Here is the hardest problem in the world. You have 300 coins. You have to insert a 300 letters message (Signal) one letter per coin. Then you flip the coins and insert a 300 letters randomly generated Noise. Then you put all the coins in a bag and scramble them. You then put the coins back on the table. The task is to flip and move the coins around until you uncover the original message. Can you do it?
16:45 *So cones are... "half full/empty"... Pareto Principle! 80/20* ________ *edit - @tbersags0278 said in the comments this: _"The top 1% represents a third of the resources. Are we surprised? Really?"_ ...and my answer was this, regarding the Pareto Principle: _"Richest 20% pay 80% of all taxes!"_
This makes me remember a shape/teaching aid a math teacher had back years ago. The middle school math teacher taught kids who were behind ( me ) and the advanced highschool kids ( highschool down the street). It always confused me. It was an upside down circular cone with a flat plane inside on a 45 dregee angle, in the middle third of the cone. Calculate the volume of each section.......
This would actually work. An optimist would think a glass is half full looking at its height, and the pessimist would remind him it's 96% air by volume
@@adarshmohapatra5058 Actually no, if a cone is half full by height, it's 87.5% air by volume. You get the 96% by turning it upside-down and looking at the new height. (wonders what this correction makes me)
@Ringcaat ahh you're right, a quick check using scaling factors tells me that 1/2 water by height is 1/8 water by volume giving 7/8 = 87.5% air by volume. . And I agree with the person who commented below you, this correction makes you a correctionist :D
One thing that wasn't mentioned is, our eyes are lying to us when we look at a cone glass because we see a 2D projection. This means on that 2/3 full glass in height we actually see 8/9 of the area is water and 1/9 is air, which makes it not intuitive to believe there's actually three times less air than that. Interestingly, if we do the math with 2D triangular glasses we get about 94% which is basically in line with the initial guess
@redbird_studios for any two dimensional shape, equal aspect scaling increases area by N^2 for N scaling, where N is the factor you're multiplying the measure by. For example if you double the measure of a polygon, it has four times as much area, doesn't matter what the actual shape is. for our example, we're scaling up by 3/2, so the new triangle has 9/4 the area. if you want some example numbers : if the small triangle has base 12 and height h, the new triangle has base 18 and height 3h/2. the small triangle area is 6h, the new triangle area is 27h/2, or 13.5h. 13.5 = 6 (9/4), as expected.
This is very mind blowing to everyone except for chemists. This is exactly how separatory funnels work! If you are a chemist, you see this extreme smallness of this volume when you drain the bottom phase and see that there's almost nothing left - it's no longer a curiosity, but a functional phenomenon!
…and for the demo in the video, they should have put in a colored non-water miscible liquid in the closed-cone instead of air. The effect would be more visible.
@@tinkeringtim7999 Another way is just to add something that lowers the surface tension (such as rubbing alcohol - soap would foam, but alcohol won't).
@Simon-fg8iz he also could've added a tiny cone on the lid with surface area just large enough to support the surface tension without leaving the cone, and explain how that water equals the displaced volume. Very meta 😉
I sometimes use this "scale factor" method to compare the areas of two pizzas based on their diameters, to understand how much more pizza I get with one size vs. another size. For example, pizza restaurants in the United States will frequently offer a 14-inch diameter pizza and a 16-inch diameter pizza. (They don't use the word "diameter" on the menu, but that's what they mean.) Without actually calculating the area of each pizza in square inches, I can compare their areas by knowing that each area is proportional to the square of the radius. So I can think of the 14-inch pizza's area as proportional to 7 squared = 49, and the 16-inch pizza's area as proprtional to 8 squared = 64. In other words, if I order a 14-inch pizza, I will get roughly 3/4 as much pizza as if I order a 16-inch pizza (because 49/64 is approximately equal to 3/4).
Yes, but one confounding factor is that the ratio of crust to normal pizza is much higher as you go for a smaller pizza (assuming the crust width is constant).
@@ThunderClawShocktrix Thin crust pizza still often has a ring of unadorned dough at the edge. Also I like a hearty combo pizza that thin crust cannot necessarily support
@@jackmandu I love it! I almost said in my original comment that my method avoids having to calculate with pi. But I didn't even think of the pi/pie pun.
I feel like going through the labor of building pyramids in Minecraft gets a little bit of the intuition going. When planning out a beacon, and how many tiers tall you want it to be, and figuring out what materials you can afford to build it with. And realizing that adding just a single tier more than doubles your material usage. If you think about the glass problem as tiers of a pyramid for a beacon, it hits really fast that, oh yeah the top bit is absolutely inconsequential.
This sort of behaviour is part of why in civil engineering, water at a weir is often directed through a triangular cross section channel to measure flow rate. Since the depth of the water increases as the flow rate increases, compared to a rectangular channel, more accurate flow measurements can be made at a wide range of flow rates, exactly what's needed in the real world for rivers that can vary from drought conditions to full flood. The other benefit is that the flow speed stays high enough that the channel is less likely to clog, even when flow rates are low.
Bernoulli’s principle is conservation of energy per unit volume. Energy in a fluid is a function of pressure and velocity; when at speeds where the flow is incompressible, smaller volume means less particles to maintain density, that lowers pressure. Since energy is conserved, lower pressure = higher velocity. These relations are called the wind tunnel equations, and we also use a semi-conical shape (more like a pyramid) to increase the speed of air in the test section of a wind tunnel.
I work as a gardener at a store that does winter storage for large potted plants, and occasionally customers want a plant in a different pot when we deliver it the next spring. It always keeps amazing me how much additional Earth we have to pack to fill a just so slightly larger pot than the original. When you go three dimensional, small increases in radius add up to a lot of volume.
As a hiker, I know there will not be enough air space in a wide-mouthed water bottle to cope with elevation changes unless you fill the bottle no higher than the shoulder of the bottle. (Also, in cold weather, keep it in the pack upside down so that it it starts to freeze, the ice won't be across the mouth of the bottle.)
Just a note about how cocktail bars work: You're not paying for the whole drink, you're just paying for the alcohol. When you order a drink, you will get a fixed amount of alcohol (usually 1.5 or 2 oz), along with mixers, ice, water, etc. If the glass you get isn't full, that just means your drink is stronger. If you get a full glass, it has more mixer, not more alcohol.
@@CliffSedge-nu5fv Depends on the beer. Some beers are supposed to be served with some amount of head. Generally speaking, though, you do overpour a little to get most of the head to run off.
I did something very similar to this demo where I asked the people around the lab the volume in the bottom of a separatory funnel. It was significantly less than anyone said, like at least half of what they said. That was after I told them that whatever they guessed, go lower. It was less than I guessed, too, but their initial guesses were waaay off.
I like the idea of doing both. The effect is more extreme when comparing height to height like in the video (I knew the answer and still was surprised to see it in decimal form), but I think it's more palpable when you see all the liquid transferred from glass to glass.
