxford math professor with personality and character. You probably (without knowing) are making a lot of students feel like they could envisage themselves at an institution with professors like yourself thereby encouraging them to apply! I can’t put into words how happy it makes me to see this.
@@TomRocksMaths Thks. One thing that springs to mind about this problem is that we know the outside surface cannot be painted, but the inside volume can be filled, so my question is, can the inner surface be painted?
@@markmcpeake715 Filling the inside (finite) volume with paint is the same as covering the (inner or outer) surface with a coat of paint decreasing in 1/x. So, both can be done with a volume Pi of paint. But painting the (inner or outer) surface with a coat of constant thickness would necessitate an infinite amount of paint!
To be honest you're actually supposed to, since the calculus you learn in A Maths is basically a more meaningful and extensive discourse in understanding and computing limits, derivatives and integrals - the key concepts remain the same, nothing new. Besides, don't forget that these questions (Oxford entry) are meant for Y13 students so... :)
I didn't understand much but I really like your voice. My boyfriend thinks I'm crazy and that you don't watch advanced maths-videos you don't understand simply because you like the voice. But here I am
I think Oxford university is very important . Especially , because of it, more students are interested in science thanks you. Mr Sir I love mathematics . Even I found some comforts from mathematics can you help me? I will wait a day
I SPENT 30 MINUTES TRYING TO FIND OUT WHAT HE MEANT BY SURFACE AREA BECAUSE THE ANSWER I HAD WAS ALWAYS INFINITY. Man I should really just sit back and watch these videos.
This video is awesome! Will there be a second video on these interview questions? I hope they will help me preparing for my Cambridge interview in December.
This Gabriel’s Horn reminds me of something else I saw! Say you have a cake, cut it in half, and then cut one of the pieces in half again and stack one of the small pieces on top of the big piece. with the other small piece, cut it in half and stack one half on top of the others, and keep doing this forever. Here, you’ve created something with infinite surface area but finite volume!
Nice idea - can you come up with a formula for the surface area after n steps? That would be how I would go about showing it tends to infinity as the number of steps does...
If you set Gabriel's horn in space with the horn's hole or orpheus facing down and it rains what water would fall and the floor of Gabriel's horn is gravity. The liquid will flow down. And collect in its opening and will fill to compasity and any excess water will fall into space. I learnt of Gabriel's horn from red pen blue pen. Several days ago. When he told me the filled paint couldn't cover the surface.
The interesting thing about this is that I would argue that it should be possible to paint the outside using a finite amount of paint, it would just take an infinite amount of time. lets say that the horn is infinitely thin, it has a volume of pi units, so that means we could fill the horn completely using just pi units of paint. When the horn is filled that implies that every part of the interior surface area is coated in paint. If the horn is infinitely thin then the outside surface area should be almost identical to the inside surface are (I think), so that implies it should take fewer than pi units of paint to entirely coat (given that the paint coating the interior is also filling the vacant space between the walls of the horn. I'm obviously wrong given the result but I'm genuinely curious about why this isn't true, or if it's just one of those quirks of infinity.
Until you stated it here, I never really saw the paradox clearly. If you fill up Gabriel's Horn with paint, you would inherently also coat its inner surface. And the inner surface would, given an infinitely thin surface (as would be created by a line) sould be the same as the outer surface area. So it actually would appear uou can in fact paint an infinite surface area using s finite amount of paint. Or, at least, you could if the paint was not made up of discrete particles. In reality the diameter woule eventually get small enough that no paint would pass any further.
I don't know why Gabriel's Horn is so hard for some people to grasp. I understand it seems counterintuitive that the higher dimensional measurement converges while the lower dimensional measurement goes to infinity, but if you're evaluating Gabriel's horn, you already understand that some integrals converge when the integrand goes to infinity. It's a very similar thing, just a dimension higher.
I don't think it's a paradox, it just shows that there is an infinite amount of 2d space in finite 3d space. The outside surface area of the horn is the same as the inside surface area since the horn has no thickness, so we already know we can paint the surface area of the horn with an upper bound of pi units of paint. Then if you imagine removing a volume from the inside of the horn using another horn that is infinitesimally smaller in volume we can see that you can paint the surface area of the horn with practically 0 units of paint.
One way to think about is is that the approximation is okay for the volume because it is only 1 of 3 lengths, whereas for the surface area it doesn't work in the same way because it is 1 of 2. of course this isn't rigorous, but I find it a useful way to think about it.
