Cones are MESSED UP - Numberphile

cheers Bray thanks for having me back it's been a while it's been a while cheers I mean we could just drink it but hey more more excitingly is that going to fit in there what do you mean what do you mean what what what are you saying I mean let's just say I need to run to the bar to fill out my glass and I want to dump that somewhere that's not going to go to away so I can just put it in your glass but would it fit what the liquid yeah if I pour that glass in there am I making a mess I mean all my intuition would say yes they both look quite full have you got a cloth okay look at that cones are messed up you didn't spill a drop I'm impressed I hope you are too the bubbles are making me nervous every time they pop to the top it's nearly overflowing but hey it fits so what do I take from what just happened there tells you that your intuition about cones is never going to be good let's talk a bit about why cones are counterintuitive I mean this is a famous demonstration that a lot of people who've worked in cocktail bars know about I still like it even though I know what's going to happen it feels it still surprises me but I came across another problem about cone recently which exploits this counter intuition even more and I like to do that with with you and the viewers if that's right well let's do it but this has made me realize how ripped off you can get in a bar if you don't get the glass full you are being cheated of literally half your drink uh so yeah walk away with your your gin and tonic or your Martini whatever uh and if it's not filled with the brim that's probably a significant fraction of what you could have paid for uh it depends whether they you know measuring their units and but it's not as fully as you think it is cuz it's a cone and I want you to imagine that the cone is upside down by which I mean flat side down so I want you to imagine it's got some water in and I will show you this for real by the way in time but I don't want you to see it for real because I want your intuition to work in your mind first uh same for the viewers imagine it's got some water but it's not full it's 2/3 full by height and that's important I'm not talking about the volume 2/3 of its height is full of water are you picturing that I am good the simple question is what happens when you turn it the other way up and the particular question is how full will it now look by height do you have any gut reactions I did do a little bit of quick processing in my head because of what you showed me before the trick at the start yeah and that makes me think that 1/3 of air is in the is in the narrow the and I've learned that not much of the volume of a glass is there so I imagine it's going to look surprisingly full so the first bit of intuition is to establish whether it's going to look fuller or less full and I think you know which way you're going and a lot of people do quite quickly because your logic is almost undeniable there right that air gap is in the narrow bit flip it it's now going to be in the white bit so surprisingly full before we do the calculations which is that's all we're going to do we're just going to solve this problem it's a number crunching exercise now do you want want to put a guess on a percentage like between Z and 100% I'm going to go I'm going to go for 95 okay he's gone High H it's going to be 95% full that way up yeah cuz I think it's going to surprise me the way your other trick surpris I'm curious about what you're doing did you pick a guess and then inflate it just to make sure that you've kind of got the surprise out you don't have to answer that we'll find out right all right and I I guess having a guess for everyone watching this is worth doing before we do the calculations I suggest also before we even draw anything if people want to stop the video and work it out now is a good time okay no but don't stop watching the video good good advice uh right let's let's talk it out so there's a long way to do this and anyone who's watching thinks there's a quicker way you're right there's almost always a cooka away but it's nice to explore what everyone does instinctively which is to do with volumes of cones so let's sketch the original problem here's a cone that bit is 2/3 that's my original setup let's call the height of it one the other thing is I didn't give any other information about like the radius or the ratio of the angle of the cone and the clue is that I don't need to although it feels like you need it so let's just give them some labels we could call this bit R and I could call that H if I wanted to and this would be 2/3 of H most number four viewers I think will be aware that it feels like they're going to cancel out in the end anyway but let's try the thing so there's the setup and the question is later on when I turn it the other way up what's that fraction so one way to do it is to do some volumes and the volume of a cone is important starting point here let's do the total glass is a thir piun r^ 2 h and that's the sort of generic formula but since I've used the letters R and H in this that's literally the volume of the entire glass now I do care about the volume of water but that's hard because it's I think they call it a frustum shape it's like a truncated cone and actually it's much easier to calculate the volume of air uh which is the top bit so I'm going to do V air and know that I could do one minus the other one to get the other one so the volume of the air is also a cone but this time it's a third because that's in the cone thing and then the height of that air bit is well the R is different the H is a third of H and I need to work out the r here but I'm going to claim that it's they're similar shapes so if this is a third of the whole thing then the radius is also a third so it's an R over three in there as well so I hope you can see it's the same equation but this time the radius is 2/3 smaller it's a third of the size and the height is a third smaller as well that's how scaling things works so I'm just going to do this going to TI it all up uh let's do third piun r^ 2 over 9 and there's another third here in h let's just this is now on over 81 Pi r^ 2 H and if I subtract these two I'll get the volume of water if I turn this into 80 81s I need to multiply by hold on three to get nine by another nine so multiplied by 27 so this is 27 81s how do you say 81s Bray 81s yeah I I still struggle with this like I'm just going to carry on regardless B want or 8 first seems to make sense so this ends up being 26 out of 81 piun r^ 2 H and what that's the volume of water that's the volume of water yeah this is not the bit of mass that gets me excited but these calculations are easy enough to do um there's a whole bunch of algebra algebraic faffing that I'm not enjoying the algebraic faf is like the means to an end but I'm going to push through here's the