This is something you learn if you ever build a pyramid one stone at a time. The top half of the pyramid's height is only an eighth of the volume. You feel like it's taking forever, and then suddenly it's done. Dunno how important that is in understanding the psychology of ancient people who built pyramids, but it's worth thinking about.
"I don't have any formal qualifications" I think, having watched you as an interviewer on this channel over the years, you definitely have better qualifications than most.
He's definitely fantastic at his job, but that doesn't mean he has qualifications. Experience, talent, and skill are all even more valuable than formal qualifications.
Cube root of 1/2 ~ 0.8 is one of those handy numbers to have in your head, like sqrt(2) ~ 17/12, log10 2 ~ 0.3, ln 2 ~ 0.7, e^3 ~ 20 and so on. Feynman has a chapter in Surely You're Joking where he shows how useful this is.
Yep and this would have made an interesting demo as well, if he'd filled the glass 1/2 by volume. It would be "80% full" of air when upside down, then "80% full" of water when flipped.
I saw an old advertisement for a company selling cardboard container for movie popcorn. They were selling how much the theater would make by using their cone containers instead of the standard box. You could give a bigger cone container but it would contain much less popcorn than a box. Buy cones not boxes and spend less on popcorn.
I don't know about the UK, but in the US "licensed" bartenders pour a specific amount of liquor to make sure one "drink serving" is the same across bars, and bars rarely go over because booze is more expensive than mixer. If your martini glass is 100% full, it means your drink is watered down
Most bars in the UK will use the standard Weights and Measures, which means any spirits must be measured in either a single (25ml, ~1 oz) or a double (50ml). If you order a martini, you're most likely to get a double of gin and a single of vermouth
Also unless you want to be sitting at the bar leaning down to sip the first half of your drink from the rim of the glass you most definitely *do not want* that 3 oz martini in a geometrically "3 oz" glass... ::edit:: As demonstrated 18:34 XD
In France all drinks must have the same amount of alcohol in it by volume, so you will be evenly drunk by a glass of whatever you take in a bar (10 grammes of alcohol per glass)
@@marcosolo6491 yes, but as this is France, shipping charges going to be massive. Can always opt for the Atlantic though. If you're REALLY thirsty, you might get the Pacific Ocean regardless.
An interesting issue I've been dealing with is closely related to cones, but they're within fixed cylinders, which made this video fascinating. If you deal with a silo of some diameter, regardless of hight or (in most cases) material property, where you fill from the top center and pull from the center at the bottom, you end up with cones in the material (or remnants of cones as material is added or removed). A popular technique to find the volume of the material is to drop a single plumbob from 1/6 the diameter of the silo containing the material, measure the distance it descends into the cylinder, and calculate a volume estimate from the results by treating the measurement as the top of a complete cylinder. Unfortunately, nobody has any proofs as to why 1/6 the diameter is the best choice, nor how close you could estimate your measurement is.
Or just scale up the model - obviously at a certain point weight becomes an issue but I bet it's possible to get to a no-cling point before it's too heavy to handle...
An important note to save the bartenders in your life some trouble: Bartenders measure the drinks they are mixing before pouring them into the glass, in no way is the drink supposed to reach all the way up to the rim. You aren’t being cheated you’re getting what you paid for in a container large enough to keep you from spilling it allover yourself.
I have some very conical hour glasses, this demo helps so much with the intensity in which they appear to "accelerate". Now I can look and say, "Oh its about 20% lower by height, half way there!"
There's a frame at 15:50 (pause the video and use < and > to find the frame) where pointy-end-up one is at 0% and pointy-end-down one is still at 17%. Now THAT is counterintuitive.
This video explains why I've always been so annoyed with my measuring cup set that has tapered sides. They're essentially chunks of cones so I have absolutely zero intuition for how to eyeball fractions of full for things like 1/6 or 1/8.
@@mrosskne The cups come in 1/4, 1/3, 1/2, 3/4, 2/3, and 1. Sometimes I want a sixth or an eighth of a cup of something. It's not a graduated cup it's just a set of individual ones.
Thinking about an illustrative medium that wouldn’t make a mess, I thought of sand or using immiscible fluids (dyed oil/water). That pointed out the way hourglass height measures time linearly most of the time, but runs out really quickly at the end.
I used to work as a waiter, and those glasses are absolutely horrible to carry on a tray. You don't want that glass full because even the customer will spill it😅
the moral of this should be that drinks arent priced based on the geometry and filling efficiency of the glass, they have calculations for these kinda things based on the cost of the liquor and markups etc lol, you're not being "cheated" out of anything because they dont fill the glass to the brim, just like you're not being "cheated" if part of the glass is occupied by ice, if they filled it twice as full it would cost... twice as much
Talking about unintuitiveness of a cone... It was puzzling when I found out the max volume of a cone is not when the cross section is a right isoceles triangle. Given a hypotenuse, a right triangle's area is max when it is isoceles. Given the same slant height, a cone's max volume is attained when the top angle is around 109.47 degrees instead of 90.
It's a funny coincidence that it is exactly the same angle as the angle in tetraeder between lines connecting center and vertices. I didn't find any reason, why they should be the same angle.
Yeah this angle is also found in the inherent structure of soap films (Plateau's Laws) because of tetrahedral symmetry. It is interesting to find here in the cone, but I think the above commenter is correct, law of small numbers. Notably the volume formula for a unit tetrahedron of edge length A is: V = 1/3 A h
@galoomba5559 You are probably right. The optimization problem leads to finding the extreme of a cubic equation with low integer coefficients, which leads to cos(a/2)= sqrt(1/3). And using cos(a) = 2cos(a/2)^2-1 formula, I get the fraction with low integers. And the other problem, when you have n+1 points in n dimensions on a sphere, with maximising the distance, then you get that the angle between them is arccos(-1/n).
I've had my run in with this visual illusion in the kitchen when using hemispherical measuring spoons. I poured a 0.5 Tbsp of liquid into a 1 Tbsp measure, and was surprised by how close to the top it was. I had always eyeballed half a measuring spoon by filling it half way or maybe 60% of the way up, but you need to go higher.
Bartender here. It's a quite annoying myth that you could be getting ripped off by some trick of glass shape or ice volume. All bars will serve you about 2-3 ounces per cocktail of the base spirit and many will measure it out exact depending on bar policies. Some drinks served in a martini glass may require more juices, like a cosmo, so the glass will get more filled on top of your 2-3 ounces of base spirit. Some drinks might have only booze/spirit ingredients like a dry martini which is basically just vodka or gin chilled. That drink won't fill the glass up very much but it'll be the same amount of booze. Same rules apply for ice displacement. If you think you are being smart by ordering with light ice, the base spirit will always be approximately 2-3 and your get more juice, soda, or mixer. You wouldn't order a whiskey neat and expect the glass to be filled to the top. You might be getting a bad deal from, for example, a corporate restaurant that has their bartenders pour 1.5 ounces of spirit per cocktail and uses too much juice, soda, and mixer. But it doesn't have to do with a trick of the glass.