@@TomRocksMaths As the 'disc' gets thinner (and thinner) then the length dl will be dx so don't you still get the same result? You integrate 2*pi*1/x so get the limit of 2*pi* ln(a) as a tends to infinity so you still get the result of the area is infinite? or have I must something fundamental?
Speaking as someone with not very much formal education in maths, with a recreational interest, it's definitely possible. if you take it slowly and one step at a time, understanding this isn't as above you as you'd think. there are lots of resources on the internet, especially in videos like these that can introduce you to concepts that you would have otherwise assumed might be beyond you. a little patience and taking it slowly, you can definitely learn how to do things like this and more by yourself.
Nice work Steve! Gabriel's horn is not an unusual phenomenon. You can't really compare volume with surface area - the units of the quantities are different. One can easily find examples of finite areas enclosed by closed fractal curves of infinite length.
That is the paradox resolved.The dimensions are different ,a bit like comparing mass and temperature.Still conceptually freaky though.The presentation was fun..
Now that's some amazing stuff im your subscriber since you were having 3k subs But it's always awesome to watch these kind of videos with 3b1b also 😁 I wish you could have a collaboration with the veritesium also
A mathematician and a sociologist talk at dinner. Mathematician: So what do you research on? Sociologist: inequality, differentiation, integration, try to understand structure Mathematician: That's what I do too! Hey! we have a lot in common! #SameWordsDifferentMeanings
The whole time I was mostly looking at the pokeball tattoo. On a serious note, what a refreshing way to look at interviews! This was so interesting! And I'm from physics background 😅
Watch part 2 of the interview on the sum of the reciprocals of the prime numbers here: ua-cam.com/video/oi4ET0KzViI/v-deo.html
I didn't know there was a part 2! I will be checking it out now.
Thanks for all the editing and the opportunity to collab. It was super cool and super fun!
looking forward to part 2!
Steve I just came across Tom's channel now. Your videos are awesome Steve
Passion is contagious and Super cool
Yay
After a long day I came to know ur real name is steve😂
Awwww, my two favorite mathematicians 🥰🥰🥰
Pls dr peyam , u should be there as well. U are my fav
An oxford professor with a Poké Ball tattoo on his arm. That, if nothing else, makes me feel old.
Not a professor, but I get your point.
@@aRskaj OK, then just an Oxford Fellow. But he actually says at 00:47 "being an Oxford professor myself". I guess he was jumping the gun a little!
Earrings.
You're only as young as you feel Martin!
I think he's trying to be down with the hood ....it's that annoying look how cool AND. clever I look .....🙄
Today I broke 1500 on my chess rating. I felt pretty smart. Then I watched these guys and I realise that I’m the TikTok to their Wikipedia.
Don't worry, man. I assure you, I can solve these question easily as well, but you'll probably defeat me in a game of chess! :)
brool
You would no doubt destroy me at chess Andrew...
similar thing happened to me except it was when i won my first tournament 1800-1900 10-0 and they broke my mind at the surface area of gabriels horn
it's because you don't get smarter by playing chess. You are getting better at playing chess.
Brilliant. It makes you realise just how good at maths you have to be to get into places like Oxford and Cambridge. It also highlights really well the difference between being able to 'do' a topic in maths and really understanding it, and once you've got the understanding how you go about applying it. It's also great to see how much fun you can have doiing maths.
With good education brought up in developed nations, you should be quite geared towards those academics, especially math, which is largely logic. Th rest of math, you can explore on your own when you're interested (dopamine rush?), but do take note that most mathematicians who are too into their games might at times, lost touch with the current world...
Well kinda not I mean those questions surely are quite hard to someone who hasn’t looked into math. However for someone who enjoys doing math (in their spare time)at least the first question should have been easy. I’d also like to note
Nah this question wasnt hard. Im 15 years Old, and I would solve it, but it is really cool that surface is infinite, but volume is finite
this is easy as shit
what level would be the people doing these interviews? like good high schoolers?
@@Philgob The level when people try to get into universities. Now you guess when that is.
Note to self: impress the examiner with your pen-wielding skills during the interview
I feel like I'd have so much fun doing this interview if I knew a whole lot more about mathematics. It`d totally feel like playing a game.
It is meant to be fun yes!
You are cooler than any math prof I've ever had, by quite a wide margin...
Turns out that even mock interviews I'm not actually part of can make me feel as nervous as an actual interview.
good practice at least!
You are seriously one of the coolest guys I have ever seen who is like genuinely into math, keep up the good work!
