# A mirror paradox, Klein bottles and Rubik's cubes

Share
Embed
• Published on Jun 21, 2015
• The Mathologer puts the latest \$2000 addition to his Klein bottle collection to work. A couple of first-ever fun mathematical stunts in this video. This video finishes with a puzzle for you to think about. We posted a video with the solution on 1 August 2015: thexvid.com/video/ZMC61C5tigA/video.html
Enjoy!
Burkard Polster and Giuseppe Geracitano
P.S.: Most of of Burkard's glass Klein bottles as well as the Klein bottle hat were made by Cliff Stoll of ACME Klein bottles (www.kleinbottle.com). The metal 3d-printed Klein bottle opener was made by Bathsheba Grossman (www.bathsheba.com).

## Comments • 2 132

• Robo Day ago

Cringey

• KCatch22 Day ago

When you said you got the Klein bottles from California, I immediately thought of Clifford Stoll :D

• Mathologer  Day ago

Yep, got most of my KBs off Cliff. According to him I'm his best customer :)

• LDiableo 2 days ago

But how?

• Edward Robaon 2 days ago

If you take a strip of paper and write the words Time Destroys All on one side and Things Come With on the other and then make a Möbius strip out of it you then have a working model of the spacetime continuum. You make a meaningless sentence mean something by repeating it.

• zdcyclops1 2 days ago

Mirror paradox?

• Bill Patterson 2 days ago

The Guy on the mobius strip was transferred THROUGH the paper to invert him. If he had travelled around a second time that would place him on the outer surface like the original and he would not be reversed. There is no reversal unless he is transferred between the two surfaces.

• Mathologer  Day ago

A mathematical MS has no thickness and the little man who lives inside it has neither. There is no transfer between surfaces.

• Ace The FireDragon 2 days ago

The 4th dimension is time.

• Ace The FireDragon 2 days ago

Wait dafuq? How did u get that Rubio cube in there?

• Tone Seeker 2 days ago

Would make a great bong

• Sarah jayne Wilkes 2 days ago

Awesome

• badwolf1913 2 days ago

so, why are there only 2 of them?

• Korasi 2 days ago

Your Chanel is cool and everything, but what is the point of making a whole video about putting a rubics cube into a Klein bottle and then never actually showing us how it’s done??? I’m totally serious, what were you thinking?? “OK now I’m going to go do the part that everyone wants to see, but without any cameras!”
I mean, you didn’t even EXPLAIN how you did it!! As far as I’m concerned, you simply didnt! I can only assume that you had an additional Klein bottle with a rubics cube already in it, that you must have asked a glass blower to make.
Disappointed...

• Mathologer  Day ago

Hmm, how about you pause the video right where I display the Rubik's cube, switch on your brain to ponder what exactly you see. Isn't it BLATANTLY obvious what is going on? Now just in case you really cannot figure out what is happening, there is a follow-up video :)

• Ding Chavez 3 days ago

someone has a klien bottle fetish

• Eden John Talbot 3 days ago

Why could you not just show us how to full it with water?

• Mathologer  2 days ago

There is already a TheXvid video like that :)

• Eden John Talbot 3 days ago

Could you full it with liquid and rotate it and show us the results of liquid distribution?

• KlaxonCow 3 days ago

A Kate Bush fan? I recognise those three chords at the beginning.
I name that tune in 3.

• Utkarsh Mishra 3 days ago

Why is Mathematics a Language of Universe?

• X YZ 3 days ago

And the purpose of a Klein bottle ? If it is actually useful for anything, why have’nt we all got one ?

• jazz Rejalde 3 days ago

Is it strange that I'm not good at math but loves things like this?

• jazz Rejalde 3 days ago

I think this explains how time travel was made...

• M.A. R 3 days ago

How in the actual hell did he get that cube in there?!

• Axel Andru 3 days ago

Sit back and go fishing please !

