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...:)
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
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?
That's pretty much it. Having said that it's important to realise that it's not a problem to write down equations that mathematically pin down these exotic 3d universes within higher-dimensional spaces. Also just because higher-dimensional spaces may not be a physical reality does not mean that these exotic spaces cannot describe the universe we live in :)
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 ruclips.net/video/-gLNlC_hQ3M/видео.html 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 burkard.polster@monash.edu. RUclips is not very good at notifying me when new subtitles are waiting for me to approve.
+Lars Höög Just having a bit of fun there. Pacman already featured in the video on the Klein bottle Rubik's cube. Check it out if you are interested (no ghosts though :)
+Mathologer Why does that cat love Quantum-Electrodynamics, so much? ;) (I understand that it's an acronym for a Latin phrase, in this case) Yes, I am a nerd.
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.
Feel free to ask questions :) A lot of very smart people roaming this comment section who are happy to help (just in case I don't get around to it first)
The secret to understanding this stuff is to not understand it. Don't be caught up in trying to get it right away. Just enjoy the non-understanding by exposing yourself to things like this and keep exposing yourself to it. Then something magical happens. One day it just clicks and you do understand it.
Less deaths of famous people, but tax cuts for 1%. But in 2018 we're looking forward to another catastrophe. Either war with NK, impeachment of the POTUS and political shitstorm afterward, or just plain Depression, after the cuts, which didn't give more jobs as promised. Which whole wide world will feel.
oooh, so that universe itself has this twist in it so by adding a twist yourself, you actually make a strip with one twist (like a mobius band) but including the twist of that universe, there are two twists in total so it's a normal band. So basically this is a universe that inverses mobius strips and cilinders. Because of it inherrit twist, it switches the odds and evens.
oh, great, then I understand it somewhat! I know it's not a literal twist, but something resembling it somehow. If you were to live on a 'globe' that has this poperty and you sailed west until you got back to where you started, would you see the entire world mirrored and would they see you as mirrored at the same time? Odd question, and possibly too much scifi, but I mean like: everything that appears to you would be what you know, but mirrored while you appear to the people who know you as mirrored, right? This is making me question reality.
Happy New Years Mr. Mathologer! Ever since you introduced me to your channel (during that quick post-lecture lunch) my appreciation for maths has increased by a heck of a lot. You've been able to explain things other RUclips channels can't, and your dedication to the craft is really inspiring. Here's to 150,000 subs!!
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?
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?
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?
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.
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.
I've read Jeff's book twice, I guess that makes me a math connoisseur. Another cool fact is that the three dimensional analogue of the projective plane is actually orientable. I love his geometry games, and I recommend taking a look at HyperRogue, which isn't Jeff's, but is an awesome tool for demonstrating the nature of the hyperbolic plane, and you can choose the projection you are playing in. It even has spherical and elliptic modes, but not in the free version.
Actually, somebody just recently recommended HyperRogue to me and I've been playing it quite a bit lately. Very cool game. I am actually tossing up whether I should feature it in a video on the hyperbolic plane :) Are you familiar with MagicTile? It's got hyperbolic twisty puzzles among many many aother things roice3.org/magictile/ :)
@@Mathologer For what it's worth, it would be great to have HyperRogue featured :) Popular mathematicians seem to like HyperRogue, there have been some blogposts/videos about the very early versions, but recently, they are like "I've been meaning to do this but I have no time" or "I think if I tried this game, I would love it so much that I would not do any work anymore". And we still do a lot of work on it, we add new ways of showing the nature of hyperbolic plane, new projections, new tilings, new hyperbolic gameplay, etc. :) Spherical and elliptic modes are free for a long time, though the "racing mode" added a week ago is currently not.
@@ZenoRogue HyperRogue is great, I've spent many, many hours playing it (the MacOS version). If I ever get around to talking about the hyperbolic plane again it's definitely something I'd like to mention.
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.
There is actually no hole. There might be more complicated self-intersections. However, these are really irrelevant as far the the Klein bottle is concerned. Of course, in general, as soon as you've got self-intersections, strictly speaking you are only dealing with an immersion of a Klein bottle and not the real thing. If you don't want self-intersections you have to either unscramble things in higher-dimensional space or go for the sort of Klein bottle in an exotic 3d space as I show at the end of the video :)
I love it, when mathematicians create 'exotic" worlds, that only exist in our brains (or maybe they do for real?!), and play with their thoughts for the love of math itself. Sp once again, Mathologer, you baldly went for us where no man has gone before. Thanks for the answer! [And sorry for that small joke from one 'chrome dome' to another :)]
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.