water here but I know the volume of that now it's is this so what I need to know now is the height I'm going to work out the volume of that water in another way even though I've already done it once because I would like to get this fraction of H in fact I'm going to call this length x and it's some fraction of the total height so here's a couple of other observations the radius of this cone was R so the radius of this cone of water in here is actually like a scale down version It's scaled down by X so this radius is going to be XR you can have a think about that but that's the way things are scaling the whole thing is gone from H to x * H and so the radius has gone from R to x * R so I've now got two ways of describing this new volume let's do it the volume of water again is normal cone formula 3 Pi r^ 2 is going to be XR s that's the radius of this cone and the H is xh that is the volume of water in this new case but I also know it's this so I've got an equation to solve now and it's kind of the algebraic bit has done its work I just need to solve this equation and find out what x equals so I'm going to do some cancelling H Pi the r s is going to get cancelled still got an X squ so I've got a thir X cubed because the X2 and X here is given us an X Cub = 26 over 81 multiply everything by 3 x cubed = 26 over 27 which means X is the cube root of that thing let's get a calculator let's do a cube root of 26 over 27 and it's actually nice to reflect on 26 over 27 is really close to one if you cube root it is it going to it closer or further away from one and that's maybe not intuitive but because we're starting less than one actually Cube rooting it is going to get us closer to one so we already know this answer is going to be really close to one which is why your initial reaction was maybe good so if I do the answer here I get 0.987 dot dot dot so to the closest percentage this apparently is 99% fall 99% 99% I'm sorry to say you you went too low your intuition was pretty good and then you like it's going to be bigger than it's possibly reasonable to expect but I did this calculation I was like I don't believe it you're telling me that if it's 2/3 full and it looks like there's a big gap and you turn upside out it's going to look 99% full and it's like well back my maths but I don't believe it unless I see it so um bearing mind I also want to show you a much quicker way through this because you kind of spoiler you may have noticed me grunting through the algebra there's a much nicer way to see this and I want to talk about why it's worth paying attention to another way to S it but before we do that should we see if if it looks like this for real yeah here is a martini glass but but I glued glued a lid on it uh so please excuse the crudity of the model channeling Brown again uh and the glue you can ignore that but I'm going to claim that it is about 2/3 full by height now I I don't think that's exact uh because some water keeps leaking out when I do this but I don't think it's going to matter for our demo this I hope you agree is the original setup of the problem are you happy with that Brady I'm happy so just going to turn upside down and I want you to be both prepared for like a good gobs smacking reveal and a massive anticlimax because nothing's going to happen but it is genuinely really full and the slight frustration is it's so full that if you want to put the camera on the top that the surface tension involved of having a lid on it and water in there messes up everything it's like it's not even full enough for it to sit away from the lid just move it around and this this bubble is moving around there's a tiny amount of air left in there because it's just about 1% uh by height and it's so it's touching the top but this does back us up we go back again 2/3 full probably a bit less than 2/3 if we were going to measure it because a bit has leaked out and when I tried to screw the the lid on chocker block yeah now the reason I didn't want to show you this at first is that I think actually when you look at this your intuition kicks in slightly better when you see it because you looking at that have a pretty good sense of the three-dimensionality of this thing I think that space is quite small now you you knew that because you were thinking about the demo with we did right at the beginning and Cones being unintuitive as well but it's so small and that space gets spread out over that big surface area now at the top which means that of course it is and the math backs us up it has to be 99% full by height it's still surprising I can't really convey to people how full it can you rotate it towards yeah so if I spin it around it depends on the lean of the table but I could fill it that way around if I lean it back you can see the Gap is in there there's a little bit of air which there has to be I think what's annoying about the demo is that there's no way of rigging this to make it look more impressive cuz it's so full that the surface tension the physical properties of water are doing weird things because it starts touching things uh which makes the demo less exciting I do think it's nice though when you have a a mathematical surprise to actually do it for real and just check that the mathematical world still still works in reality I would like to talk though about another way of solving this which is much cleaner and we'll try and do it without any brown paper we'll do some handwavy things on the demo if you're up for that I'm going to stand up to do that and there's one bit of 15-year-old Matt that comes to sort of our rescue and here it is two cones here one is the total glass and one is the air they are similar mathematical term for same proportions right but that means they have a scale factor they can scale up and that is the crucial property here because other shapes don't have that if you chop the bottom off a cylinder you don't get a similar cylinder at the top because it's different ratios cones you do and that's crucial that's what's making this work so this air cone is similar to the big cone and I'm going to claim I hope obviously that the scale factor is a third that is a third of the whole thing because this is 2/3 full by height happy with that so far so Hand wve the length scale factor is a third and here's the bit of GCS maass what's the volume scale factor if you know the length scale factor you can get other scale factors like the area scale factor and the volume scale factor by squaring for area or cubing for volume so the volume scale factor between the two cones is a thir cubed otherwise known as 1 over 27 which tells me straight away maybe it's a familiar number that that air gap accounts for 127th of the volume already small number so the water is 26 27 and that I hope is a familiar number from uh something we wrot down earlier oh look that one and that that was there all along in those calculations