Hypertwist: 2-sided Möbius strips and mirror universes

You’re watching a Mathologer video and that that probably means you’re eating Klein bottles and Mobius strips for breakfast and you know that these tasty mathematical surfaces have just one side. Except, and only real mathematical connoisseur seem to know this, they are Klein bottles and Mobius strips that have two sides. Let me explain. Quick revision: this strip of paper has two edges and two sides. To make it into a Mobius strip what I have to do is to bring the ends together such at the edges combine into one long edge. Every Mobius strip has just one edge but as you can see something else happens here. As I bring the ends together also the two sides combine into just one side, so this is a Mobius strip that has one side. Now there are actually a couple of different ways to bring the ends together by twisting them to make this strip into a Mobius strip. You can just do one twist and glue, that gives you a mobius strip, three twists or five twists or any odd number of clockwise or counterclockwise twist that will yield a Mobius strip. Now all these Mobius strips have just one side. If you do an even number of twists and then glue you get one of these surfaces. They all have two edges and two sides. Now these are not Mobius strips these are called topological cylinders or just cylinders. Now I claim there are ways to bring together the ends into Möbius strips that are two-sided. Hard to imagine how is that miracle possible? Well, it turns out that the number of sides of a Mobius strip or actually of any 2d surface depends on which 3d space which 3d universe it is contained in and how exactly it is contained in this 3d universe. Now most people think that 3d just means xyz space what you deal with in school or at university. But there are actually infinitely many 3d universes, mathematical 3d universes and we actually don’t know which one of these mathematical universs describes the universe that we live in. Now quite a few of you will actually have heard that our universe may be a pacman universe which means that there may be a direction, special direction. If i head off in this direction and just keep going straight I’ll get back to where I started from. So let’s just assume we are inside a mathematical universe that has this property and let me introduce you to my math cat maskot the QED cat. Well, actually, there’s a bit of dispute at home whether it is a cat or chihuahua but no matter let’s just launch it in this special direction on its space surfboard and see what happens. So as the cat travels along the surf board actually generates a strip. Ok now keep going, keep going. Eventually it gets back to where it started from, there. And now it wants to turn the strip into Mobius strip and you can see that to create this one long edge what we have to do is have to kind of flip upside down and keep moving forward. Ok, so we’re creating a twist like this and you can see the Mobius strip that we’ve actually created here is just one of those one-sided Mobius strips, so nothing new here. Now, in a more fancy universe something else can happen, so let’s just do this again. So QED cat heads off again, we leave one of those ghost images behind. It gets back to its starting position but now something else has happened, it’s actually turned into its mirror image. That’s unusual and now you see to create this one long edge, to create the Mobius strip the cat has to just keep on going, it doesn’t have have to do any of those upside-down acrobatics. So just like this and we’ve created a Mobius strip and obviously that Mobius strip is two-sided. So if you put a anti-cat on the other side and have QED and the anti-cat run around on those two sides they’ll never meet. Once you’ve got one of those strange mirror reversing trips all sorts of other nice things start happening so for example the cat gets back to the starting position, it’s mirror reversed and it wants to eat some of its cat food but actually that’s no longer possible because the mirror reversing happens at a molecular level and so the food and whatever processes the food in the stomach won’t match anymore, won’t happen. So to unscramble itself the cat actually has to either backtrack or just do a second round and then it can eat. Also once you have a two-sided Mobius strip like this you can extend it into something solid and this solid corridor is actually a real 3d counterpart of a 2d Mobius strip. A lot of you may have wondered whether something like this exists, well there you go. Finally what I want to show you is how you can use a mirror reversing path like this to create a one-sided cylinder. Now, usually, cylinders are two-sided, right? So now let’s just look at this situation again, we can now turn this strip here into a cylinder by just doing a twist, a twist like this creates two edges, we’re dealing with a cylinder but as you can see this is a one-sided surface. So at this point what you really want me to show you is one of those mirror versing path in our real universe or at least in a mathematical universe that i can hold in front of you. But that’s actually very hard to do because no matter what you doing in xyz-space round-trip wise you’ll never mirror reverse yourself. That also means that we cannot have a copy of one of those reversing universes inside xyz- space, makes it hard to describe. But what I can do is I can show you the analog of one of those one-sided cylinders, a 2d analog and for that I need a flat cat. Now what’s a counterpart of a cylinder in a 2d world, it’s just a circle. So what I