I'm here from watching a Star Talk clip about mobius strips and damn this is some mindblowing stuff! Glad I decided to look into it more and that I came across this video. Makes me want to go for another degree in math/physics after I'm done with my current one (international relations)! Thanks for this both clear and totally fascinating video
5 × 0 = 0 as we say. 0= 0÷5 that is again zero. But, 5×0= 0 then, 5=0÷0 that is infinity. then the statement " ANYTHING MULTIPLIED BY 0 IS 0. This is same for all zero applications like 0+0 0-0 0÷0 & of course 0×0
Maybe also watch this video by Jeff about the most basic of these exotic 3d universes ruclips.net/video/-gLNlC_hQ3M/видео.html Really very, very good :)
Cool video! I realized that two-sided Mobius strips were possible when I was an undergraduate math major at the University of Texas at Austin (during the late 1990s). In fact I used a thought-experiment very similar to that described at the beginning of this video. For a while I thought I had made an original discovery in this, but eventually I chanced upon a topology book in the Physics-Math-Astronomy Library that actually discussed all these ideas. Unfortunately, I don't remember the author or the title of the book, but it was not the book referenced in this video.
What would happen to a 2d piece of paper if both sets of opposite edges were glued together in a mirrored fashion? ___>___ |.......| ^......d |.......| ___
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 : www.jp-petit.org/science/f300/astrophysics_and_space_science_1995.pdf
I have a Question, Hope somebody answers this one day.... Imagine a Regular Tube... connect the Two ends we get a Torus... Now a Torus have 2 sides, the outside one and the inside one... Remember this Now take Another Identical Tube, and conne.. but wait, how about we twist the Tube in some higher dimension so that the inner surface is the outer one, and outer surface is inner, and then connect the ends? What shape is this, Torus with 1 side? Does it even have any Volume? That higher dimensional twist, is it even possible? I mean, it's not the same as Mirroring the tube. It's something different.
Well, I am in negotiations with a 4D company to make these Mobius flakes a reality. I'll let you know when they hit the shelves in your local supermarket :)
Just a tidbit for those who don't know: QED stands for "quod erat demonstratum", which is latin for "which is what had to be proved." It's usually put at the end of proofs.
Why not the bigger 628 gram packet? ;-) When I see things like all the Earths my mind gets drawn to what it would be like if we could make an E8 where we could stand on a point and view all the connections jumping from one to another. Once again, thank you. The tee is great.
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.
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.
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
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!
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.
Reye's configuration. Protectively equivalent to the figure consisting of the twelve edges of a cube and it's four cube diagonals, in which three points are at infinity.
Lying to oneself is one of the worst things that you could do to yourself it doesn’t make you smarter to deny the truth same as 1 + 1= 2 is the same with objective truth it is the correct answer to the question of reality. What I’m saying is that fact is based on reality and vise verse it simply is so and cannot truely be disputed it is what it is 3 is 3 , 4 is 4 end of discussion. So never lie to yourself let your conscience be your guide listen to your gut adhere to reality.
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
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.
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.
If the cat was to hover above the MS it would be (locally) "on the other side" after one roundtrip. However, the cat is not hovering, it is part of the zero-thickness MS, and as such has zero thickness itself. And so after one roundtrip it appears reversed. Maybe it helps to not worry about the movement and just focus on the cat at the beginning and at the end of the roundtrip.
Physics speculation: I once heard the the universe is expanding like the surface of a balloon (everything is getting further away from one another with no obvious center). Then this made me think that maybe that's more true then we realize. Like we are living on the surface of a 4 dimensional sphere expanding.
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?
In terms of surfaces as long as what you are dealing with is part of R^3 2-sided is the same as orientable. In general, orientability is what is referred to as an intrinsic property of a surface or manifold whereas sidedness is an example of an extrinsic property that only makes sense in the context of a surrounding space. I'd say have a look at the book I recommend in the movie. Really very easy to read, especially for someone with your background :)
Out of curiosity, is there any mathematics related to infinite dimensional space? Could anything even happen in infinite dimensional space or would the lack of special dichotomy cancel everything out?
