Hypertwist: 2-sided Möbius strips and mirror universes

by birtanpublished on August 29, 2020

You watching mythology video and that probably means your eating plan bottles and movie strips for breakfast and you know that these tasty mathematical surfaces all have just one side except an only real mathematical connoisseur seem to know this there are client

Bottles and mobius strips that have two sides let me explain click revision this strip of paper has two edges and two sides to make it into movies to what I have to do is to bring the ends together such that 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 the strip into a mobius strip you can just go one twist and glue and that gives your movie strip three twists or five twists or any odd number of

Clockwise or counterclockwise twists 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 end into mobius strips that are two-sided how to imagine how is that miracle possible well it

Turns out that the number of sides of a movie strip or actually of any 2d surface depends on which 3d space which 3d universe it is contained and how exactly it is contained in this 3d universe now most people think that 3d

Just means XY that space what you deal with in school or at university but there's actually infinitely many 3d universes mathematical 3d universes and we actually don't know which one of these

Mathematical universe's describes the universe that we live in now quite a few of you will actually have heard that our universe may be a pac-man universe which means that there may be a direction special directions if I head off in this

Direction does keep going straight I will get back to where I started from so let's just assume we are inside a mathematical universe it has this property and let me introduce you to my math cat mascot QED cat well actually

There's a bit of dispute at home whether it is a cat or a 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

Surfboard actually generates a strip okay now keep going keep going eventually it gets back to where it started from there and now it wants to turn the strip into a mobius strip and you can see well to create this one long

Edge what we have to do is we have to kind of flip upside down and keep moving forward okay 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 movie strips so nothing new here now in a more fancy universe something else can happen so let's just do this again so QD cat heads off again we leave one of those ghost images behind it gets

Back to 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 to do any of those upside-down acrobatics so just like this and we've created a movie strip and obviously that mobius strip is two-sided so if you put an anti cat on the other side and have QED and the anti cat

Runaround 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 mirror reversing happens at a molecular level and so the food and whatever processes the food and 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 all so once you have a two-sided movie 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 you know a lot of you may have wondered whether something like this exists well there you go and 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 the situation again we can now turn this strip here into a cylinder by just doing a twist 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 reversing paths in our real universe or at least in a mathematical

Universe that I can just kind of fold in front of you but that's actually very hard to do because no matter what you do 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 mirror reversing universes inside XYZ space so it 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 the

Counterpart of a cylinder in a 2d world it's just a circle so what I want to show you is a one-sided circle now usually circles are two-sided right here two-sided circle with respect to this 2d world the cat is living in and now

Instead of using this off-the-shelf to the universe I use a mobius strip universe that's a 2d universe the cats living inside it the circle is part of this universe and I'm going to chase the cat around it but what's really

Important here is actually to emphasized that a real mathematical surface has zero thickness just like the XY plane inside XYZ space has zero thickness so that movie strip is zero thickness the cat sliding around and it

Has zero thickness let's just see what happens when it runs around a circle okay so it's completed its round trip any case as you can see with respect to this 2d universe it's 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 the 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 within 3d space it doesn't make any sense to ask how many sides 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 does doesn't work so for example we could put something like this in four dimensional space and just have a floating there doesn't make any sense to ask how many sides one of those things has now I also promised you

Some two-sided klein bottles how do we get those have a look at this so QED is flat so it can't really see a mobius strip but what wants to play with it anyway so it can do this out of pac-man it's not ideal but it's good

Enough to to visualize what's going on okay so what you do is you just kind of draw a flat rectangle and QT 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 glued to itself in a

Way and that certain way I can actually show you very easily also with arrows goes like that so what 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 client 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 and I'm just going to bring corresponding points of the edge together like this and there

You've got your client bottle now obviously once you've found one of those mirror reversing parts 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 client bottle and

It's exactly what happens all right now we've got two pictures of a client bottle here and just like QED can use the square to describe a client bottle we can use a solid cube to describe a solid counterpart of a client bottle so

A basically a solid client bottle and this is also done by these fancy arrows here the what the fancy arrows show you is how opposite faces of the solid acute was supposed to be glued together for example these two points we're gonna put

Together these two points those two points and so on should be pretty obvious and actually this solid client bottle there is one of those mirror universes and if you have a really very close look you can see that this here is

A two-sided client bottle within this 3d mirror universe very very fancy very very pretty it's absolutely beautiful client bottle much nicer than the ones that I've shown you before the ones I showed you before

Has this strange sort of self intersection which is just really annoying this one doesn't have any of this so so much much nicer in this respect okay now I learned about all this 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've really recommend it to everybody here Jeff's also created some amazing

Of software totally free that you can download from the website I linked in from the description and they allow you to you know play chess on client bottles on tour I may pull all kinds of other things but it also has pieces of

Software that allows it to kind of fly around in strange 3d universes so for example here's a view of a very small version of this solid client bottle universe which just basically a 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 kind of figure out how exactly this

Works how how exactly the pattern of earth and mirror earth comes about now there's a lot more to be said about all this or four dimensional stuff I may say a little bit about this in the description also I'll definitely come

Back to these strange 3d universe and 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

You you

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