want to show you is one-sided circle. Now, usually, circles are two-sided right. Two-sided circle with respect to this 2d world that the cat is living in. Now instead of off using this off-the-shelf 2d universe I use a Mobius strip universe, that’s a 2d universe. The cat’s living inside it, the circle is part of this universe and I’m going to chase to cat around it but what’s really important here is actually too emphasize that a real mathematical surface has zero thickness just like the xy-plane inside xyz-space has zero thickness. So that Mobius strip has zero thickness, the cat sliding around in it has zero thickness. Let’s just see what happens when it runs around the circle. Ok so it’s completed its roundtrip and as you can see with respect to this 2d universe its living in it’s actually mirror-reversed itself and it seems to be locally on the other side of the circle but, as you can see, when we do a second trip around it actually gets back to the beginning and what this means is that this circle here has just one side. On the other hand, if I take away the Mobius strip and surround this circle by this ring here, then the circle is actually a two-sided circle. So what that also tells you is that without the 2d context it actually doesn’t make any sense to ask how many sides this circle has. Just kind of floating with in 3d space it doesn’t make any sense to ask how many sizes this thing has and similarly if you’ve got a surface you need a 3d context to be able to ask and to answer how many sides one of these surfaces has. Otherwise it just doesn’t work. For example, we could put something like this in four-dimensional space and just have it floating there, it doesn’t make any sense to ask how many sides one of those things has. Now I also promised you some 2-sided Klein bottles. How do we get those? Have a look at this. So QED this flat so it can’t really see a Mobius strip but it wants to play with it anyway, so it can do this a la pac-man. It’s not ideal but it’s good enough to visualize what’s going on. Ok, so what you do is you just kind of draw a flat rectangle and QED can run around in there and then you just indicate how the ends are going to be glued together with arrows like this so. The arrows here basically tell you that these two points get glued together and these two points get glued together and so on and now a Klein bottle is actually just a Mobius strip whose edge has been route to itself in a certain way and that certain way I can actually show you very easily also with arrows, goes like that. So we have to do is, we have to glue these two points together, we have to glue these two points together, and so on and that will give you a Klein bottle and well since we have 3d beings I can actually show you this construction in space. So here I’ve got a Mobius strip. Now I’m just going to bring corresponding points of the edge together, like this, and there you’ve got your Klein bottle. Now obviously once you’ve found one of those mirror reversing paths and a two-sided Mobius strip it’s pretty easy to imagine that we might be able to extend this strip into a two-sided Klein bottle and this is exactly what happens. All right, now we’ve got two pictures of a Klein bottle here and just like QED can use the square to describe a Klein bottle we can use a solid cube to describe a solid counterpart of a Klein bottle, so basically a solid Klein bottle and this is also done by these fancy arrows. What the fancy arrows show you is how opposite faces of the solid cube are supposed to be glued together. For example, these two points get glued together, these two points, those two points, and so on, should be pretty obvious and actually this solid Klein bottle there is one of those mirror universes and if you have a really close look you can see that this here is a two-sided Klein bottle within this 3d mirror universe, very very fancy, very, very pretty. It’s an absolutely beautiful Klein bottle much nicer than the one that I showed you before. The one I showed you before has this strange sort of self-intersection which is really annoying. This one doesn’t have any of this so so much much nicer in this respect. Ok, now I learned about all the stuff for the first time from this book here, The Shape of Space by Jeffrey Weeks. This is an amazing accessible introduction to two- and three-dimensional universes manifolds. I really recommend it to everybody here. Jeff’s also created some amazing pieces of software, totally free that you can download from the website I’ll link in from the description and they allow you to you play chess on Klein bottles on tori, play pool, all kinds of other things but he also has pieces of software that allow you to fly around in strange 3d universes. So, for example, here’s a view of a very small version of this solid Klein bottle universe which just basically has space for one Earth and as you kind of look around because of the way it kind of connects up to itself you can actually see yourself, see Earth over and over, not only Earth but also the mirror image of Earth and you know I leave it to you to kinda figure out how exactly this works how how exactly the pattern of Earths and mirror Earths comes about. Now there’s a lot more to be said about all this, e.g., four dimensional stuff. I may say it’s a little bit about this in the description. Also I’ll definitely come back to these strange 3d universes, make another video about that but for the moment I just like to say thank you very much for all your support throughout 2016 and Happy New Year to all of you and I’ll see you again soon.