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.
An electron when rotated 360 is not the same ... one must rotate it 720! See: blog.sigfpe.com/2007/04/curious-rotational-memory-of-electron.html (no I don't understand this ...)
You could definitely realise these exotic 3D universes inside higher-dimensional spaces. However, for our universe to be such a mirror universe a 4d surrounding space is not necessary :)
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.
It would be surprising if you did not have problems figuring this out (explicitly). Basically, in this respect we are all facing exactly the same problem as the flat cat trying to figure out how there can possibly be a 2D mirror-reversing path. We as 3d creatures can see easily what the flat cat has problems figuring out. Now, of course the mathematicians of the cat's 2d world also have no problems writing down equations that describe the mirror-reversing Mobius strip universe space in (for them) abstract xyz-space. Similarly, if there were higher-dimensional creatures they could easily construct a solid 3D Mobius strip corridor winding through higher-dimensional space and we 3d mathematicians can at least write down the equations for such a space. Really the best way to develop some intuition for this sort of stuff is by really understanding what happens one dimension down and then extrapolating from that.
That's the whole twist: the universe itself has a twist so if you have a strip and twist it once, you actually make a cilinder. You have a twist in a twist. However, unlike an actual cilinder, it's still non-orientable.
The paper Mobius strips are really just (slightly misleading) real-life approximations for the real mathematical deal. 2D implies 0 thickness. Anything that has non-zero thickness is not a surface anymore :)
All over the place, I am just doing a comprehensive search on math t-shirts every once in a while and get everything good in sight. And about half of them I make myself :)
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?
Not sure I understand how exactly you want to glue things. If you twist your prism and then glue wouldn't you automatically get AB, BC, CA? There is a comment by a sculptor further down in the comments with a linked in picture that shows a shape like this (cross-section is a square not a triangle though :)
not sure how to label things properly. I did this with a paper tube formed into a triangle and it gave me what appeared to be one Möbius side and one regular side.
You mean the solid Klein bottle space at the end? No, that one is different from a toroidal space. To get a toroidal space the arrows on the front and the back have to also point in the same direction :)
+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?
Not really sure what you are worried about. Yes, when you draw your two parallel lines you are creating a Mobius strip inside another Mobius strip. However in that context it does not make sense to speak of how many sides this new MS has--you need a surrounding 3d space for that :)
well, it seems preposterous on a mobius strip, do the same on a klein bottle and you've just outlined one of its component mobius strips... so why is it magically valid when you step one level higher? the mirroring effect is the same! it happens because space bends back over itself, and its really a one-sided mobius band, your cat just got mirrored in 3 ways, not 1. reflect something vertically and front-back, you get a cat looking backwards from the bottom of the band, the third mirroring is the one you mentioned in the video, you only see the cat on top of the band because it rotated 180 degrees. it is a one sided mobius strip.
Reminds me of that one joke that goes, "Which side of a cat is the softest? The outside."
:)
The first time i misread "cat" as "car"
@@t0k4m4k7 That would be the opposite, then. Maybe, the car could be inside-out. 😅
:)
I like the mathcat reference for reasons that may seem obvious. Love your channel!
:)
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...:)
Definitely also check out this video by Jeff ruclips.net/video/-gLNlC_hQ3M/видео.html :)
Thank you sir:)
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
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?
That's pretty much it. Having said that it's important to realise that it's not a problem to write down equations that mathematically pin down these exotic 3d universes within higher-dimensional spaces. Also just because higher-dimensional spaces may not be a physical reality does not mean that these exotic spaces cannot describe the universe we live in :)
So what would happen if one set of arrows had black and white reversed? :)
You get a different sort of 3d space :) Maybe also check out this introductory video by Jeff ruclips.net/video/-gLNlC_hQ3M/видео.html
Mathologer Interesting! Thanks! :)
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 ruclips.net/video/-gLNlC_hQ3M/видео.html
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 burkard.polster@monash.edu. RUclips is not very good at notifying me when new subtitles are waiting for me to approve.
Mathologer kudos for including Pac-man and ghosts in the video and happy 2017!
+Lars Höög Just having a bit of fun there. Pacman already featured in the video on the Klein bottle Rubik's cube. Check it out if you are interested (no ghosts though :)
Mathologer, I've for sure seen it and, by the way, where do you get all those t-shirts from.