• John Rainey 3 days ago +1

Great moves, glass bottle, keep it up! I’m proud of you.

• oak 4 days ago

As a young lad it was "prophicized" that I would find a Klein's bottle in a thrift store for 10 cents. Years later I found a Double Klein's Bottle for 25 cents. Strange world isn't it....

• Mathologer  23 hours ago

+oak The overall shape is similar to this one here? www.kleinbottle.com/double_and_triple_Klein_Bottles.html
Actually come to think of it I talk a little bit about double Klein bottles in this video: thexvid.com/video/ixduANVe0gg/video.html

• oak Day ago

Yes it is wrapped in tissue as a valued piece in my desk. about 1960 I would say. Possibly a glass blower's idiocyncratic piece...Found it in Moutain View California.

• Mathologer  Day ago

You really found a double KB for 25 cents? That's amazing :)

• Ram Posham 4 days ago

smoke out of it

• Jing350 4 days ago

Shout out to Cliff Stoll, he does a good job explaining this concept as well! as making them lol

• Mathologer  4 days ago

I got most of my Klein bottles off Cliff. Even the big one is one of his :)

• Naresh Gupta 5 days ago

Im so confused how this is possible

• Deven Dong 5 days ago

I had to watch this in x1.5 speed...

• Dio PM 6 days ago

The bottle definetly got an edge

• Marko d Nivel 6 days ago

dont watch second video, really sucked at 1 minute your heart will stop.

• Stephen Spreckley 7 days ago

Eine kliene bottle musik

• myc riddle 7 days ago

What?

• Sam Tux 7 days ago

A mirror image of you will likely starve to death because good old amino acids don't match your new chirality.

• Mathologer  7 days ago

That's definitely one of the challenges of the new world order that you are facing :)

• holyholo1 8 days ago

So my guess that the Rubik's cube is attached with magnets rather than screws so you introduce one piece at a time and the magnets which are in a specific place will only attract the right ones.

• P W 8 days ago

I'll bet he has an "aluminum foil kline bottle hat" for when aliens comes .

• P W 7 days ago

+Mathologer paradox's give me a headache . But harnessing plasma lights my mind .

• P W 7 days ago

+Mathologer 😉

• Mathologer  7 days ago

That one's still missing in my collection :)

• David Gonçalves 8 days ago

The Klein bottle isn't imperfect. The bottle's "missing" surface is just in the 4th dimension, ya?

• Gregory Lynch 8 days ago

This guys nuts

• Summer Crow 8 days ago

A complete waist of 16 min

• Summer Crow 8 days ago

Magnets

• Rheinmetall 8 days ago

So is it a secret how you did it, or you were bored to show the process? Or you were just trolling us?

• Roy Kennet 9 days ago

And....what is the point actually?

• Kris McMillen 10 days ago

Very cool video! And I feel I have finally.found someone who can answer this question for me;
Given this rule: (please correct me if I'm wrong)
1. Any positive Number (whole or fraction) when divided by 2 will result in another positive number (whole or fraction), correct? In other words 2÷2=1, 1÷2=.5 so on and so forth.
Isn't it possible to move 2 objects towards each other, cutting the distance between them in half (or dividing it by 2), without the 2 objects ever touching each other?
Would this be considered a paradox or is there a mathmatical explaination for this? I know what the theory of relativity states which would kind of prove my point to an extent, I think? Help me lol!!!

• David none 10 days ago

Its a special rubric cube. they are 8 magnets he puts in one at a time

• ! ! 10 days ago

What about the dreidel? Not all the toy's there!

• Grant Hutchins 12 days ago

Cool bong, bro.

• Jay Cie 12 days ago

@3:16 a mobius strip has only 1 edge.

• Xehemoth 12 days ago

Is that Rubik's cube really just magnets/metal that form that shape? that is the only way that is possible. physical manipulation inside of that Klein bottle is impossible from the outside.