100 thoughts on “Hypertwist: 2-sided Möbius strips and mirror universes

  1. Just back from a hiking trip to beautiful New Zealand (just in case you’ve been wondering about the extended “radio silence” on my part or why I may look a little bit sunburnt in this video 🙂

    For a really nice introduction to finite 3d spaces check out this video https://youtu.be/-gLNlC_hQ3M

    Oh, just in case you are wondering why my cat mascot is called QED: in maths QED stands for "quod erat demonstrandum" which is something people used to write at the end of proofs. It's Latin for "What had to be demonstrated/proved". In physics QED stands for quantum electro dynamics which has nothing to do with our cat. Also, the QED cat mascot was originally invented by my colleague and friend Marty Ross. The flat version on the cereal box has been our (the Maths Masters) mascot for decades (check out www.qedcat.com).

    As usual, if you contribute a translation into a language other than English, could you please let me know by sending an e-mail to [email protected] YouTube is not very good at notifying me when new subtitles are waiting for me to approve.

  2. Just wanted to let you know that this channel is a large part of the reason I'm considering a masters. I'm about the finish my bachelors but I feel like I've barely scratched the surface of mathematics. Your videos have to exposed me to many fascinating areas of mathematics that I didn't even know existed! So just wanted to thank you for that.

  3. Coming from a physics background, I've used so much mathematics that I now think I want to get into mathematics instead. This sort of abstract stuff always interested me more than experiments, to be honest, and got into physics with hopes of studying about these things in the first place.

  4. An electron when rotated 360 is not the same … one must rotate it 720! See:
    (no I don't understand this …)

  5. When I heard 2 sided möbius I thought of a long rectangular prism with a twist and connected, turning 4 sides into 2, and I was thinking if there is a way to turn 2 Möbius strips into one of these möbius prisms.

  6. great video extending what we are already familiar with regarding Mobius strip's and Klein bottles and adding another dimension to them.

  7. Every time I see a new video of yours it makes my day! You should collaborate with 3Brown1Blue. You two are my favorite YouTube channels 🙂

  8. Nice video as usual.

    Here is an outrageously under rated youtube channel that explains the same thing:


    Lot of people are waiting for part 3… if someone can help the author that would be great

    Thanks again, looking forward for more funny maths in 2017

  9. yeah but a mobius stip isnt really 2d. it may have similar properties as a plane but it is not 2d. the only way the mobius strip exists is if it was a 3d object. us living in a world that doesnt "mirror flip" could probably mean that we arent just some beings only experiencing 3 out of 4 dimentions, because the only way that the mirror flip works is if we lived in a world that did. this makes the 2 sided mobius strip and klien bottle purely hypothetical.

  10. Assuming the situation that an event occured, and that the strip has a person on it.
    That the person moving on the strip is able to return to the position before, say an event occured on the script. How can this happen without interfering with the past events?

    This question is based on an experience i have, whereby i went to bed, and the following day woke up and i was 4 years back in the past, i could remember what happened the day before. I have relived my life from 5 january 2013 through the same events and have constant "already seen" memories.

    I thought my question could give me some insight to the nature of what happened.