+Mathologer Why does that cat love Quantum-Electrodynamics, so much? ;) (I understand that it's an acronym for a Latin phrase, in this case) Yes, I am a nerd.
SpudHead I can't tell if you're mocking him, or anyone who would level that insult, thereby lowering the quality of our discourse.
Mouse flavoured cat food. How have they not thought of this?
Because it exists naturally.
So does veal, lamb, salmon and chicken.
> Mouse flavoured cat food. How have they not thought of this?
cats do not buy catfood! you can not market the food to cats, you market it TO HUMANS!
Cats don't actually specifically like the taste of mice I don't think. They like playing with their food which is why they catch mice.
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.
Admit it, you're secretly making all those awesome videos just to show off your nerdy t-shirts :D
Happy New Year!
Just ordered about 20 more to keep me going this this respect. The math t-shirt collection now comprises well over 100 :)
and this one says...? "But J can on your side"? I don't get it.
I think it says "But I _am_ on your side!" (which of course is true!)
I love your videos, even if I might not completely understand them.
(I will probably have to watch this again).
Feel free to ask questions :) A lot of very smart people roaming this comment section who are happy to help (just in case I don't get around to it first)
Mathologer Thanks very much for the friendly reply!
The secret to understanding this stuff is to not understand it. Don't be caught up in trying to get it right away. Just enjoy the non-understanding by exposing yourself to things like this and keep exposing yourself to it.
Then something magical happens. One day it just clicks and you do understand it.
***** Thanks, I just hope it clicks sooner rather than later ;)
Let's hope 2017 is not a mirror image of 2016
Agreed.
Antonio Lewis wouldn't that make everything better?
No it would only make it backwards.
It won't be, because Obama, and Trudeau, and Merkel, and Cameron are no longer going to be in charge.
Less deaths of famous people, but tax cuts for 1%. But in 2018 we're looking forward to another catastrophe. Either war with NK, impeachment of the POTUS and political shitstorm afterward, or just plain Depression, after the cuts, which didn't give more jobs as promised. Which whole wide world will feel.
oooh, so that universe itself has this twist in it so by adding a twist yourself, you actually make a strip with one twist (like a mobius band) but including the twist of that universe, there are two twists in total so it's a normal band.
So basically this is a universe that inverses mobius strips and cilinders. Because of it inherrit twist, it switches the odds and evens.
Pretty much :)
oh, great, then I understand it somewhat!
I know it's not a literal twist, but something resembling it somehow.
If you were to live on a 'globe' that has this poperty and you sailed west until you got back to where you started, would you see the entire world mirrored and would they see you as mirrored at the same time? Odd question, and possibly too much scifi, but I mean like: everything that appears to you would be what you know, but mirrored while you appear to the people who know you as mirrored, right?
This is making me question reality.
That tiny klein bottle universe with the earth in it kinda looks like the inside of the black hole in Interstellar, doesn't it?
ElectroBlood that's what I thought too
I tried to watch this video, but I was too busy admiring the phrase "topologically fortified".
:)
Happy New Years Mr. Mathologer!
Ever since you introduced me to your channel (during that quick post-lecture lunch) my appreciation for maths has increased by a heck of a lot. You've been able to explain things other RUclips channels can't, and your dedication to the craft is really inspiring.
Here's to 150,000 subs!!
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?
314 grams? that's too much for me, I usually eat 271 grams of these for breakfast (2.71828...)
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?
Thank you for making all these entertaining and interesting videos!
I love to watch them!
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?
You used brown paper strip! Such a Parker Square of a Numberphile!
numberfile sucks.
+Silly Sad Yeah, I also hate Numberfile. They rip off Numberphile so much.
NO!!!!!!!! #Numerphile(2^√ 2^√ 2^√ 2^√ 2^√ 2^√ 2^√ 2^√ 2^√ 2^√ 2...)days
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.
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.
I still remember coming across this insight for the first time and being really taken by it :)
if anyone wondered QED stands for Quite Explosive Decoration
It means one of the following:
Quod Erat Demonstrandum, a popular latin phrase.
Quantum Electrodynamics, a field of Science.