But when you order a pint of cider or ginger beer from a bar in Australia and they offer you ice, they WILL fill that sucker with ice and give you maybe a half-pint of drink. So the answer is always 'no'
You are saying, that bars don't actually break the law, and if they say "x ml of stuff" they serve "x ml of stuff". We are saying, that everything is designed to trick consumers to pay more for less, which is not that much of an accusation, as bars sell entertainment, not "smashing yourself" generally.
@@HeroDarkStorn I mean you can say that, but it doesn't sound sensible to anyone who works a bar or drinks cocktails. Cocktails served in martini glasses don't have any less of their base spirit then cocktails served in other kinds of glasses. And no one orders a cocktail- or a glass of whiskey for that matter- expecting the bartender to fill up the glass. If you order something short and strong, you understand (or should understand) that your not paying for volume. And martini glasses weren't designed to trick anyone. They replaced earlier glasses- like coupes- that were more curvy and less "deceptive". But the cone shape wasn't designed as a trick- it just fit in with the popular Art Deco aesthetic of the 1920s. And in the early days martinis were served in smaller glasses meant to be filled all the way, but most bars eventually started using larger martini glasses because they're less prone to spilling. And a lot of drinks have since gone back to being served in more curvy glasses; only a few are still served in martini glasses due to tradition. Many bars have phased out martini glasses altogether. In any case no one is trying to trick anyone. That's a silly notion.
I was thinking “imagine how much worse this would be for a 4d hypercone”, and suddenly realized that’s exactly what you’re dealing with in astrophysics with future-past light cones.
Not quite. If special relativity lived in 4D Euclidean space that would be the case, but it lives in Minkowski space which has metric signature + - - - (or - + + + sometimes). This leads to counterintuitive results (for example, the unit sphere does not have a finite volume). Hypervolumes in Minkowski space need a treatment with differential 4-forms, which is something you don't generally encounter in a physics degree until final year or postgrad level.
@@davidgillies620 ...what? :( I literally have multiple books within arm's reach pertaining to different dimensionality in geometry, and I didn't understand 80% of what you just said xD
@Smoth48 The "unit sphere" is a hyperbolic surface. Rather than the usual x²+y²=1 circle stuff in a Euclidean plane, it's like a x²-y²=1 hyperbola because one of the coordinates is imaginary. Basically a so-called circle x²+(iy)² is where the negative term is hiding out. Anyway, you know sin(), cos(), and tan() on a circle... for complex numbers and hyperbola, you get sinh(), cosh(), tanh() Minkowski spacetime is the Argand complex plane with two more real axis. Velocity then is the rotation angle of your time vector... since velocity is dx/dt and the x and t axis together are a complex plane.
@@Smoth48 Dimensionality has got very little to do with it. The point isn't that spacetime is 4D, it's that it's non-Euclidean. Pythagoras doesn't work when you include time coordinates. This corresponds to things called Lorentz boosts which is where the weirdness of special relativity such as time dilation manifests itself. From a mathematical standpoint you can delve as deep as you like, and before long you'll encounter things like Lie Algebras and the Laplace-Beltrami operator and it all gets very complicated very quickly.
The few moments after Ben Sparks says, "You can have a think about that..." are reliably some of the most enjoyable. Thank you to all involved, I loved that.
Very excited i intuitively came up with 1/24. Not spot on but close enough for me. Also if you happen to see this and ever want to do this demo again consider using isopropyl alcohol/ it has much lower surface tension. You could also put a drop of dish liquid in but that may cuase bubbles when you turn it.
Another nice feature of cones: Place a cone radius r, height r, point up and a hemisphere, radius r flat side up, parallel to the plane. Take a horizontal cross-section through both shapes at height y above and parallel to the plane. How does the area of the cross section vary with y? . . . . . . . . . . It doesn't - it is always πr².
One way to do the demo would be to add a plate one third up, add the water on top of that, then remove the lid somehow and watch the water flow down, but seemingly not go lower. For this the lid meeds to be watertight and have some way to be (re)moved, so probably stick pointing up or something.
You could have poured in a drop of dish-washing liquid, to break the surface tension. Maybe some grenadine, as well, to colour the liquid, and make it stand out more.
This helps explain the problem that arises in a bunch of fields when dealing with many dimensions. The volume of the corner gets aggressively smaller the more dimensions you add
Ben Sparks is one of my favorites on this channel. It is a pleasure to listen to him. And what a mind blowing video this is. Easy to understand, but simply never thought of this.
Great video! Always love when Ben Sparks comes. I guess a hydrophobic coating on the top part of the closed glass would solve the surface tension problem.
This is actually a really important property of martini glasses. Turns out if your glasses are twice as big as they need to be to hold a specific amount of drink, it's much harder to spill them; which is particularly handy if you run a bar and don't want your patrons to spill their drinks as they get on with martini night.
Not sure if they are harder to spill if you need to fill the glass so close o its full height in order to make it half full. For a cylinder to make it half full you would only need to fill it to halb of its height. That would make spilling less likely.
@@skyscraperfan Martini glasses are notoriously easy to spill. A lot of bartenders don't like them. There are alternative glasses that you can serve a martini in, like a coupe, that are honestly better. But everyone likes that angular glass because it makes you feel like James Bond. But come on, a cylinder? You want to drink your martinis out of a whiskey glass? Have *some* class.
@@warron24 I never understood why different drinks need different glasses. They always talk about the surface area. Some drinks need to "breathe" more than others. Not sure if that really makes a difference. I do not drink any alcohol anyway.
@@skyscraperfan I'm sure someone could explain all the science to you of the different glasses, but I don't know that much about it. Obviously you can't serve anything with ice in a martini glass. I think a whiskey glass warms a bit faster, encouraging the ice to melt and making for a smoother drink. On the other hand for drinks without ice you want a glass with a stem so it doesn't warm as quickly from handling it. But then again there are drinks (like the Sazerac) that don't have ice but are still served in a whiskey glass. So I don't really know. A lot of it just comes down to history and tradition. Functionally coupes, Nick and Nora glasses, and martini glasses are all pretty much the same, yet they're all favored for different drinks.
@@warron24 Even red wine and white wine have different glasses. I have a more practical approach. For example I use a heavy butcher's knife even for a normal meal, because I love how sharp it is and how much force it allows me to apply. Such a large and heavy knife feels better in the hands. So I love using it. I also never understood why people reserve their favourite clothes for special occasions. If I like a certain pullover, I prefer to wear it quite often. And then there are the Christmas traditions. Certain food is only available in the weeks and months before Christmas, although it tastes great the whole year. It annoys me that I can't even get it in summer if I wanted. For example "Spekulatius".