  11. That doesn't make sense….you pretty much said "ok, we're going to have QED cat suddenly become the mirror image, and bam, 2 sides to the mobius strip." You didn't explain how the cat became the mirror, and if there is an exact point at which he does become the quantum mirror, then that point ends the strip like a scissor cut….it no longer follows the rules of space but rather stretches into a higher dimension, in this instance "mirror space." You've thread the surface through this upper dimension, so it is no longer a mobius strip, but rather coinciding with another coefficient to tell the coordinate whether it is the mirror or normal.

  12. Thank you. Never before have I thought about what would happen if you put a n-dimensional object into a n+2-dimensional space, and how you actually need some kind of a n+1-dimensional surface to analyze the object's sides.

  13. Hey! A circle has one edge and two sides! I can imagine twisting one.

    No, I don't get typology.

    And Happy New Year!

  14. I'm so glad you're making these videos. Math itself is a bit complicated for me and I never tried to learn it seriously, but I find these videos quite entertaining. It might take huge amount of time to make one episode with such high quality.

  15. I really dont like the 3D concept. It some how confuses my perception. I would rather think space as a 2D moving in open space with no discriminate dimensions.

  16. Well, I'm a bit concerned about that lovely QED-cat that can't eat anymore because of topology. I mean Schrodingers cat was kinda dead, but at least it could eat. (And don't eat at the same time…)

    But here's my question: If you twist your strip 3 times instead of once, and make a Klein Bottle of it, would it still have only one 'hole'? I mean since 3 turns are homeomorphic to 1, there should result a homeomorphic Klein Bottle. Or not? Really confusing stuff.

  17. Basically you can't see 3D Möbius strip properly if you can't see in 4 dimensions much like the normal Möbius strip which can't be visualized in 2d but is a 2 dimensional object. Am i right?

  18. oh boy…there's something im starting to figure out…there's still a long run, but I believe one day we all will get to know about reality…thanks for all your wonderful videos…happy new year, btw…:)

  19. I've made a shape which I call a möbius. However, I'm not sure whether or not is is. Could you please c if it is? https://www.shapesofstonemasonry.com/#/mobius/

  20. Unless I'm misunderstanding, edges are understood to continue at the edge of the universe and sides are not. But why?

  21. Great video! Would you ever consider making an episode on the subject of projective space? It's an area I find fascinating that I think could benefit from the Mathologer treatment.

  22. The movie linked in the description, The Shape of Space, is by the Geometry Center, the same group behind Outside In (aka "how to theoretically turn a sphere inside out") and Not Knot. They are good at animating this stuff

  23. I have a question for you. Given that if you had a long flexible triangular figure, such as a prism shaped figure such that their edges marked as A-A, B-B, C-C, when twisted would be matched as A-B, B-A, and C-C, would make an object with two sides, only one of which would be a Möbius strip within the deformed figure. Are there any figures of this type with any number of sides where combination of matching edges will l produce more than one Möbius strip within the figure?

  24. Have you ever gone into this pac-man universe scenario before? I've never seen a good explanation of this so far, would be delighted to see you going through it.

  25. Hi, i really like the channel and appreciate the videos:) this might be off topic, but im interested in the theoretical basics of imaginary/complex numbers could you guys recommend some litterature of this topic to an applied mathematician, since there seem to be endless books written on this topic.

  26. If the Mobius strip was made of magnetic material and small enough, the magnetic field would be a sphere which would be like your Klein bottle.. It would have strange properties, in mathematical terms.

  27. If the Mobius strip was made of magnetic material and small enough, the magnetic field would be a sphere which would be like your Klein bottle.. It would have strange properties, in mathematical terms.


    i have too calculate the sum from -inf to inf of 1/k^4, with the equation of paserval, but i can't find a function, with fourier coifficients = 1/k^2

  29. 5 × 0 = 0
    as we say.
    0= 0÷5
    that is again zero.
    5×0= 0
    that is infinity.
    then the statement
    This is same for all zero applications like
    & of course 0×0