This is such an awesome collaboration between two of my favourite maths UA-camrs. Unfortunately I was rejected before interview this year which sucks especially considering I was rejected after interview last year haha, but that was so much fun to watch!
I'm glad you enjoyed it, but sorry to hear about your application. Keep your head up, there are SO many other brilliant universities out there that would love to have you :)
@@TomRocksMaths Thanks for the kind words! I wasn't expecting a response haha. At the end of the day, there is still the opportunity for a masters degree at Oxford should I choose to pursue it, and I will be sure to make the most of wherever I go. One things for sure, I will stick with watching your videos for as long as I can!
Same thing just happened to me now , do you reckon its worth it to take a gap year and re apply , or should i go to warwick and apply for masters. Are you doing a masters now or next year?
man I think u have an impressive personality(happy , cheerful) for a mathematician
But, I have the best job in the world so of course I'm happy :)
wow, this was so amazing! thank you!
I kinda always thought all professors at Oxford were really old and closed-minded, thinking only they are right, keeping minimal interaction with "normal" students or others, this has been really eye-opening for me to see how such fun ppl are at Oxford uni.
Also, it was great to see these interviews for real and the things that go on in them!
I'm glad you enjoyed it Jatt :)
This was so awesome to watch! 😁 Sad I've only just seen it now. I had 6 interviews for Chemistry at Oxford but unfortunately didn't get in. I cherish that experience because it really taught me about what interesting questions actually are.
I've thought a lot about the answers I gave and the likely answer expectations, but my GCSE grades were meh compared to the average applicant and there were no entrance exams for Chemistry when I applied about 12-13 years ago.
It's always been a dream to study at Oxford. I have a Chemistry degree. I'm 2/3 through a maths degree and doing well and loving it. One day maybe I could come back to Oxford and ace the interviews.
Well done Steve for making that look easy! In the normal case there would typically be a lot more give and take between interviewer and interviewee I presume... it depends very much on the experience of the interviewee (and interviewer to a degree). Plus the passion in the way Steve communicated, written and verbally, would surely have out-shone anything incorrect he may have written accidentally on the day.
Then you have me who blurted out an incorrect answer to the first question I was given in my first interview. 🤣 I'll never forget the bond angle of CO2 ever again though! 🤣 Luckily idiocy is often a very useful quality! 🙃
I'm forwarding so many of these to my youngest. He's a budding STEM major in his freshman year of high school. Hopefully will give him the confidence to consider some of the top schools seeing what the interview questions involve.
awesome - best of luck to him!
*Loved the collaboration, this was amazing*
Glad you enjoyed it - and thank you!
fellow andrew dotson commenter
It's analogue to a model of the universe, geodesics on the surface (light traces) travel for ever without ever coming back to the same point so we say the Universe is infinite but the circumscribed volume is finite so therefore there's finite mass in the Universe.
blackpenredpens so humble and a quality maths proffesor and all round person I wish nothing but the best for him...
Amen
This is how uni professors should be like, casual, approachable, smiling. i feel jealous i hate my professors
If the paint is infinitely thin it would actually be possible to cover the whole surface with LESS paint than pi units. The only reason Gabriel’s horn seems paradoxical is because we instinctivily apply physical properties, eg. like paint having thickness, to a mathematical construct which is impossible to build in the real world.
But it is actually possible to imagine paint with finite thickness being used to cover the surface. Create two horns, one bigger than the other, fill up the big one with paint and then insert the smaller horn inside it. Remove the bigger horn and you are left with a horn that is both filled AND covered by the same amount of finite paint :)
Once again we witness that people are cool, when as feel confident about something. Not being able to solve a problem makes us all nervous. Great video and wow, what a knowledge man.
i love this video so much you can really see how he feels happy after he gets the right answer
This was fun to watch, can't wait for the next one
Glad you enjoyed it Sanne. Part 2 should be up in the next few weeks.
@@TomRocksMaths I know that this is quite late, but any updates on part 2?
@@dusscode as luck would have it I started editing last weekend so hopefully will be online in the next few weeks :)
@@TomRocksMaths can't wait
For more special guest appearances check out the 'Interviews' playlist including videos with 3blue1brown, Hannah Fry and several famous mathematicians: ua-cam.com/video/UsRfECCPsCY/v-deo.html
Wonderful Steve. You're admitted into oxford
@Tom Rocks Maths i love the fact that you break the traditional visual presentation of teachers, while being good at the subject.
okey this is giving me such wholesome vibes, you two are awesome
If all Oxford mathematics interviews are like this, I'll definitely encourage my son to apply. Maths should be fun! Do you ever teach computer science students?