I love this demonstration. I really wish I hadn't already seen something like this because Ben does such a great job. I would have loved to be surprised. That's a great argument for not going to a bar to drink martinis
“Come, every frustum longs to be a cone, And every vector dreams of matrices. Hark to the gentle gradient of the breeze: It whispers of a more ergodic zone.” - Lem, Cyberiad
interesting (probably coincidental?) appearance of the 80-20 phenomenon here. if you fill the cone 80% one way, you've filled it only about 20% the other way, since cube root of 1/2 is approximately 0.7937
Possible ways of dealing with the meniscus problem: * Put some detergent it the water (although that could lead to a problem with foaming) * Hydrophobic coating on the inside of the glass and lid * Use a bigger glass
In order to solve the problem, I imagined a hollow cube balancing on a corner, with another cube inside. That way, getting to (1-(⅓)³)¹ᐟ³ is really intuitive. If the inner cube (representing the air) has ⅓ of the side length, it must be (⅓)³ the volume. Then I imagined an inner cube filling up 1-(⅓)³ of the volume (now representing the water). Since for this cube the volume is also the side length cubed, to get back to the side length I took the cube root, so (1-(⅓)³)¹ᐟ³. The first thing I always do when π shows up in a geometric problem (but cancels out in the formula) is to get rid of anything round and think in squares and cubes.
I saw this fact about cones on a the Mind Your Decisions UA-cam channel and it blew my mind. Glad to see Numberphile do an in depth video on the topic 😊
No, you haven't, as the glasswear probably wasn't meant to be 100% filled up. That would be a nightmare for wait staff, or a drunken customer. Especially since even if you get the same style of glass, a manufacturer might make it support a different volume Even non-cone glasses are meant to not be filled up 100% of the time
OK guys, one of my favorite videos you've done. Great math, involved, but not too complicated. And the physical, tangible objects used to demonstrate the idea are perfect. Great work!
You need to have Cliff Stoll and his friend make you a few of these completely sealed, all-glass, martini glass demos with the water in them. (Maybe even with a liquid that won’t grow anything over time, turning it into a gross terrarium.) Give one to Ben!
You could make a few and fill them to various important levels. (2/3 filled by height, inverted; 1/2 filled by height, inverted; etc). While the live demo of the 50% full by volume=79.3% by height, standing is more impressive live (pouring the liquid between the open-topped glasses) - it would be interesting if Cliff and his friend could do a cone, with a partition at 79.3% high, then some sort of tube connecting the two sealed regions and like an hourglass, you could transfer the liquid from one side to the other to make either side full
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Here is the hardest problem in the world.
You have 300 coins.
You have to insert a 300 letters message (Signal) one letter per coin.
Then you flip the coins and insert a 300 letters randomly generated Noise.
Then you put all the coins in a bag and scramble them.
You then put the coins back on the table.
The task is to flip and move the coins around until you uncover the original message.
Can you do it?
16:45
*So cones are... "half full/empty"... Pareto Principle! 80/20*
________
*edit -
@tbersags0278 said in the comments this:
_"The top 1% represents a third of the resources. Are we surprised? Really?"_
...and my answer was this, regarding the Pareto Principle:
_"Richest 20% pay 80% of all taxes!"_
This makes me remember a shape/teaching aid a math teacher had back years ago. The middle school math teacher taught kids who were behind ( me ) and the advanced highschool kids ( highschool down the street). It always confused me. It was an upside down circular cone with a flat plane inside on a 45 dregee angle, in the middle third of the cone. Calculate the volume of each section.......
This was on MindYourDecisions 8 days ago. It is not hard to guess once you realise volume goes with the cube of height.
@@johnjameson6751 I was just about to post a question, asking, whether this video was inspired by that one 😅.
Optimist: The glass is half full.
Pessimist: The glass is 96% empty.
This would actually work. An optimist would think a glass is half full looking at its height, and the pessimist would remind him it's 96% air by volume
Pissimist: the glass is now overflowing
@@adarshmohapatra5058 Actually no, if a cone is half full by height, it's 87.5% air by volume. You get the 96% by turning it upside-down and looking at the new height.
(wonders what this correction makes me)
@@Ringcaat A correctionist?
@Ringcaat ahh you're right, a quick check using scaling factors tells me that 1/2 water by height is 1/8 water by volume giving 7/8 = 87.5% air by volume.
.
And I agree with the person who commented below you, this correction makes you a correctionist :D
Optimists: the glass is half full
Pessimists: the glass is half empty
Mathematicians: hold my martini 🍸
Mathematicians: By height or by volume?
an optimist mathematician: the glass is full
Chess players: hold my Queen.
Engineers: The glass is twice as big as it needs to be.
Physicist: **ducks**
It's 99% because that's 66% upside-down.
it really do be like that XD
Perfect. Just perfect.
hahaha
One thing that wasn't mentioned is, our eyes are lying to us when we look at a cone glass because we see a 2D projection. This means on that 2/3 full glass in height we actually see 8/9 of the area is water and 1/9 is air, which makes it not intuitive to believe there's actually three times less air than that. Interestingly, if we do the math with 2D triangular glasses we get about 94% which is basically in line with the initial guess
Exactly this. Even the pictures are 2D here, while there is one whole more dimension for the trick!
Can you write out the math for the 2d triangular glasses?
I wonder if seeing it in person would help in the estimation somewhat.
@redbird_studios for any two dimensional shape, equal aspect scaling increases area by N^2 for N scaling, where N is the factor you're multiplying the measure by. For example if you double the measure of a polygon, it has four times as much area, doesn't matter what the actual shape is. for our example, we're scaling up by 3/2, so the new triangle has 9/4 the area.
if you want some example numbers : if the small triangle has base 12 and height h, the new triangle has base 18 and height 3h/2. the small triangle area is 6h, the new triangle area is 27h/2, or 13.5h. 13.5 = 6 (9/4), as expected.
Thank you. That was the first question that came to my mind--whether it works out the same in two dimensions, and if not, how does it work out.
I've been telling everyone that cones are messed up for years. Finally some vindication.
Vin-di-caaaaaation
yeah but that's only because you keep taking them home from the street after a night out
A stoners experience
Totally messed up. One cone actually killed my dog for no reason
They won't get it even after watching this video because their math skills are messed up.
This is very mind blowing to everyone except for chemists. This is exactly how separatory funnels work! If you are a chemist, you see this extreme smallness of this volume when you drain the bottom phase and see that there's almost nothing left - it's no longer a curiosity, but a functional phenomenon!
…and for the demo in the video, they should have put in a colored non-water miscible liquid in the closed-cone instead of air. The effect would be more visible.
@@Linus_Beckerthat would've been cooler than my idea, which was a sufficiently hydrophobic lid.
@@tinkeringtim7999 Another way is just to add something that lowers the surface tension (such as rubbing alcohol - soap would foam, but alcohol won't).
@Simon-fg8iz he also could've added a tiny cone on the lid with surface area just large enough to support the surface tension without leaving the cone, and explain how that water equals the displaced volume. Very meta 😉
@@Simon-fg8iz : So what you're saying is that he should have put a martini in the glass?