0:10 Now, I want some of those Möbius flakes. 😋
I've read Jeff's book twice, I guess that makes me a math connoisseur. Another cool fact is that the three dimensional analogue of the projective plane is actually orientable. I love his geometry games, and I recommend taking a look at HyperRogue, which isn't Jeff's, but is an awesome tool for demonstrating the nature of the hyperbolic plane, and you can choose the projection you are playing in. It even has spherical and elliptic modes, but not in the free version.
Actually, somebody just recently recommended HyperRogue to me and I've been playing it quite a bit lately. Very cool game. I am actually tossing up whether I should feature it in a video on the hyperbolic plane :) Are you familiar with MagicTile? It's got hyperbolic twisty puzzles among many many aother things roice3.org/magictile/ :)
I can't wait to try it.
@@Mathologer For what it's worth, it would be great to have HyperRogue featured :) Popular mathematicians seem to like HyperRogue, there have been some blogposts/videos about the very early versions, but recently, they are like "I've been meaning to do this but I have no time" or "I think if I tried this game, I would love it so much that I would not do any work anymore". And we still do a lot of work on it, we add new ways of showing the nature of hyperbolic plane, new projections, new tilings, new hyperbolic gameplay, etc. :) Spherical and elliptic modes are free for a long time, though the "racing mode" added a week ago is currently not.
@@ZenoRogue HyperRogue is great, I've spent many, many hours playing it (the MacOS version). If I ever get around to talking about the hyperbolic plane again it's definitely something I'd like to mention.
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.
There is actually no hole. There might be more complicated self-intersections. However, these are really irrelevant as far the the Klein bottle is concerned. Of course, in general, as soon as you've got self-intersections, strictly speaking you are only dealing with an immersion of a Klein bottle and not the real thing. If you don't want self-intersections you have to either unscramble things in higher-dimensional space or go for the sort of Klein bottle in an exotic 3d space as I show at the end of the video :)
I love it, when mathematicians create 'exotic" worlds, that only exist in our brains (or maybe they do for real?!), and play with their thoughts for the love of math itself. Sp once again, Mathologer, you baldly went for us where no man has gone before. Thanks for the answer! [And sorry for that small joke from one 'chrome dome' to another :)]
All under control then :)
This video hypertwisted my brain.
Mission accomplished :)
so true!
i nearly quit watching, right before i realized that it makes sense.
Hey Mr. Mathologer do a video on Wau (F)!!
To hard to compete with Vi Hart in this respect :)
If you can compete with Numberphile you can compete with Vi Hart!
I can compete with her in terms of the maths no problem but not in terms of her charm :)
Nice animations and content as always
Ok I'm confused now.
Back from New Zealand. You can have two New Years.
New Zealand is actually just around the corner from where I live (it's only a 3 hour trip to Christchurch from Melbourne here in Australia :)
newcastle here.
Hey my fave channel is back yuppyyyy
This phenomenon is exploited in Dark, two parallel universes that emerged from one
hi im from italy; I have watched all of your videos, cant find anyone else explaining maths like you do, amazing job!! hard work pays out! :D
Glad the videos work for you and thank you for saying so:)
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.
I'm here from watching a Star Talk clip about mobius strips and damn this is some mindblowing stuff! Glad I decided to look into it more and that I came across this video. Makes me want to go for another degree in math/physics after I'm done with my current one (international relations)! Thanks for this both clear and totally fascinating video
Morbius
"it's morbin time"
- Michael Morbius 2022
5 × 0 = 0
as we say.
0= 0÷5
that is again zero.
But,
5×0= 0
then,
5=0÷0
that is infinity.
then the statement
" ANYTHING MULTIPLIED BY 0 IS 0.
This is same for all zero applications like
0+0
0-0
0÷0
& of course 0×0
the geometrygames site is so cool, it even has chess in finite unbounded 2D spaces
Maybe also watch this video by Jeff about the most basic of these exotic 3d universes ruclips.net/video/-gLNlC_hQ3M/видео.html Really very, very good :)
0:10 I know everyone saw this, but there's 314 grams on the box, as in Pi!
Cool video! I realized that two-sided Mobius strips were possible when I was an undergraduate math major at the University of Texas at Austin (during the late 1990s). In fact I used a thought-experiment very similar to that described at the beginning of this video. For a while I thought I had made an original discovery in this, but eventually I chanced upon a topology book in the Physics-Math-Astronomy Library that actually discussed all these ideas. Unfortunately, I don't remember the author or the title of the book, but it was not the book referenced in this video.