• Pete Miller 12 days ago

Interesting video, thanks for posting. Cheers.

• Blind Man Bert 12 days ago

No Klein bottle collection is complete without this handy Klein Bottle Opener. :-)

• Lazy Pharaoh 12 days ago

If you came back as a mirror of yourself, what would the normal world appear to be like from your perspective?

• Jeremy Deutch 13 days ago

14:59 wait, all this putting things in other things seems strangely.... sexual?

• Sir Vival 13 days ago

7.34 if your man carries on walking the WHOLE way round on the same side,and not jumping sides halfway round,he will be absolutely the same.

• Mathologer  Day ago

The MS has zero thickness and the little man who is living inside the MS too.

• xyz xzy 13 days ago

Perelman.

• Yours Truely 13 days ago

And the dimension thing is still 3 dimensions because it has walls inside. And because he can hold it in his hands.

• Yours Truely 13 days ago

When he dies he probably wants his ashes to be put in a klein bottle too.

• Albert Bazinyan 13 days ago

My mans really did not just say thousands of dollars

• Jack Arnold 14 days ago

How many edge , corner, and center pieces would a 4D Rubik's cube contain?

• Marduk Slayer 14 days ago

Könnte echt ne coole Bong sein.

• El Tomas 14 days ago

There is a bit of relativity involved here. It's all about perspective.
In this video the rectangle is depicted as having a volume so it's 3d. But it works the same as if it were 2d.
Once we have a 3d rectangle, something could walk continuously on the surface of the 3d rectangle. Imagine if the rectangle were a balloon, we could inflate it enough to deform the rectangle into a sphere. So we know something could travel over the entire surface in any direction. This includes something walking the length of the rectangle down the surface of the end and countinue on to the underside of the rectangle.
The two sides of the rectangle having two different colors gives the illusion that there is more than one surface. But there is only one continuous surface no matter the shape. It is the skin of the object. The color of the continous skin means nothing.
Because we can make the rectangle into a tube and get the same results and since there is only one surface of rectangle, putting a twist in the rectangle prior to joining the ends does nothing.
This reminds me of the paradox of why we can travel either east or west forever with no issue, but of we traveled north to the north pole and we keep going on the same course, we would change to traveling south at the north pole.
But what is going on here is the masking of the two ends of a single hollow cylinder.
The rounded and flared out base of the bottle rests on a tangent plane which is the table top.
Where the bottle and the table top plane it sits on both touch forms a circle.
If we made a cut along that circle we will have us a deformed hollow cylinder.
Anything outside of the cut is one end of the hollow cylinder and anything inside the cut is the other end of the cylinder. If the glass was flexible, we could reach into the bottle and pull out the end that is inside the larger end.
After that, it really doesn't matter where we relocate the end we pulled out of the bottle.
But by choosing to place the end of the gooseneck cylinder seamlessly where the opposite end of the cylinder lays, it gives the illusion of a path to a fourth dimension.Once you can visualize the tube being cut at the base and separating the two ends, you can visualize the continuous tube being cut anywhere and the Klein bottle morphing back into a simple hollow cylinder.It is just a little difficult for anyone to imagine a tiny part of a rectangle like the very end surface being blown up into a very large cylindrical shape.The Klein bottle may not have any edges, but if it did, all you would need to eliminate the edges is to just round the edges off. If you do enough rounding, a square can become a circle.

• Anonymous 14 days ago

Yeah we are familiar with hugh mungus now in 2018 ..

• lee Roberts 14 days ago

Hey man I love your video and I also love how polite you are to Halt this stupid people on the internet. like you're really smart and you're really good at explaining stuff but don't worry about people in the comments who just say negative stuff there intentionally trying to make you mad or they're just like 12 year olds who are retarded and dont get huged enough you are doing great never read the comments

• Mathologer  Day ago

Been around for long enough to not be fazed by this :) The visitors from the shallow end of the pool don't get anything, declare this video for a waste of time and leave, never to be seen again. As far as I am concerned this is a very good thing. Keeps the comment sections of the rest of the videos civilised :)

• lee Roberts 14 days ago

Yeah but a mirror image of oneself is a reflection of only two Dimensions there is no such thing as a three-dimensional mirror

• Alien Nomad 14 days ago

For \$2k I'd exoect to get something with two sides.