I sometimes use this "scale factor" method to compare the areas of two pizzas based on their diameters, to understand how much more pizza I get with one size vs. another size. For example, pizza restaurants in the United States will frequently offer a 14-inch diameter pizza and a 16-inch diameter pizza. (They don't use the word "diameter" on the menu, but that's what they mean.) Without actually calculating the area of each pizza in square inches, I can compare their areas by knowing that each area is proportional to the square of the radius. So I can think of the 14-inch pizza's area as proportional to 7 squared = 49, and the 16-inch pizza's area as proprtional to 8 squared = 64. In other words, if I order a 14-inch pizza, I will get roughly 3/4 as much pizza as if I order a 16-inch pizza (because 49/64 is approximately equal to 3/4).
Yes, but one confounding factor is that the ratio of crust to normal pizza is much higher as you go for a smaller pizza (assuming the crust width is constant).
@@dylangergutierrez unelles you get the objectlivy better type of pizza, thin crust
@@ThunderClawShocktrix Thin crust pizza still often has a ring of unadorned dough at the edge. Also I like a hearty combo pizza that thin crust cannot necessarily support
Yes, but even using only the scale factor you still have deal with pie.
@@jackmandu I love it! I almost said in my original comment that my method avoids having to calculate with pi. But I didn't even think of the pi/pie pun.
I feel like going through the labor of building pyramids in Minecraft gets a little bit of the intuition going.
When planning out a beacon, and how many tiers tall you want it to be, and figuring out what materials you can afford to build it with.
And realizing that adding just a single tier more than doubles your material usage.
If you think about the glass problem as tiers of a pyramid for a beacon, it hits really fast that, oh yeah the top bit is absolutely inconsequential.
Hey this means that when they were making the pyramids in Egypt, they had to spend to spend more time on the base, than almost the entire top!
@@adarshmohapatra5058 And it means that every bit of making the sides steeper has/would have saved them incredible volumes of material.
Upvote if you want Ben Sparks to explain the monster group. Whenever he explains something, I get it, no matter how complicated it is.
Please yes. That original video went over my head lol
I’d watch just him all day. His Golden Ratio video is my favourite on UA-cam.
Talking about our intuition, they are making matrix resurgence 😊😊😊 +Sadhguru channel equals this happy mathematician
Sometimes, I go back and binge his episodes. He's my favorite at presentation and math concepts.
Ben simply sparks (hope it makes sense in English cause it's not my native lang), cheers. Great video thx.
This sort of behaviour is part of why in civil engineering, water at a weir is often directed through a triangular cross section channel to measure flow rate. Since the depth of the water increases as the flow rate increases, compared to a rectangular channel, more accurate flow measurements can be made at a wide range of flow rates, exactly what's needed in the real world for rivers that can vary from drought conditions to full flood. The other benefit is that the flow speed stays high enough that the channel is less likely to clog, even when flow rates are low.
Bernoulli’s principle is conservation of energy per unit volume. Energy in a fluid is a function of pressure and velocity; when at speeds where the flow is incompressible, smaller volume means less particles to maintain density, that lowers pressure. Since energy is conserved, lower pressure = higher velocity. These relations are called the wind tunnel equations, and we also use a semi-conical shape (more like a pyramid) to increase the speed of air in the test section of a wind tunnel.
👌👌👌
I work as a gardener at a store that does winter storage for large potted plants, and occasionally customers want a plant in a different pot when we deliver it the next spring.
It always keeps amazing me how much additional Earth we have to pack to fill a just so slightly larger pot than the original.
When you go three dimensional, small increases in radius add up to a lot of volume.
As a hiker, I know there will not be enough air space in a wide-mouthed water bottle to cope with elevation changes unless you fill the bottle no higher than the shoulder of the bottle. (Also, in cold weather, keep it in the pack upside down so that it it starts to freeze, the ice won't be across the mouth of the bottle.)
PET bottles can withstand such pressures easily. What's the concern?
@@jurajvariny6034 Not having the lid leak and dribble all over everything else in your pack. The expansion is plenty to loosen the lid.
Just a note about how cocktail bars work: You're not paying for the whole drink, you're just paying for the alcohol. When you order a drink, you will get a fixed amount of alcohol (usually 1.5 or 2 oz), along with mixers, ice, water, etc. If the glass you get isn't full, that just means your drink is stronger. If you get a full glass, it has more mixer, not more alcohol.
Beer glasses should be filled to overflowing though, with no visible head.
@@CliffSedge-nu5fv I worked at a bar part-time for a little bit, we fill until the boundary between the head and the beer itself sits on the pint line
@@CliffSedge-nu5fv you psycho
@@CliffSedge-nu5fv Depends on the beer. Some beers are supposed to be served with some amount of head. Generally speaking, though, you do overpour a little to get most of the head to run off.
@@CliffSedge-nu5fvdepends on the country, especially UK vs Germany. Germans love the head, and UK does not
picking 2/3 empty at the start and then turning upside down and seeing ~90% full is even more interesting.
I agree, that's even more counter-intuitive👍
100%
I did something very similar to this demo where I asked the people around the lab the volume in the bottom of a separatory funnel. It was significantly less than anyone said, like at least half of what they said. That was after I told them that whatever they guessed, go lower. It was less than I guessed, too, but their initial guesses were waaay off.
20 one third filled glasses emptied into one would have been a better demonstration
I don't know about better, but it definitely would have been another cool one to do.
Honestly that is a great idea
I like the idea of doing both. The effect is more extreme when comparing height to height like in the video (I knew the answer and still was surprised to see it in decimal form), but I think it's more palpable when you see all the liquid transferred from glass to glass.
Brilliant!
I look forward to seeing it on your UA-cam channel.
Thank you in advance.
Not exactly practical for a bar trick, though.
This is something you learn if you ever build a pyramid one stone at a time. The top half of the pyramid's height is only an eighth of the volume. You feel like it's taking forever, and then suddenly it's done. Dunno how important that is in understanding the psychology of ancient people who built pyramids, but it's worth thinking about.
That's why the "Jeddah Tower" is already more than half completed in volume, although it only reached a little more that a quarter of its height.
@skyscraperfan ahh so all modern building shapes kinda behave like that
It feels faster if you build the pyramid from the top down
@@mmo5366 Oh man, that would be awful. You'd feel like it's going to be done in no time, and then your progress seems to grind to a halt.
That demonstration at the start is well worth a like all by itself. Absolutely amazing!
"I don't have any formal qualifications"
I think, having watched you as an interviewer on this channel over the years, you definitely have better qualifications than most.
He's definitely fantastic at his job, but that doesn't mean he has qualifications. Experience, talent, and skill are all even more valuable than formal qualifications.
“IT’S ABOUT THE CONES” - Ben Wyatt
I was looking for this comment
Cube root of 1/2 ~ 0.8 is one of those handy numbers to have in your head, like sqrt(2) ~ 17/12, log10 2 ~ 0.3, ln 2 ~ 0.7, e^3 ~ 20 and so on. Feynman has a chapter in Surely You're Joking where he shows how useful this is.
My favorite is 355/113
Yep and this would have made an interesting demo as well, if he'd filled the glass 1/2 by volume. It would be "80% full" of air when upside down, then "80% full" of water when flipped.
for the demo: use something without surface tension. I'm sure some fine-grain sand could work really well.