but if I flip it nothing's changed that air gap is still 127th and the volume is 26 27 but to turn it into a height calculation what I'm claiming is the two cones now the the total cone and the water cone are similar and their scale factor is sorry their volume scale factor is 26 27th cuz that's the amount of volume in one compared to the whole one so to get back to a length scale factor I have to cube root 26 27ths which is again the last step I did in the calculation but this time I didn't have to write any of the pi r s h business all the algebra is gone because actually it canceled out anyway as you saw and those scale factors you go from a length scale factor to a volume scale factor and then you flip it and then you go back to the length scale factor just cube root 26 27 and we already know that's close to one it's going to get even closer we end up with a number ridiculously close to one about 98.7% and that's without any writing down I mean everyone's sad that I didn't use the brown paper but it's nice to have an elegant argument through a question when you've done it the long way and there's one other thing I wanted to mention is that cones are counterintuitive you saw that in that first demo we did where we we did the sort of the bar glass advice make sure it's full but even though I knew that and even though you knew that you'd seen the demo your guess is still not extreme enough for this puzzle for some reason cones really mess with our heads and our intuition and I think it's because a bit like in the video we did years ago Brady with the speeding cars and in that there was a counterintuitive result which is coming from energy being proportional to your speed squared and because it's a nonlinear relationship our intuition sort of gets out of kilter and this is the same thing happening it's just even worse because there's some cubing going on so it's nonlinear but there's also because of the way cones work we're doing it twice we're having to cube one way and then cube root and that combination of two nonlinear relationships just messes up our intuition so like this is a good puzzle even though the maass is not difficult it's a good instructional outcome about how to tackle problem solving and don't stop when you've solved it go back and ask the question is there another way is there another Insight I can find all right joj time so there's the calculation we just did when this is 2/3 full that is 99% full it becomes really intuitively obvious particularly if you spin this around and you get the three-dimensionality of it that does not look like a lot of air and so if as soon as you see this for real you're like oh yeah it's going to be super full which you intuitive without seeing it well done I think it does look like a fair bit of air I think you're downplaying how much air that well maybe it's cuz I've been thinking about this trying to justify why the answer is so surprising to me but you can see there is a tiny air gap in this one over here it's just spread ACR that area so it's it's small but this is true for other Heights so I want you to imagine if it's half full by height which we can do I'm going to hide the inverted cone and just set it to half full okay this is going to go down to 0.5 even when it's half full by height flip it up it's still 96% full by height the other way up yeah you can see that sort of air gap floating around there but it's still 96% full and that's again because our intuition about the nonlinearity and if if we animate this you can see how it does not change linearly so you can see that Beyond half full it just can't tell any difference anymore but if it goes down below half full you start to eventually see this change in a nonlinear way and by the end it's going to change super quick it sort of Zips away and vanes at the end but this thing is moving linearly and that thing is moving definitely not linearly which is why our intuition about how it scales isn't very good so here's my question this is the question do you know what the question is the question is how full to have half the volume uh here is a picture of my Marty you appreciate the scale model this time is actually in 3D it's not just a picture but here's what it looks like half full that is half full now by volume should we do the calculations just to finish this off because this is basically the answer to how full did you have to fill those glasses for our initial little demo let's do the calculation hand wavy style right it's half full by volume so the total cone is half full the volume scale factor between the total cone and the water cone is a half so what's the length scale factor cube of a half CU of half 79.3% so the answer is if you want to make your glass look like it's full but actually in reality only be half full of volume 80% by height I really love mathematics and science as you probably guessed because well I make a lot of videos about them but it's been a long time since I was at high school I have haven't got any formal qualifications so where do I or someone like you go to get a grounding to get the basics and then maybe from there launch to a higher level start digging deeper into these subjects well I'm going to suggest today's episode sponsor brilliant as a great place to start they've got a huge collection of superbly designed courses covering a range of topics everything's really interactive which I love it's always fun to move the sliders around to play with the pictures to become part of it all they're all designed by experts it's smart it's witty there's always new stuff being added as well just have a look at some of these great stuff to get your teeth into now you can try everything on brilliant for free for a full 30 Days by going to brilliant.org Slumber file there's also a link in the description and a QR code on the screen by the way you'll get 20% of an annual premium subscription by using that link now thanks to brilliant for supporting number file go and check him [Music] out so I think I'm just going to fill this one full for the sake of it should work I haven't thought about whether the bubbles Chang like the surface tension or anything like that this is where my looks pretty close physics don't look too close to the paint on the wall though so if it does spill it doesn't go on the paint cuz that it's hard to fix now I can't even move the thing without sping light it I don't mind spinning on the wood that's all right oh my god oh that's not so good you know that thing that you said it'll be fine it was always going to happen hang on so much for the b-roll being a slick sleep back up oh mate that b roll just became twice as valuable to me yeah everyone loves it yeah that's going to go roll after your sponsor sponsor clip isn't it like

Channel: Numberphile

Views: 406,908

Keywords: numberphile

Id: Mkn3PzdaByY

Length: 18min 53sec (1133 seconds)

Published: Tue May 28 2024