Afraid not, just maths. I do try to make my interviews fun though for sure :)
When you start to study it is not than entertaining most of the time because you have to learn the basics of higher mathematics. It is great when you start to see the patterns and when you can apply the rules easily to solve different problems.
The paradox isn't really a paradox. If you paint a surface in the real world, you would need to apply a layer of paint with some thickness, and below a certain thickness you wouldn't consider it to be properly painted.
But the horn gets narrower and narrower, so no matter what thickness you pick for the paint layer, the horn will be thinner than that at some point, meaning a finite ammount of paint wouldn't "properly" paint the surface of the horn in the real world.
Any volume can be split into infinitely many 2d surfaces however, giving a volume an infinite ammount of surface area. So mathematically, a volume of paint can cover an infinite surface area.
It is kind of like how you can travel a finite distance by halving the distance to your destination an infinite number of times. So long as the time steps also become infinitely small, you can do this in a finite time, but if you needed a set time per halving, you could never finish.
Such a humble guy man.
Steve's fab isn't he?
Wow great to see Steve as a student for once !
He did well don't you think?
@@TomRocksMaths Of course, as a good teacher, he was certainly a good student as well.
This was so lovely to watch! I like Steve as a student :)
Absolute joy this. It's great on so many levels and especially in helping the kids I teach to see some of the interview process. Thanks guys.
Thank you very much for this video. Before I always treated dx like part of notation and just did integration as though it was algebra but now this video actually explained what dx is.
Best collaboration ever
Thanks Oscar - glad you enjoyed it!
Thanks for the insightful problem - I've heard of Gabriel's horn and did these calculations a couple years ago, and now looking at problems to hopefully have some questions for an interesting school math competition, and just to practice my own maths, this is pretty amazing to watch. Right now my high school is new [only around 2 years old right now], and I really don't have much "history" to go off of, so these are the types of problems I hope to add... problems that require no more than a basic intuition of derivatives, integration, and a bit of cleverness since at the end of the day, cleverness is what separates the great from the best ya know. If I'm smart enough to apply - it would be pretty fun to have you or a similar interviewer, since games are a lot more fun than a 80 year old staring you down lmao.
tldr; tysm. i needed this, and it was pretty fun to follow along
I appreciate you Dr. Tom
And I appreciate that!
Got to catch them all (all the equations).
Isn't anyone gonna talk about how beatiful the answer to the volume question is?
'unexpected pi'
Magic of Infinities. (Infinity might have been a strange function rather than an exceedingly large number...)
I have only just discovered this channel. I absolutely love that you are an Oxford math professor with personality and character. You probably (without knowing) are making a lot of students feel like they could envisage themselves at an institution with professors like yourself thereby encouraging them to apply!
I can’t put into words how happy it makes me to see this.
amazing - thank you!!
Steve is very good! Tom really rocks! I really enjoyed this as an ex Uni maths lecturer decades ago.👏👏👏👏
pretty proud of myself for managing to solve them along side backpenredpen, just took my calculus 1 final yesterday.
I hope it went well!
@@TomRocksMaths it did, I'm taking calc 3 now
Steve, you have been cordially accepted to the University of Oxford!
I feel honored!
This was great. I loved the real world interview question. I guess it's removed from the interview question list now! Best of luck. Be safe all.
Glad you enjoyed it Mark. And yes, I can confirm I will no longer be asking this question!
@@TomRocksMaths Thks. One thing that springs to mind about this problem is that we know the outside surface cannot be painted, but the inside volume can be filled, so my question is, can the inner surface be painted?
@@markmcpeake715 Filling the inside (finite) volume with paint is the same as covering the (inner or outer) surface with a coat of paint decreasing in 1/x. So, both can be done with a volume Pi of paint. But painting the (inner or outer) surface with a coat of constant thickness would necessitate an infinite amount of paint!
@@Jooolse As I thought, but wanted to hear it definitively. Many thanks.
Great content. I am grateful for both of you for your commitment, modesty and high quality of your video content. 👍
Steve (bprp) more of a Cambridge guy I reckon... but this video was a delight. Thanks both!!