Or another liquid with lower surface tension like pure alcohol.
I think he realized the problem with the surface tension after gluing the lid on and there was no going back
A comical, canonical, conical video.
a frustrum is frustration, an infundiblium is uber fun!
I love the way that Ben goes at the speed of an average person on camera, despite being a next level mathematician. ❤️
I saw an old advertisement for a company selling cardboard container for movie popcorn. They were selling how much the theater would make by using their cone containers instead of the standard box. You could give a bigger cone container but it would contain much less popcorn than a box. Buy cones not boxes and spend less on popcorn.
Animation @ 8:20 Very well done!
A nice example of how wrong maths can get you to the right answer. You can solve this problem by just turning around the 66.
I don't know about the UK, but in the US "licensed" bartenders pour a specific amount of liquor to make sure one "drink serving" is the same across bars, and bars rarely go over because booze is more expensive than mixer. If your martini glass is 100% full, it means your drink is watered down
Most bars in the UK will use the standard Weights and Measures, which means any spirits must be measured in either a single (25ml, ~1 oz) or a double (50ml). If you order a martini, you're most likely to get a double of gin and a single of vermouth
Also unless you want to be sitting at the bar leaning down to sip the first half of your drink from the rim of the glass you most definitely *do not want* that 3 oz martini in a geometrically "3 oz" glass... ::edit:: As demonstrated 18:34 XD
In France all drinks must have the same amount of alcohol in it by volume, so you will be evenly drunk by a glass of whatever you take in a bar (10 grammes of alcohol per glass)
@@marcosolo6491 yes, but as this is France, shipping charges going to be massive. Can always opt for the Atlantic though. If you're REALLY thirsty, you might get the Pacific Ocean regardless.
@@marcosolo6491how does this follow from anything he said? if you order water, they serve you a glass of water. obviously.
An interesting issue I've been dealing with is closely related to cones, but they're within fixed cylinders, which made this video fascinating.
If you deal with a silo of some diameter, regardless of hight or (in most cases) material property, where you fill from the top center and pull from the center at the bottom, you end up with cones in the material (or remnants of cones as material is added or removed).
A popular technique to find the volume of the material is to drop a single plumbob from 1/6 the diameter of the silo containing the material, measure the distance it descends into the cylinder, and calculate a volume estimate from the results by treating the measurement as the top of a complete cylinder.
Unfortunately, nobody has any proofs as to why 1/6 the diameter is the best choice, nor how close you could estimate your measurement is.
A drop of washing-up liquid will solve those surface tension woes!
And some food coloring would be nice.
that was a thought I had
I have a suspicion that pure alcohol instead of water would also do the trick and it would be "in the theme".
Soap would just cause bubbles. But I think pure alcohol has much less surface tension.
Or just scale up the model - obviously at a certain point weight becomes an issue but I bet it's possible to get to a no-cling point before it's too heavy to handle...
An important note to save the bartenders in your life some trouble:
Bartenders measure the drinks they are mixing before pouring them into the glass, in no way is the drink supposed to reach all the way up to the rim. You aren’t being cheated you’re getting what you paid for in a container large enough to keep you from spilling it allover yourself.
I have some very conical hour glasses, this demo helps so much with the intensity in which they appear to "accelerate". Now I can look and say, "Oh its about 20% lower by height, half way there!"
This feels like classic Numberphile ❤
There's a frame at 15:50 (pause the video and use < and > to find the frame) where pointy-end-up one is at 0% and pointy-end-down one is still at 17%. Now THAT is counterintuitive.
This video explains why I've always been so annoyed with my measuring cup set that has tapered sides. They're essentially chunks of cones so I have absolutely zero intuition for how to eyeball fractions of full for things like 1/6 or 1/8.
if you have to eyeball an amount, it's not a measuring cup
@@mrosskne The cups come in 1/4, 1/3, 1/2, 3/4, 2/3, and 1. Sometimes I want a sixth or an eighth of a cup of something. It's not a graduated cup it's just a set of individual ones.
@@ngwoo is it that important anyway? oh my, i added 10ml too much water...
@@paradiselost9946 You know that could be huge in certain scale
@@mateussilva635 yeah, but generally not when cooking...
iunno. i work fine on the "about that much" technique. noone complains!
Thinking about an illustrative medium that wouldn’t make a mess, I thought of sand or using immiscible fluids (dyed oil/water). That pointed out the way hourglass height measures time linearly most of the time, but runs out really quickly at the end.
I used to work as a waiter, and those glasses are absolutely horrible to carry on a tray. You don't want that glass full because even the customer will spill it😅
Exactly. There's no conspiracy here
And that's why you serve based on a measurement. It's not a beer where you fill the glass to the top or a line.
The moral of the video is "Don't buy drinks that come in martini glasses."
I can't help wondering if the martini glass was designed with this in mind.
@@kjh23gk this guy's thinking outside the box
or outside the cone in this case
the moral of this should be that drinks arent priced based on the geometry and filling efficiency of the glass, they have calculations for these kinda things based on the cost of the liquor and markups etc lol, you're not being "cheated" out of anything because they dont fill the glass to the brim, just like you're not being "cheated" if part of the glass is occupied by ice, if they filled it twice as full it would cost... twice as much
@@JustCallMeCharlie *WOOSH*
Did you see that? That was the joke going over your head.
@@RadioactiveLobsterNothing goes over their head. Their reflexes are too fast. They would catch it.
Talking about unintuitiveness of a cone... It was puzzling when I found out the max volume of a cone is not when the cross section is a right isoceles triangle. Given a hypotenuse, a right triangle's area is max when it is isoceles. Given the same slant height, a cone's max volume is attained when the top angle is around 109.47 degrees instead of 90.
It's a funny coincidence that it is exactly the same angle as the angle in tetraeder between lines connecting center and vertices. I didn't find any reason, why they should be the same angle.
@@jakubkocak887what is tetraeder?
@@jakubkocak887 It could be just law of small numbers. The angle is arccos(-1/3), a pretty simple formula.
Yeah this angle is also found in the inherent structure of soap films (Plateau's Laws) because of tetrahedral symmetry. It is interesting to find here in the cone, but I think the above commenter is correct, law of small numbers. Notably the volume formula for a unit tetrahedron of edge length A is:
V = 1/3 A h
@galoomba5559 You are probably right. The optimization problem leads to finding the extreme of a cubic equation with low integer coefficients, which leads to cos(a/2)= sqrt(1/3). And using cos(a) = 2cos(a/2)^2-1 formula, I get the fraction with low integers. And the other problem, when you have n+1 points in n dimensions on a sphere, with maximising the distance, then you get that the angle between them is arccos(-1/n).
Hey! When is flipped is a super sensitive instrument to measure small volumes
I've had my run in with this visual illusion in the kitchen when using hemispherical measuring spoons. I poured a 0.5 Tbsp of liquid into a 1 Tbsp measure, and was surprised by how close to the top it was. I had always eyeballed half a measuring spoon by filling it half way or maybe 60% of the way up, but you need to go higher.
