sorry dude but above time there isnt dimension anymore. Dimension means: borders, boundaries, mathematical dependency and can be explained by formula (only from a higher dimension into a lower one) every try to explain higher dimension from a lower is a CONCEPT and isnt true but only approaches the truth eg klein bottle. Concept: a lower dimension thinking about a higher one Definition: a higher dimension observing a lower one. a mobius strip is the two dimensional infinity and can only exist by staying in a higher dimension! 6:15 - is what we see as a donut shape. The torus. The klein bottle is the three dimensional infinity existing in a higher dimension (=something like a proof there is at least one higher dimension above 3D, what i called CONCEPT above)
en.wikipedia.org/wiki/Curse_of_dimensionality more views means a better idea of the concept of the dimension that is above 3D (4D i think is the time dimensionality)
Want a Mobius strip? I'll sell them for half price of the Kline bottle. Some assemble will be required because the Mobius strip is delivered as a post card to you and you have to tape the two ends of the postcard together yourself. The instructions are on the post card. In fact, the instructions appear on both sides of the post card after you have taped the ends together to form the Mobius strip.
For some reason, nobody has ever explained to me a cube on 2d intersecting itself is like a Klein bottle on 3d intersecting itself. Idk why. I got the concept a while ago, but never had it explained like that.
Mathematically, this is a visual of the failure of a certain property of functions.. For an object to embed in n-dimensional space, there must be a function that brings the object into n-dimensional space that is, among other requirements, one-to-one. The object intersecting itself is a manifestation of the function's failure to be one-to-one. Points that were previously distinct on the object are brought to the same point.
Tgon Mwort you don’t need to see it. no one can imagine 4 dimensions, and a taurus is embedded in 4d. but you can imagine a cube, and you can imagine a drawing of a cube, and you can see how certain point have to intersect on the paper to represent it. despite these apparent intersections, you know that none of the vertices of a cube actually overlap, and you can compare that to a drawing of a klein bottle. remember, you don’t need to imagine it. just understand the comparison.
That feeling when you see a square with arrows that you drew in five minutes for a Wikipedia article back in 2006 show up on a RUclips thumbnail in 2020…
There is something about the thorus that took me quite some time to understand which is the difference between it and a sphere. I took a while to understand why the universe of the asteroids game couldn't be a sphere, since it is also true that in the sphere when you travel all the way to the left you come out on the right and the same is valid for up and down (and in the same orientation). The answer I found is that two perpendicular lines on the thorus only intersect once, whereas on the sphere they intersect twice. So if two spaceships would start at the same point in the universe of astroids and one travels up and the other travels to one side, let's say right, then they would meet again in the same point, the first comming from below and the second comming from the left (assuming equal lengths). However if the universe was a sphere, they would have to meet in another point, which would be the diametrically opposite to the one they started, whic is not the case. I just thought this would be something nice to point out
Yeah, the video also had me think "what about spheres?" at several points. The thing about spheres is you can't topologically embed them in a plane, so it's hard to compare. If we use something like Mercator projection (how globe maps are usually drawn), yes if you go west you come out on the east end, but the same is not true for north-south. As you move towards a north-south edge of a map, once you're there, you come back down from the same edge but in opposite direction (on the map) at a different point that isn't even connected to the previous one. That's because essentially, if you embed a sphere in a plane, you have to turn the poles from points into edges (the north and south edge of the map). So basically, if such a 2D game were to have spherical-ish topology, if you go through the left border you'd come out on the right, but if you move through the bottom border (going down) you come out on the bottom border going up but exactly half a screen length displaced.
Interesting thoughts. I did think asteroids is a sphere. The issue I had with a torus was, I thought if your spaceship was on the inner ring, and pointing along the mid plane of the torus, if you press up then your spaceship would move in a circle on the screen (or perhaps run along a single vector line and back again without going off screen). Ultimately, the entire surface of a sphere cannot be laid flat in 2D to make a SQUARE, but with a torus it can... I think... The bend radius would have to be very small. Ok now I need some paper and scissors! It would have been good if this was mentioned in the video.
@@gmweb1304 I don't know if this is relevant, but the position of the observer is different in the two examples. When mapping a sphere we do so from a position above the flat surface to be mapped. With the game of Asteroids it appears as though the point of observation is from below what could be described as the flat surface of their environment.
The sphere can be pictured with a rectangle. Just look at a world map. If you travel to the top (or bottom) you still come out to the top (or bottom) just at a different point - left or right.
I remember when I was in grade 8 my math teacher spent a class talking about infinity and topology. That was the first time I felt math is interesting and I wanted to be a mathematician. 11 years later, unfortunately, pi is no longer pi to me. pi=e=3
I think it was not a sphere because you can make a sphere using only one kind of circles that continues to go in one direction anywhere. That creates a sphere successfully. The answer was torus because you cant only do it with one sphere going one direction and you need two so that the hole of the torus can be made. And i know i explained that very terribly but im tired.
Honestly, yeah, that is sorta what a torus is, a loop traced along another loop. Although I suppose the bead and necklace idea that I associate with that idea does make think of those oops as separate from each other I'd more readily picture them as dangling from each other, but I'm pretty sure that's just a "me" thing. It's probably why I'd try to describe the loops of a torus as embedded into each other, but I'm pretty sure I'm just being pedantic.
All kidding aside, this is blowing my mind. I'm getting old so I don't like to talk about myself too much, but the cartesian product as you're describing here is the way I stumbled upon for describing a simple 4D object as a teenage stoner. Of "stacking" spheres upon one another in the same way you've described, to "extend" the sphere out into 4-space.
Awesome! