Hunt for the Elusive 4th Klein Bottle - Numberphile 

Klein Bottles just aren't the same without the Klein bottle guy.

Apparently all topologists are obsessed with ants.

The videos this week are very one sided...

Easy, it's under Cliff's house. You need to ask him.

worst invention ever: mobius strip seatbelts

The "Klein Bottle" that we can hold is really only the 3d shadow of a 4d shape. Think of it this way, if you had a flat drawing of a Mobius strip, it would appear to "go through itself" in order to connect backwards with the other end. That drawing would only be a 2d shadow of the 3d shape that the Mobius strip is. The Klein bottle as we can see it passes through itself. In the 4th dimension it would be a single uniform shape like the Mobius strip. Think of the Klein bottle he's holding as a 3d drawing of a 4d shape.

The biggest question is if someone can find a groß bottle.

If a Klein bottle has no Volume, are you allowed to take one with you in a plane???

19:58 disney should try making klein bottle roller coasters. Yeah, there's an idea.

Oh come on, that's such a one-sided view of a mobius loop!

6:47 "oh surface tension, we meet again old enemy"

So Mario Circuit in Mario Kart 8 is essentially a Mobius Band? That's really cool! :D

As a chemist this video on chirality is so beautiful, and in fact highly applicable to the so-called 'real world'. Indeed there are many molecules with built in chirality analogous to these mobius strips, and they can show many amazing properties. Chiral molecules (which we call enantiomers) are denoted R and S depending on their 'handedness'. The interesting thing from a chemistry perspective is chemically, the R and S molecules behave identically. In virtually any chemical reaction/interaction, the R or the S molecule will behave in the same way (the only exception is if there is another chiral influence in the system, which will in turn interact differently with the R molecule compared to the S). The cool thing here is if we move to the analogous molecular Klein bottles, where we now have a single molecule with two built in chiral components (termed 'diastereomers', which can adopt configurations of RR, SS, RS and SR), these molecules behave differently in virtually any chemical system, regardless of whether the external influence is itself chiral or not. I'm not an expert maths or topology, but I would be fascinated if someone who is would be able to tell me if these '4 unique' Klein bottles are in fact analogous to the 4 typically unique configurations of diastereomers.

The reason the Stein is smaller is that it's a Klein Stein, and "klein" means "small"

if you had posted this a day earlier, i would have had enough time to put it into my essay on multiple dimensions and non-oriented surfaces...

I just realized that I once made a klein bottle out of a slinky.

Why did the chicken cross the road? - To get to the same side!

Can you make a video about turning a sphere inside out? That stuff keeps showing up in my suggested videos but I would like it to be explained properly. EDIT: Interestingly.

Have always found these to be awesome topological objects. It's so cool seeing the projections of them in three dimensions.

Haha waldo in the thumbnail "The SEARCH for..."

Why is that people who study Mobius bands and Klein Bottles turn out to be really cool dudes?

More videos on Klein Bottles: bit.ly/KleinBottles

And he made a paper about it to clear it up? That is awesome, we'll need it when we learn to travel to the fourth dimension ;)

That is not coffee, that is tea: Greetings from Finland.

17:16 "This thing" -> *fart* Yeah, that thing...

Great video. Many thanks to Brady and Carlo. Keep making videos on topology, it's a very interesting area of mathematics that brings the numbers and the reality together in a really cool way.

Damn that's some watery coffee

I love professor Séquin's lectures, he is a really great speaker and topology is great for this channel.

Got the Klein Bottles, now I see I need a 3D printer!

I love the way he says 'liquid'. "Leek-vid"

I love this guys sincerity and teaching method with words. Would make a great teacher.

These days we know those are 3D printed, but in the 80s and 90s, these shapes would blow peoples minds, based on how plastics were moulded.

That has to be the weakest "coffee" on the planet.

ein Klein Stein?

If you start with a square sheet, and you orient both pairs of opposite edges in parallel, and join them according to those orientations, you get a torus. If you orient one pair of opposite edges in parallel, and the other pair in opposing directions, and join them according to those orientations, you get a Klein bottle. [You can see this on the leftmost figure in the frame at 22:55.] But if you orient both pairs of opposite edges in opposing directions, and join them according to those orientations, you get a cross cap. I read this in an early Mathematical Games column by Martin Gardner, in Scientific American, but I've never seen a cross cap actually realized. What does it look like? It obviously must be self-intersecting in 3D. BTW, there's a neat way to visualize a non-self-intersecting Klein bottle in 4D, that might be a bit too hard to describe here, but maybe I can give it a go. If you conceptualize 3 dimensions as being drawn on a 2D sheet of paper (a common occurrence, so this should be reasonably easy), then just start drawing the bulb portion, continuing into a narrowing neck, which terminates as an open end. On the bulb part, you continue the drawing to show it 'folding' into itself, and then terminating in another open end, inside the 'bulb.' OK, now cut out that drawing, and bend the neck end back and over the bulb, and connect it to where the 'infolded' open end is. The idea is that the actual 3rd dimension (above the paper) is representing the 4th dimension, and you can make the tube you're bending, appear in the 3D space represented within the paper, anywhere within that 3D space, so you just make it 'pop' into there where the infolded part of the bulb is, to join them together. And voilà! - A non-self-intersecting Klein bottle!

Genuinely brilliant. Great stuff from the good doctor.

So watching this, naturally my first thought is I NEED A KLEIN STEIN! But how on earth do you clean such a thing? :P

4:45 Kleinstein. Found in every Cave in the German Pokémon Translations

Carlo's vids on either weird topology or higher dimensional fancyness are amazing.

Ah this is soo cool! I read Carlo Séquin's article about the number of Klein bottle types. It's amazing to basically hear the paper from him!

Welcome to Kleinphile, the channel all about Klein bottles.

21:17 I could've sworn I heard crickets in the background after he finished that sentence! xD

This is one of my favourite Numberphile videos so far =)

Thank you for explaining so well a very hard concept, you`re a great teacher !

6:24 I can't quite put my finger on it, but… * puts on detective hat * …something about it tells me that wasn't coffee…

I want to know what happens with you join the edges of two Möbius bands that are each twisted three times. (Three half-rotations). Would it still form a regular Klein bottle?

Heard about the Klein Stein and just had to get one. Thank you for the information.

ahh i love carlo sequin, his twisted torus video is legendary

It actually seems a little straightfoward in hindsight why the 4th klein bottle is in a sense a mirror image/identical to the first type after following his methodology. the first is a combination of a left and a right mobius, the second is a combination of two right mobii, the third is a combination of thwo left mobii and the fourth is a combination of a right and a left mobius, it can be though of as different, but it is essentially the same in a way. do I have this right?

Where can I buy the Klein stein?

I love all the 3D printed shapes this guy has. They are all so beautiful. I’d like them just hanging around. They are so beautiful

Thankyou so much for clarifying around the 4:00 part the differences between a mathmatical 4D klein bottle and a real physical klein bottle.

400 years ago the Puritans would stone to death anyone found in possession of a Klein bottle.

Can a German company please produce these little mugs and place pictures of famous mathematicians on them? Because I'd really love to buy one for myself and a friend and share... ...mein und dein klein Einstein Klein Stein

I LOVE IT! You have taught not only mathematics but also sociology in your approximation of understanding a demographic of those who have the ability to understand the Mobius stripe and the Klein bottle, surely with your effort multiplied we can reach a more reasonable 1 to 10 ratio than 1 to 100! I believe it can be done eventually. Great presentation.

I really liked the knotted structures.. reminded me of celtic knots

Ok, so I wonder, is there an object, like the mobius band and the klein bottle, that is based in 4 dimensions? Let me explain. in 2d: draw a 2 lines parallel, and connect the ends of the lines to each others beginnings, without crossing: you can't, but a mobius band (3 dimensions) does exactly that. in 3d: take a hollow tube, and connect the end to the beginning, in a way that the outside of the tube, becomes connected to the inside of the tube: you can't, but a klein bottle (4 dimensions) does exactly that. so, what would the 4 dimensional step be?

Is this the guy who bought the first klein bottle for 100 bucks? :D

The best Klein bottle video I have ever watched!

5:08 I realize the potential for the KleinStein is that you could fill the outside with water, stick it in the freezer, and once it's frozen, you fill the inside with beer. Now you have ice cold beer that doesn't get watered down, but as the ice melts into water while you're drinking your beer, you now have a little bit of water you can drink to counteract the beer, because smart people drink responsibly.

You figure someone working at a university would have some better coffee. Damn that looks more like tea then coffee.

I want a Klein Stein now...

I like how he accidentaly spilled like half a cup of coffee on the ground and just let it soak it in.

I love his voice and that poem was awesome

how to clean that cup???

Okay, so if you take two Möbius bands together you get a Klein bottle, and the dimensions needed jumps from 3 to 4. But if you take two Klein bottles, and 'glue' the two sides together, what shape would you get? And how many dimensions would it require? Furthermore, can I keep doing this forever?

Séquin always does great videos!!

Worked in a robotic hospital laundry. Rubberised conveyor belts wear out on one side ( for transferring flat sheets) so what they did was twist one side over before attaching it so it was a mobius strip. The effect was that every other time the belt made a complete turn it was on the other side.

17:17 was that a fart? :P

How would one wash a kleinstein?

Understanding that video about turning a sphere inside out, helps with this.

Such a soothing voice

11:05 I'd call that a Klein Pretzel rather than Bottle.

people with ocd will go insane watching this video

More topology!!! This is good stuff, Numberphile :)

I like this guy. I could listen to him talk for hours, for some reason.

That can't be coffee, surely.

Wouldn't it be better if the Klein Stein's handle went the other way (i.e. the bottom end of the handle to the inside and the top end to the space between inside and the outside)? I'm quite certain that much less air would be stuck in between the walls...

For the "Kleinstein" to work better you have to change the connections on the handle. Meaning: inside to bottom and outside to top and by the way the intersection is like an edge (like going into a tunnel)

I liked a lot to hear this mathematician!

That was some nasty looking coffee.

Klein stein sounds like Einstein.

I started out searching what a klein bottle is. Now I can't stop watching these videos

This guy is awesome. You need to do more videos with him.

what if i were to sew two klein bottles together in the fifth dimension? because a klein bottle is simply sewing together two mobious strips in the third dimension.

That's some weak looking coffee lol

Really enjoyed this

Brady!! check out how to turn a sphere inside out without braking the rules of regular homotopy!! the solution is pretty stunning!

that's some manky coffee. Now I know why they complain about funding.

7:58 Calvin Klein Underpants.

This is amazing!!!!! Now I want a Klein Stein.

Mobius Strip: Mindblown everytime I'm reminded of them.

this guy looks like if darth sideous was a jedi instead of sith

what happens if you fill a *regular* klein bottle with water

That was the most lack-luster end to a video ever... but I still loved all the explanations!

Wonderful video, this is a good instructor.

This is interesting, but it makes my brain very mad.

Someone should make a video game that allows you to navigate through a 4 dimensional world. lol

Love this topic

his voice is amazing.