Bartender here. It's a quite annoying myth that you could be getting ripped off by some trick of glass shape or ice volume. All bars will serve you about 2-3 ounces per cocktail of the base spirit and many will measure it out exact depending on bar policies. Some drinks served in a martini glass may require more juices, like a cosmo, so the glass will get more filled on top of your 2-3 ounces of base spirit. Some drinks might have only booze/spirit ingredients like a dry martini which is basically just vodka or gin chilled. That drink won't fill the glass up very much but it'll be the same amount of booze. Same rules apply for ice displacement. If you think you are being smart by ordering with light ice, the base spirit will always be approximately 2-3 and your get more juice, soda, or mixer. You wouldn't order a whiskey neat and expect the glass to be filled to the top. You might be getting a bad deal from, for example, a corporate restaurant that has their bartenders pour 1.5 ounces of spirit per cocktail and uses too much juice, soda, and mixer. But it doesn't have to do with a trick of the glass.
But when you order a pint of cider or ginger beer from a bar in Australia and they offer you ice, they WILL fill that sucker with ice and give you maybe a half-pint of drink. So the answer is always 'no'
Fancy drinks are always extremely overpriced, so it's not a myth that you get ripped off in bars 😂.
You are saying, that bars don't actually break the law, and if they say "x ml of stuff" they serve "x ml of stuff".
We are saying, that everything is designed to trick consumers to pay more for less, which is not that much of an accusation, as bars sell entertainment, not "smashing yourself" generally.
@@HeroDarkStorn I mean you can say that, but it doesn't sound sensible to anyone who works a bar or drinks cocktails. Cocktails served in martini glasses don't have any less of their base spirit then cocktails served in other kinds of glasses. And no one orders a cocktail- or a glass of whiskey for that matter- expecting the bartender to fill up the glass. If you order something short and strong, you understand (or should understand) that your not paying for volume.
And martini glasses weren't designed to trick anyone. They replaced earlier glasses- like coupes- that were more curvy and less "deceptive". But the cone shape wasn't designed as a trick- it just fit in with the popular Art Deco aesthetic of the 1920s. And in the early days martinis were served in smaller glasses meant to be filled all the way, but most bars eventually started using larger martini glasses because they're less prone to spilling. And a lot of drinks have since gone back to being served in more curvy glasses; only a few are still served in martini glasses due to tradition. Many bars have phased out martini glasses altogether. In any case no one is trying to trick anyone. That's a silly notion.
Thou protest too much
Scale factors tied a satisfying ribbon on this demonstration. Thank you for sharing
I was thinking “imagine how much worse this would be for a 4d hypercone”, and suddenly realized that’s exactly what you’re dealing with in astrophysics with future-past light cones.
Not quite. If special relativity lived in 4D Euclidean space that would be the case, but it lives in Minkowski space which has metric signature + - - - (or - + + + sometimes). This leads to counterintuitive results (for example, the unit sphere does not have a finite volume). Hypervolumes in Minkowski space need a treatment with differential 4-forms, which is something you don't generally encounter in a physics degree until final year or postgrad level.
@@davidgillies620 ...what? :(
I literally have multiple books within arm's reach pertaining to different dimensionality in geometry, and I didn't understand 80% of what you just said xD
@Smoth48 The "unit sphere" is a hyperbolic surface. Rather than the usual x²+y²=1 circle stuff in a Euclidean plane, it's like a x²-y²=1 hyperbola because one of the coordinates is imaginary. Basically a so-called circle x²+(iy)² is where the negative term is hiding out. Anyway, you know sin(), cos(), and tan() on a circle... for complex numbers and hyperbola, you get sinh(), cosh(), tanh()
Minkowski spacetime is the Argand complex plane with two more real axis. Velocity then is the rotation angle of your time vector... since velocity is dx/dt and the x and t axis together are a complex plane.
@@Smoth48 Dimensionality has got very little to do with it. The point isn't that spacetime is 4D, it's that it's non-Euclidean. Pythagoras doesn't work when you include time coordinates. This corresponds to things called Lorentz boosts which is where the weirdness of special relativity such as time dilation manifests itself. From a mathematical standpoint you can delve as deep as you like, and before long you'll encounter things like Lie Algebras and the Laplace-Beltrami operator and it all gets very complicated very quickly.
n-dimensional generalisation when, Ben?
The few moments after Ben Sparks says, "You can have a think about that..." are reliably some of the most enjoyable. Thank you to all involved, I loved that.
Very excited i intuitively came up with 1/24. Not spot on but close enough for me.
Also if you happen to see this and ever want to do this demo again consider using isopropyl alcohol/ it has much lower surface tension. You could also put a drop of dish liquid in but that may cuase bubbles when you turn it.
2:59 Even though Brady's guess was a bit off, I'll still count it as another win for Brady's superb mathematical intuition!
The first demonstration blew my mind tbh. Loved learning about this!
Another nice feature of cones:
Place a cone radius r, height r, point up and a hemisphere, radius r flat side up, parallel to the plane.
Take a horizontal cross-section through both shapes at height y above and parallel to the plane.
How does the area of the cross section vary with y?
.
.
.
.
.
.
.
.
.
.
It doesn't - it is always πr².
One way to do the demo would be to add a plate one third up, add the water on top of that, then remove the lid somehow and watch the water flow down, but seemingly not go lower.
For this the lid meeds to be watertight and have some way to be (re)moved, so probably stick pointing up or something.
Now the old philosophical question "is the glass half full or half empty" has to be clarified : is it a Martini glass?
That was great exposition.
Especially when they are on the M25. That's messed up
???
"The Cone-nundrum of Cones" would've been a better title, imo.
"Cones are messed up". There's a Numberphile merch line just waiting to be started.
Yep same issue as the task of generating a uniformly distributed sample of points over the unit disc using polar coordinates.
Presh from MindYourDecisions did a video on the same subject literally the other day. What a coincidence.
5 days ago
I presumed that was the problem Ben saw?
it is a coincidence - I'm sure this video was filmed, or at least planned, more than five days ago.
It's a conecidence
Big Martini PR working overtime
You could have poured in a drop of dish-washing liquid, to break the surface tension. Maybe some grenadine, as well, to colour the liquid, and make it stand out more.
what a nice little video
Wow, I never thought of this, I'm fascinated!
First, we have Parker Squares. Now, we have Sparks Glass ! A drop is outside, hanging on the glass...
This helps explain the problem that arises in a bunch of fields when dealing with many dimensions. The volume of the corner gets aggressively smaller the more dimensions you add
nice animation when you flipped the code and 66 :)
Came here to point that out as well!
Ben Sparks is one of my favorites on this channel. It is a pleasure to listen to him. And what a mind blowing video this is. Easy to understand, but simply never thought of this.
you should fill the martini glass with vodka , less surface tension , but perfect lore
There is something very satisfying about the number 66 turning into 99 as you rotate the glass.
1:25 you are paying for a set amount. the size of the glass has nothing to do with that so please, please don't complain about this to your server.