Glad you enjoyed it Andrew :)
This man single handedly got me my degree
As a year 12 student doing AS maths I'm suprised how I actually understood nearly all of this
To be honest you're actually supposed to, since the calculus you learn in A Maths is basically a more meaningful and extensive discourse in understanding and computing limits, derivatives and integrals - the key concepts remain the same, nothing new.
Besides, don't forget that these questions (Oxford entry) are meant for Y13 students so... :)
I didn't understand much but I really like your voice. My boyfriend thinks I'm crazy and that you don't watch advanced maths-videos you don't understand simply because you like the voice. But here I am
These 2 makes a guy feel about as smart as a rock
To me, the real paradox is why the integral from 1 to infinity of x^-1 dx diverges but the rotated one converges to pi.
Is ‘Gabriel’s Horn’ what happens when watching Equations Stripped?
Nice Marc, nice.
🥴
I love watching both your channels and I never realised you had a collab! I love how happy red pen black pen looks when he knows the answer😂❤
At 14:00 you just have to say that sqrt(1 + 1/x^2) > 1, so (1/x).sqrt(1 + 1/x^2) > 1/x, so the integral diverges.
Nice
Exactly!
Yes, that's another way by observing the reciprocal of x and then extending the relationship by logic.
I think Oxford university is very important . Especially , because of it, more students are interested in science thanks you. Mr Sir I love mathematics . Even I found some comforts from mathematics can you help me? I will wait a day
Alternative solution for the final integral is substituting u=x^4+1, giving the integral of sqrt(u)/4 which clearly diverges
4:30
BPRP: technically it should be going on forever
T: right, so it's an infinite horn
BPRP: *draws end of horn*
T: 👁 👄 👁
Is it blackpenredpenbluepen now?
That's what I've been thinking all along dude😅
This was fun to watch!
Thanks Abhishek!
I SPENT 30 MINUTES TRYING TO FIND OUT WHAT HE MEANT BY SURFACE AREA BECAUSE THE ANSWER I HAD WAS ALWAYS INFINITY. Man I should really just sit back and watch these videos.
This video is awesome! Will there be a second video on these interview questions? I hope they will help me preparing for my Cambridge interview in December.
Yes, part 2 will be out soon (and hopefully before December)
The Horn of Gabriel, Fascinating!
I don't understand much, but love tgeir genius.
Don't worry, you'll get there if you keep working hard :)
This Gabriel’s Horn reminds me of something else I saw!
Say you have a cake, cut it in half, and then cut one of the pieces in half again and stack one of the small pieces on top of the big piece. with the other small piece, cut it in half and stack one half on top of the others, and keep doing this forever. Here, you’ve created something with infinite surface area but finite volume!
Nice idea - can you come up with a formula for the surface area after n steps? That would be how I would go about showing it tends to infinity as the number of steps does...
If you set Gabriel's horn in space with the horn's hole or orpheus facing down and it rains what water would fall and the floor of Gabriel's horn is gravity. The liquid will flow down. And collect in its opening and will fill to compasity and any excess water will fall into space. I learnt of Gabriel's horn from red pen blue pen. Several days ago. When he told me the filled paint couldn't cover the surface.
This was soo much fun!!
The interesting thing about this is that I would argue that it should be possible to paint the outside using a finite amount of paint, it would just take an infinite amount of time. lets say that the horn is infinitely thin, it has a volume of pi units, so that means we could fill the horn completely using just pi units of paint. When the horn is filled that implies that every part of the interior surface area is coated in paint. If the horn is infinitely thin then the outside surface area should be almost identical to the inside surface are (I think), so that implies it should take fewer than pi units of paint to entirely coat (given that the paint coating the interior is also filling the vacant space between the walls of the horn.
I'm obviously wrong given the result but I'm genuinely curious about why this isn't true, or if it's just one of those quirks of infinity.
My Further Maths teacher told me that only kids would say 'oval' , but 'ellipse' for mathematicians.
Take this guy to Oxford!
Yes, the volume of the pi of the formula and axis. Definitely.
For the second part I just sais that S>= lim b->+inf (integral from 1 to b of (2π/x dx)) which is actually infinity so S>=+inf so S=+inf
Amazing video! My first time hearing about Gabriel's Horn. Really fascinating.
It's one of my favourites for sure
6:02 Cavalieri's principle 😍 I have been told about that in my uni
I’m excited to apply in 2021! Hopefully I can get in lol.
p.s. great channel keep up the great work
Thanks Zane - and best of luck!