What would happen to a 2d piece of paper if both sets of opposite edges were glued together in a mirrored fashion?
___>___
|.......|
^......d
|.......|
___
MAKE MÖBIUS FLAKES 314 G/PACKET A REAL THING, PLEASE!
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 :
www.jp-petit.org/science/f300/astrophysics_and_space_science_1995.pdf
I have a Question, Hope somebody answers this one day....
Imagine a Regular Tube... connect the Two ends we get a Torus...
Now a Torus have 2 sides, the outside one and the inside one... Remember this
Now take Another Identical Tube, and conne.. but wait, how about we twist the Tube in some higher dimension so that the inner surface is the outer one, and outer surface is inner, and then connect the ends?
What shape is this, Torus with 1 side? Does it even have any Volume?
That higher dimensional twist, is it even possible? I mean, it's not the same as Mirroring the tube. It's something different.
Dislike, because the Mathologer Mobius flakes are not something I can actually buy
Well, I am in negotiations with a 4D company to make these Mobius flakes a reality. I'll let you know when they hit the shelves in your local supermarket :)
Nice :3
Just a tidbit for those who don't know: QED stands for "quod erat demonstratum", which is latin for "which is what had to be proved." It's usually put at the end of proofs.
Why not the bigger 628 gram packet? ;-)
When I see things like all the Earths my mind gets drawn to what it would be like if we could make an E8 where we could stand on a point and view all the connections jumping from one to another.
Once again, thank you. The tee is great.
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.
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.
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
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!
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.
Reye's configuration. Protectively equivalent to the figure consisting of the twelve edges of a cube and it's four cube diagonals, in which three points are at infinity.
Lying to oneself is one of the worst things that you could do to yourself it doesn’t make you smarter to deny the truth same as 1 + 1= 2 is the same with objective truth it is the correct answer to the question of reality. What I’m saying is that fact is based on reality and vise verse it simply is so and cannot truely be disputed it is what it is 3 is 3 , 4 is 4 end of discussion. So never lie to yourself let your conscience be your guide listen to your gut adhere to reality.
That's not a cat and it isn't a chihuahua.
*That's a demon.*
As a lifelong cat lover, I can confidently say (tongue in cheek): cat... demon... not so sure that there's a difference... 😉 😆
Could you make a video on why Euclid and the Pythagoreans did not consider "1" to be a number?
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
Will check it out :)
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.
happy new year
I'm greatly impressed by this video. Loving all the props and the animations. Possibly your best video so far in that respect.
Had quite a bit of fun making playing with the 3d cat in this video. Very time-consuming though :)
after zooming out of the known universe and returning mirrored, your main concern is food?
Almost at 150,000 subs
Almost :)
Practically zero considering how many natural numbers there are.
Wut is with mathematicians obsession with cereal?
vid about 1/998001 please! btw +1 sub
That cat's food would be anti-matter.
wow I'm blown
I just started reading that book about a month ago!
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.
If the cat was to hover above the MS it would be (locally) "on the other side" after one roundtrip. However, the cat is not hovering, it is part of the zero-thickness MS, and as such has zero thickness itself. And so after one roundtrip it appears reversed. Maybe it helps to not worry about the movement and just focus on the cat at the beginning and at the end of the roundtrip.
Physics speculation: I once heard the the universe is expanding like the surface of a balloon (everything is getting further away from one another with no obvious center). Then this made me think that maybe that's more true then we realize. Like we are living on the surface of a 4 dimensional sphere expanding.
Actually, Einstein did think that the universe is the 3d surface of a 4d sphere :)
New Zealand? I'm envious!
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?
In terms of surfaces as long as what you are dealing with is part of R^3 2-sided is the same as orientable. In general, orientability is what is referred to as an intrinsic property of a surface or manifold whereas sidedness is an example of an extrinsic property that only makes sense in the context of a surrounding space. I'd say have a look at the book I recommend in the movie. Really very easy to read, especially for someone with your background :)
Out of curiosity, is there any mathematics related to infinite dimensional space? Could anything even happen in infinite dimensional space or would the lack of special dichotomy cancel everything out?
there are many infinite dimensional spaces in topology, bearing some nice properties. Hilbert's cube, for example, is compact
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.
Our universe is obviously shaped like a regular infinitahedron (I know that it's impossible XD).
you spelled CAT wrong.
An electron when rotated 360 is not the same ... one must rotate it 720! See:
blog.sigfpe.com/2007/04/curious-rotational-memory-of-electron.html
(no I don't understand this ...)
Maybe I can help with this in an upcoming video on the Dirac belt trick :)
So would it be right to view one of those mirroring universes as a 3D surface embedded on a 4D mobius like structure?
You could definitely realise these exotic 3D universes inside higher-dimensional spaces. However, for our universe to be such a mirror universe a 4d surrounding space is not necessary :)
What does QED stand for? Quantum Electro Dynamics jaja?
Mauricio Huicochea Toledo Quid Erat Demonstrandum.
The cerals alone are worth a like :-)
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.
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"?
are electrons mobius strips?
Am I the only one who can't figure out how things being mirrored in three dimensions would work?
It would be surprising if you did not have problems figuring this out (explicitly). Basically, in this respect we are all facing exactly the same problem as the flat cat trying to figure out how there can possibly be a 2D mirror-reversing path. We as 3d creatures can see easily what the flat cat has problems figuring out.
Now, of course the mathematicians of the cat's 2d world also have no problems writing down equations that describe the mirror-reversing Mobius strip universe space in (for them) abstract xyz-space.
Similarly, if there were higher-dimensional creatures they could easily construct a solid 3D Mobius strip corridor winding through higher-dimensional space and we 3d mathematicians can at least write down the equations for such a space. Really the best way to develop some intuition for this sort of stuff is by really understanding what happens one dimension down and then extrapolating from that.
Nah that is kind of stupid because the mirroring is kind of a twist in a mobios band
That's the whole twist: the universe itself has a twist so if you have a strip and twist it once, you actually make a cilinder. You have a twist in a twist. However, unlike an actual cilinder, it's still non-orientable.
Marty! You're not thinking fourth-dimensionally!
Wouldn't the thickness of the paper count as a second face and therefore constitute a 2 sided mobius strip?
The paper Mobius strips are really just (slightly misleading) real-life approximations for the real mathematical deal. 2D implies 0 thickness. Anything that has non-zero thickness is not a surface anymore :)
Almost at 150,000! He deserves it!
Happy New Year BTW
The Möbius strip. The original mindbender. Thanks.
I'd definitely have mobius flakes for breakfast 😂
Where do you get your shirts?
All over the place, I am just doing a comprehensive search on math t-shirts every once in a while and get everything good in sight. And about half of them I make myself :)
Mathologer Thanks! They're really cool! The one in this video really prompted me to ask!
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?
Not sure I understand how exactly you want to glue things. If you twist your prism and then glue wouldn't you automatically get AB, BC, CA? There is a comment by a sculptor further down in the comments with a linked in picture that shows a shape like this (cross-section is a square not a triangle though :)
not sure how to label things properly. I did this with a paper tube formed into a triangle and it gave me what appeared to be one Möbius side and one regular side.
So... The universe twists and the circle stays the same? You get the same result? Sorry I'm new at this, but very interested
Yes, that's one way to do it :)
Or the universe could just have an odd number of twists in it (up to infinite), and so long as it's an odd number, any circle will have two sides?
Is it normal that I totally understood everything you just said?
Well I tried extra hard to make this one as accessible as possible :)
is the Pac-Man universe the same as a toroidal space? or are they different?
You mean the solid Klein bottle space at the end? No, that one is different from a toroidal space. To get a toroidal space the arrows on the front and the back have to also point in the same direction :)
+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?
Not really sure what you are worried about. Yes, when you draw your two parallel lines you are creating a Mobius strip inside another Mobius strip. However in that context it does not make sense to speak of how many sides this new MS has--you need a surrounding 3d space for that :)
well, it seems preposterous on a mobius strip, do the same on a klein bottle and you've just outlined one of its component mobius strips... so why is it magically valid when you step one level higher? the mirroring effect is the same! it happens because space bends back over itself, and its really a one-sided mobius band, your cat just got mirrored in 3 ways, not 1. reflect something vertically and front-back, you get a cat looking backwards from the bottom of the band, the third mirroring is the one you mentioned in the video, you only see the cat on top of the band because it rotated 180 degrees. it is a one sided mobius strip.
I missed your vidios so mush 😢