  30. So rather than an actual euclidean two sided mobius strip, it's really an artifact of a hyper klein bottle surface universe wherein traveling in a direction could result in a mirror image of yourself back at the original position due to the equivalent of zero thickness for higher dimensions right? In other words, this can't exist in a euclidean space, but is a result of a specific "trail" traveling a round trip in a mirror universe. Is there any way this kind of thing can be embedded in a euclidean 3 space? I'm trying to think of this from a 4D (euclidean) person's perspective. A 4D person could imagine a 4D hyper klein bottle surface-like universe (similar to how we imagine a 2D mobius strip universe or a klein bottle's surface 2D universe) and then imagine a 3D person (or perhaps a 3D QED cat) leaving a 2D rectangular trail. Then, just imagine the trail. I think this is like thinking of a klein bottle surface, drawing a line along it treating it as a 2 space, and then trying to embed that in a euclidean 2 space. This would just appear to be a self intersecting line with no mirror properties. Does the act of imagining an n-1 D klein-surface space's trail and embedding it into a euclidean n-1 space delete the mirroring property? I am uncertain because I may have made some error in imagining or in dimensional analogy. Correct any of my mistakes. Thanks

  31. The last time i did psychedelic mushrooms, I was in that mirror universe that is shown on the end. Though instead of earth/-s there were a bright, colorful – like out of this world color, indescribable and impossible geometric pattern. I'm of course not talking my physical body but it came to me in my mind. I'm not talking something as defuse as imagination, that which you experience when you try to picture something in your mind. It was as vivid as this world, if not more. It was really as alien to me as can be, I have no reference point to anything except this layout mirror pattern which these objects were in and the aspect of bright and dark.

  32. +Mathologer read this in an angry tone but know these are happy thoughts: I've realized something. Draw two parallel lines along a mobius strip. You have made a two sided mobius strip in two dimensions on that mobius strip. See the flaw? The two sided object is only a mobius strip because you're INSIDE one! This entire premis seems ridiculous to me. (normal tone:) Could you help me understand?

  33. In my experience when mathematicians say that something is 1-sided, they are not talking about the number of sides, but about orientability.

  34. Great video! reminds me of a book called Shape of Inner Space (authored by Yau & Nadis), maybe Mathologer would like it. It's a geometrical approach to what space is

  35. I have a question, please. You say that it doesn't make sense to ask how many sides something has without first embedding it in an ambient space. I guess I am confused because I have learned about orientations as local (relative) cohomology classes, which is a setting that is quite independent of embedding. Could you (or someone else) please clarify the relationship between these two concepts?

  36. I'm confused here.

    Is there actually a twist in the "special" 3d universe? or is the very act of traversing to the same point the "twist"?

  37. Well it has inclinated non plane surfaces whatever that means. In the extreme case an inclinated sphere (which is a self-similar object in every point)! ((suppose there would be a solution in the non-self-similar case, but Ockhams racer would tell you that there is no need for such a thing if you only exist on one side which then isn't "a side" rather just a universe(to live in) which we already knew. Sorry though "an inclinated sphere" hmmph

  38. Our universe is obviously shaped like a regular infinitahedron (I know that it's impossible XD).

  39. Those images of Möbius Flakes fill me with inexplicable sadness. And I do mean inexplicable; I have no idea why this is.

  40. Interesting ! Among the existing Physical Theories, the closest to what you describe is the Twin Universes Model of French Cosmologist J. P. Petit.
    You can find published papers at :

  41. If a mirrored person and a non-mirrored person copulated, they wouldn't need contraception as pathogen agents couldn't infect the other personbecause of their DNA twisted in the wrong direction. For the same reason, the sperm and the ovum wouldn't be able to produce an egg …

  42. If you come back reversed in a "mirroring universe", would you still be able to breathe? Also, are all digestion processes really chiral like Mathologer claimed? Could that universe really exist? It surely could in a virtual world, couldn't it? Fascinating stuff.