Until you stated it here, I never really saw the paradox clearly. If you fill up Gabriel's Horn with paint, you would inherently also coat its inner surface. And the inner surface would, given an infinitely thin surface (as would be created by a line) sould be the same as the outer surface area.
So it actually would appear uou can in fact paint an infinite surface area using s finite amount of paint.
Or, at least, you could if the paint was not made up of discrete particles. In reality the diameter woule eventually get small enough that no paint would pass any further.
I don't know why Gabriel's Horn is so hard for some people to grasp. I understand it seems counterintuitive that the higher dimensional measurement converges while the lower dimensional measurement goes to infinity, but if you're evaluating Gabriel's horn, you already understand that some integrals converge when the integrand goes to infinity. It's a very similar thing, just a dimension higher.
Horn is f(x² + y²)
For ex.: z = f(x,y) = ln(x² + y²)
I don't think it's a paradox, it just shows that there is an infinite amount of 2d space in finite 3d space. The outside surface area of the horn is the same as the inside surface area since the horn has no thickness, so we already know we can paint the surface area of the horn with an upper bound of pi units of paint. Then if you imagine removing a volume from the inside of the horn using another horn that is infinitesimally smaller in volume we can see that you can paint the surface area of the horn with practically 0 units of paint.
I remember in primary school when i learned my timetables and believed i had cracked maths. Halcyon days.
These question looked really fun
interesting question. one point i couldn't understand, when calculating the surface area, why is it dL but not dx just like the volume?
One way to think about is is that the approximation is okay for the volume because it is only 1 of 3 lengths, whereas for the surface area it doesn't work in the same way because it is 1 of 2. of course this isn't rigorous, but I find it a useful way to think about it.
You can also think about the lenght of a curve: it's always dL as the hypothenuse of the right triangle with sides dx and dy = f'(x) dx
Study Jacobian Coordinates and you will realise that there are way more than the usual cartesian coordinates.
@@TomRocksMaths As the 'disc' gets thinner (and thinner) then the length dl will be dx so don't you still get the same result? You integrate 2*pi*1/x so get the limit of 2*pi* ln(a) as a tends to infinity so you still get the result of the area is infinite? or have I must something fundamental?
Oxford interview made easy
I respect him ❤️
It’s amazing you can fill the paint but you can’t pain🥶🥶
This knowledge is so overwhelming I wish I could understand like this
The key is hard work and persistence, but don't worry if you stick at it you'll get there eventually :)
Speaking as someone with not very much formal education in maths, with a recreational interest, it's definitely possible. if you take it slowly and one step at a time, understanding this isn't as above you as you'd think. there are lots of resources on the internet, especially in videos like these that can introduce you to concepts that you would have otherwise assumed might be beyond you. a little patience and taking it slowly, you can definitely learn how to do things like this and more by yourself.
Wow no wonder I never thought about applying to Oxbridge I can do everything at home but I was nervous just watching this
Pi value as a decimal is represented as endless so we call it infinite even though it's finite value must be between 3 and 4 .......
Nice work Steve!
Gabriel's horn is not an unusual phenomenon. You can't really compare volume with surface area - the units of the quantities are different. One can easily find examples of finite areas enclosed by closed fractal curves of infinite length.
That is the paradox resolved.The dimensions are different ,a bit like comparing mass and temperature.Still conceptually freaky though.The presentation was fun..
This stuff is gold
I Like the way you present your videos and explain things. Kind reminds me of a math teacher I had. Very cool.
Awesome thanks :)
Now that's some amazing stuff im your subscriber since you were having 3k subs
But it's always awesome to watch these kind of videos with 3b1b also 😁
I wish you could have a collaboration with the veritesium also
yes I love veritasium!
wow, my two favourite mathematicians together 😀😀
A mathematician and a sociologist talk at dinner.
Mathematician: So what do you research on?
Sociologist: inequality, differentiation, integration, try to understand structure
Mathematician: That's what I do too! Hey! we have a lot in common!
#SameWordsDifferentMeanings
this man is so smart
May be value of pi goes on forever that is why surface area is infinity?
Saw this on numberphile and found this now xd
Possible coincidence I'm in both...
The whole time I was mostly looking at the pokeball tattoo. On a serious note, what a refreshing way to look at interviews!
This was so interesting! And I'm from physics background 😅
Glad you enjoyed it!
Imagine having the confidence of BpRp doing your exam
Nunca pense decir esto en el colegio, pero me entretiene ver videos de matemáticas y calculo.
All the way from Asia !! Nepal 🇳🇵🇳🇵🇳🇵