Cones are MESSED UP - Numberphile 

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 😅.

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

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**

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.

it really do be like that XD

Perfect. Just perfect.

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.

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.

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.

👌👌👌

…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?

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.

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

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 RU-vid.

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.

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.

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.

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 RU-vid channel. Thank you in advance.

Not exactly practical for a bar trick, though.

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.

a frustrum is frustration, an infundiblium is uber fun!

I agree, that's even more counter-intuitive👍

100%

I was looking for this comment

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...

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

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.

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!

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.

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?

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).

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.

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.

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

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.

do away with the lid when upright altogether, and demonstrate air pressure when inverted as well?

???

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

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

Came here to point that out as well!

I'm going to make the server fill the rest with vodka.

You missed the point he tried to make.

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".

Watch ‘till the end! You won’t believe!! Top Google engineer fired over his discovery!

is the answer 1 day?

@@jakedotnet Yes :)

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

Kandel interpretation, I presume? The poetry AI story is one of those where translation must have involved some major creative liberties...

First! 😂

*you're* a video about cones 😤

There is a serious lack of ice cream in this video, the best of the cones.

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