• Dave Rogers 14 days ago

What a let down , l was waiting pationaly for him to show us how he got the cube into the bottle 😬😬😬😬😬

• Dr. Wier 15 days ago

15:00 "HI HI" LOL!!

• Celtcetra Pyde 15 days ago

I like this klein bottle showing. I am no math wiz either but I am fascinated by the illusion. It took me sometime to figure out the design before you demonstrated it. Thank you for sharing however I wouldn't buy one for that price, haha. its when I noticed the top and bottom excluding the (connection where they cross) is how it comes in. The top is open and the bottom is open. When I take the time to follow the tube it is like a long spout. When I pay more attention to how the spout connects is when the illusion kicks in. BUT it is very cool.. You got to look at it as if it is not "clear" in order for it to be clear... Love this paradox. however, I disagree with 6:42 of this video because it should't be parallel at all unless your talking about something else which is confusing for me as a viewer focusing on your topic of your Klein bottles...

• Joe Hall 15 days ago

Have never heard of a Klein bottle before I saw this. There will be ceramic Klein bottles on my shelf next year. Verrryyy interesting

• dead pool 15 days ago

well if the fourth dimensional beings where to pull the rubix cube out of the 3rd dimension it would shrink, then you could put it in and have them put in back into the 3rd dimension and make it normal size.(tho there is no way of connecting with a fourth dimensional being)
im wondering if this is true. i kinda conected idead with the action lab talking about what it would look like to see a 3d object pass through 4d or something like that.
this is his vid as to what i meant thexvid.com/video/_4ruHJFsb4g/video.html
hopefuply you can understand me now.

• Thomas Kurth 15 days ago

What is our "mirror" image? 😲

• Romarain Games 16 days ago

"Klein bottles, they don't really live in our world. They're like in a 4th dimension" : that's where you're mistaking. Our world IS a klein bottle.

• Hiruko Winter 16 days ago

See, maths like this I understand. I understand things about dimensional space, but yet I'm so, so bad at maths, especially with numbers. I don't understand. I've tried and tried, even tutoring but to me maths is like us trying to see colours that don't exist, or exist to our perceived colour spectrum.

• Wilf Wilson 16 days ago

Why does "it turn on the spot"?

• Camren Rounthwaite 16 days ago

But how did he get it in there?

• R N 17 days ago

Dude, you are awesome. Everyone wants you as a neighbour.

• 13r4nd0 17 days ago

Never forget who? That did what?

• gerald West 18 days ago

I have already forgotten.

• Ryan Smith 18 days ago

I like these videos. Very fascinating visualising mathematic applications.

• David Svarrer 18 days ago

A klein bottle does not have any particular purpose in terms of twisting the bottle neck in through the bottle again. There is no difference between the bottle neck just going out in the open, and then going through its own base. Basically the bottle neck is just connected to a pipe or tube, and where that tube is, does not matter. Seen from the bottle's perspective a tube is going through the bottom of the bottle, and as it does not get into the bottle, it has no effect.
This is a nulle innovation. Not a scam, just a useless twist of a bottle-neck. Please let us know the purpose of this twist of the bottleneck?

• George Montes 18 days ago

I loved his excitement on the klein bottle in the klein bottle 😂
Also I'm convinced he's able to pull 3 Dimensional objects into the 4th dimension because he got the 2x2x2 in that klein bottle

• Fire TBon 18 days ago

Interesting!

• this is so funny xD

• theonlykraglord 19 days ago

Kl approved

• Reverend Wolfshade 20 days ago

You tricky Rick!