  43. You might not have no idea of the implication of the 2 d flat and it needs to have a reference 3d to make it. The actual context of the form of a 2d flat mirroring itself in a half-cycle of the Mobius strip is, in essence, the mirror of God in us and God seeing himself as he is the Creator.
    This 1/2 cycle is a complete cycle, the let us say the two half-cycle cycle (two turns) is going into another universe(parallel) as it is another location from the first location. It is as the other 2d position but parallel universe.
    The existence of the extra ring to have the single ring be recognized is the effect of the multi-universes being able to exist and support each other as the other universe makes the other be in the context of location. Being the Mobius an effect of the one God mirroring himself through the universes all at once. Quite a paradox is it not?

    Thanks, for the explanation and you made a bit clearer.

  44. I fail to see how the cat "reversed" itself. The plane has zero thickness, but zero thickness doesn't mean nonexistent. Globally, it has only one side, but locally it has two–and each "side" is halfway around that particular Möbius universe. The cat that's "mirrored" isn't actually in the same spot or right next to the original, it's as far distant is it can be on the loop.

  45. I have a question to ask you.
    You are I am sure familiar with the game Portal.

    Imagine the following 2D test chamber.
    🔲⬛⬛⬛🔲 Walkable area is ⬛
    🔳⬛🔳⬛🔳 No portal surface 🔲
    🔲⬛⬛⬛🔲 Portal surface 🔳

    Imagine this is our guy 🔰 who has a chiral counterpart and can be distinctly oriented.

    Now obviously when you play Portal, you always pop out as your same self, not your mirror image. Sometimes the place appears upside down but that's due to the "top" of the portal.

    So from left to right I'm going to call the surfaces A, B, C, and D. Now obviously if you walk through a portal on A, a portal placed on B or D will produce the same results.

    Obviously you're going to have to extend into 3D space to actually connect the points. Whether you use a strip or bend the universe itself topologically would be the same although the actual universe would probably bend rather than using a strip. For simplicity a strip is better.

    Now if a portal is placed on A and another on D, no Mobius strip is needed to connect these points, provided the portals are oriented the same way.

    If you walk through A and step out through C, a Mobius strip is needed.

    If one portal is flipped upside down the opposite happens.

    Also the Chevron can sometimes appear on the opposite side of the test chamber universe.

    Now here's where my mind rambles. If we let the strip turn into a tube after the chevron enters the "worrmhole" and then back into a strip as it touches the other portal, such that the Chevron can move in 3D space bounded by the surface of the tube, won't the chevron end up in a different orientation somehow?

    I could probably think this out more and figure it out if I draw it out but I have to sleep for work.

    Further more how does this extend to the 3D portal universe, the standard game? What sort of Klein bottles, tube, and twists would occur?

    What would occur if the Aperture universe was not topologically Cartesian?

    There's just so much I'm wondering about its math.

    Oh also are there such things as Mobius Strip analogues in noninteger dimensions? Fractal Mobius strips?

  46. I just understand Avengers end game tell us that no paradox the universe bring the mirror small changed to the large changed mirror again in the large change create another time line.

  47. Interesting video but some points are a bit hand-wavey, the cat is reversed but what is actually happened along its trip, we see the outcome but not the reason, is it in a 3D "Möbius space" or what's happening?

  48. I've just come across this video. Its very interesting, but I have one question (well, technically two). In order to create these mirror universes, you clearly have to take torsion of the manifold into account (hence the twist), so what does the metric look like for one of these spaces? And what is the curvature of your example?

  49. Fairly well spoken but keep working on getting rid of that accent….to the English ear it is quite thick yet even though you do a very good job speaking. Otherwise, good video and thank you and I gave you a thumbs up too!

  50. 1John:3:2: "Beloved, now are we the sons of God, and it doth not yet appear what we shall be: but we know that, when he shall appear, we shall be like him; for we shall see him as he is."
    If we born again Christians will see God "as he is"….then we MUST be in the same dimensionality (or outside of the dimensional) as God is. We will have the same characteristics of motion, sight, hearing, touch, everything we sense would be enhanced to God's senses.
    That does not make us gods but compared to our present abilities, the improvement would be impossibly enhanced to a degree we cannot comprehend.

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