I'm going to make the server fill the rest with vodka.
You missed the point he tried to make.
The first 45 seconds of this video absolutely blew my mind. Like Tree 3 levels of mind melting.
This goes back to just how much larger a large pizza is than a medium pizza.
But in three dimensions, not just two
@@dielaughing73 I love those 2 dimensional pizzas.
@@kwanarchive haha the sizes only vary by area, not height, so it's a square relationship not cubic
very elegantly explained all 3 intuitions
Bartenders everywhere now need to be prepared for angry Numberphile viewers.
16:58 but you still ask the questions that are on point every time!
Try that using sand instead of water, perhaps
Great video! Always love when Ben Sparks comes.
I guess a hydrophobic coating on the top part of the closed glass would solve the surface tension problem.
This is actually a really important property of martini glasses. Turns out if your glasses are twice as big as they need to be to hold a specific amount of drink, it's much harder to spill them; which is particularly handy if you run a bar and don't want your patrons to spill their drinks as they get on with martini night.
Not sure if they are harder to spill if you need to fill the glass so close o its full height in order to make it half full. For a cylinder to make it half full you would only need to fill it to halb of its height. That would make spilling less likely.
@@skyscraperfan Martini glasses are notoriously easy to spill. A lot of bartenders don't like them. There are alternative glasses that you can serve a martini in, like a coupe, that are honestly better. But everyone likes that angular glass because it makes you feel like James Bond.
But come on, a cylinder? You want to drink your martinis out of a whiskey glass? Have *some* class.
@@warron24 I never understood why different drinks need different glasses. They always talk about the surface area. Some drinks need to "breathe" more than others. Not sure if that really makes a difference. I do not drink any alcohol anyway.
@@skyscraperfan I'm sure someone could explain all the science to you of the different glasses, but I don't know that much about it. Obviously you can't serve anything with ice in a martini glass. I think a whiskey glass warms a bit faster, encouraging the ice to melt and making for a smoother drink. On the other hand for drinks without ice you want a glass with a stem so it doesn't warm as quickly from handling it. But then again there are drinks (like the Sazerac) that don't have ice but are still served in a whiskey glass. So I don't really know.
A lot of it just comes down to history and tradition. Functionally coupes, Nick and Nora glasses, and martini glasses are all pretty much the same, yet they're all favored for different drinks.
@@warron24 Even red wine and white wine have different glasses. I have a more practical approach. For example I use a heavy butcher's knife even for a normal meal, because I love how sharp it is and how much force it allows me to apply. Such a large and heavy knife feels better in the hands. So I love using it.
I also never understood why people reserve their favourite clothes for special occasions. If I like a certain pullover, I prefer to wear it quite often.
And then there are the Christmas traditions. Certain food is only available in the weeks and months before Christmas, although it tastes great the whole year. It annoys me that I can't even get it in summer if I wanted. For example "Spekulatius".
I did almost exactly these calculations to work out some path-lengths in a photo-spectrometer a few months ago.
"The video licenced liquor vendors don't want you to see"
Watch ‘till the end! You won’t believe!! Top Google engineer fired over his discovery!
I love this demonstration. I really wish I hadn't already seen something like this because Ben does such a great job. I would have loved to be surprised.
That's a great argument for not going to a bar to drink martinis
A video about cones
*you're* a video about cones 😤
There is a serious lack of ice cream in this video, the best of the cones.
Beautiful, and thought provoking. Thank you for posting.
0:40 this made me laugh so hard. Just two seconds of silence then 'cones are messed up :/'
That's actually seriously crazy.. man, I love this channel. Applied math's pretty cool.
I never before saw "1 minute ago" in the upload time of a youtube vod! Early as ever :)
First! 😂
these first 40 seconds are the best intro I've seen anywhere in a while
where's my cones are messed up T-shirt
My favorite thing about cones is that slanted slices of them can be used to define ellipses (which are also messed up in some ways)
“Come, every frustum longs to be a cone,
And every vector dreams of matrices.
Hark to the gentle gradient of the breeze:
It whispers of a more ergodic zone.”
- Lem, Cyberiad
Kandel interpretation, I presume? The poetry AI story is one of those where translation must have involved some major creative liberties...
It had to happen...
That outro was perfect! Cheers guys.
It's the return of the 80:20 rule!
I'm even more amazed just about the steepness not mattering how you might think it should !
interesting (probably coincidental?) appearance of the 80-20 phenomenon here. if you fill the cone 80% one way, you've filled it only about 20% the other way, since cube root of 1/2 is approximately 0.7937
Possible ways of dealing with the meniscus problem:
* Put some detergent it the water (although that could lead to a problem with foaming)
* Hydrophobic coating on the inside of the glass and lid
* Use a bigger glass
do away with the lid when upright altogether, and demonstrate air pressure when inverted as well?
In order to solve the problem, I imagined a hollow cube balancing on a corner, with another cube inside.
That way, getting to (1-(⅓)³)¹ᐟ³ is really intuitive.
If the inner cube (representing the air) has ⅓ of the side length, it must be (⅓)³ the volume.
Then I imagined an inner cube filling up 1-(⅓)³ of the volume (now representing the water).
Since for this cube the volume is also the side length cubed, to get back to the side length I took the cube root, so (1-(⅓)³)¹ᐟ³.
The first thing I always do when π shows up in a geometric problem (but cancels out in the formula) is to get rid of anything round and think in squares and cubes.
I saw this fact about cones on a the Mind Your Decisions UA-cam channel and it blew my mind. Glad to see Numberphile do an in depth video on the topic 😊
The moral of the story: You’ve been cheated every time that you ordered a drink that came in a Martini glass that wasn’t filled to the brim. 🍸
Then again, a martini glass that full would be inclined to spill.
Or it could be watered down to get that full @@GeorgeDCowley
No, you haven't, as the glasswear probably wasn't meant to be 100% filled up. That would be a nightmare for wait staff, or a drunken customer. Especially since even if you get the same style of glass, a manufacturer might make it support a different volume
Even non-cone glasses are meant to not be filled up 100% of the time
Love any video with Ben Sparks. Such enjoyable lessons
heeeyoo
I love when the videos have the questions I would ask.
OK guys, one of my favorite videos you've done. Great math, involved, but not too complicated. And the physical, tangible objects used to demonstrate the idea are perfect. Great work!
You need to have Cliff Stoll and his friend make you a few of these completely sealed, all-glass, martini glass demos with the water in them. (Maybe even with a liquid that won’t grow anything over time, turning it into a gross terrarium.) Give one to Ben!
You could make a few and fill them to various important levels. (2/3 filled by height, inverted; 1/2 filled by height, inverted; etc). While the live demo of the 50% full by volume=79.3% by height, standing is more impressive live (pouring the liquid between the open-topped glasses) - it would be interesting if Cliff and his friend could do a cone, with a partition at 79.3% high, then some sort of tube connecting the two sealed regions and like an hourglass, you could transfer the liquid from one side to the other to make either side full