• carolyn schrotenboer 20 days ago

thankyou sir. fascinating

• mike brannon 21 day ago

Interesting he never in 16 minutes shows what he says he will. I'm guessing theres a special 'cube' that's separated into its smallest pieces and each piece is magnetic and reassembles inside the bottle as each component is added. Now the challenge would be to get it out! good luck w/ that.

• Aria Ferranew 21 day ago

His explanation of a man on the moebius strip is incorrect, if you where to complete the strip, he would still face the same direction, and same exact shape. Try it yourself

• Aria Ferranew 16 days ago

+Mathologer but I'll give u a lil credit, in the other side of the moebius strip, yes he would Appear mirrored, but u also flip him upside down, which is still not the same, once he circles around more, and returns to his origenal starting point, he'll be upside down, flip him over, and he is still the same

• Aria Ferranew 16 days ago

I have the scene pulled up now, your treating it as if the moebius strip has the properties of a circle, that he disappears at the top, and reappears at the bottom like a wormhole or a portal from the top, if u where to follow the strip on the reverse side of the strip, u will find he stays the same, the entire time, if a moebius strip has 1 surface, it doesn't matter how u bend or twist that surface, something on that surface can never reverse or Mirror.

• Aria Ferranew 16 days ago

+Mathologer the way u drew it was as if the moebius strip was a 2d surface, even in a 2d universe,the man would continue on the other side of the moebius strip and return right side up.....and guess what, I did do that, in 3rd grade science, your also trying to argue the existence of a 3d model, in a 2d plain, as a moebius strip out of paper, is a 3d model.

• Mathologer  20 days ago

Yes, by all means try it for yourself. However, keep in mind that an ideal MS does not have any thickness and that the little man is part of the MS and is not somehow sliding around on its surface :)

• MF 21 day ago

Erster teil intelligent. der zweite etwas sinnlos...

• Dirty Daniel 21 day ago

“Bong collection”

• Vasilis Narlis 21 day ago

Gamers: If we could do this video in 4K...
Mathematicians: If we could do this video in 4D...

• THE MAN CHANN 21 day ago

It's a 4th dimension bong

• FlyBoy97 7682 21 day ago

New drinking game. Take a shot every time he says “a special way”

• Steve Daniel 21 day ago

Hang on a sec...are you Herr Puzzle?!

• a friedrich 21 day ago

Sure, you can put a Klein bottle in a Klein bottle, but can you put a Gross bottle in a Klein bottle? This is a rhetorical question. Please don't reply.

• Steampunker X 22 days ago

Doesn't the twist of the strip to the left @6:30 require 3 dimensions for it to work? I mean in 3 dimensions the "paradox" never happen. It relies on twisting a 2-dimensional objekt in 3 dimensions, no?

• tatonilo 22 days ago

I am sorry to say, but despite the good start, this video utility is to waste around 12' of the full 16' of my (and of other 4429 people @today) lifetime about your insane passion about Klein bottle unusefull gadgets . You know that there are Doctors able to help you about to exit your "Klein-bottles tunnel" ? (also if it is an endless one). (This latest statement comes from the 16-12= 4' above).

• Jose Martini 22 days ago

7:22
Now you know how left-handed people are made...

• Steve Barnes 22 days ago

Now my Brane (sic) hurts

• Adam Pitt 22 days ago +2

Memes are less wast of time then this. . . Witch I will watch now.

• Adam Pitt 22 days ago +1

Gotta love ClickBait, tells you he can do something, but never actually does it on camera. . . First and last Ep. I will watch.

• Adam Pitt 22 days ago +1

ClickBait, that's all it is, even if there is another video that "explains" it, I'm not going to wast my time to watch another commercial to give you more TheXvid money if I don't even agree with your explanation in the first place.

• Kai Jackson 22 days ago

I don’t even do maths anymore but I find this channel really interesting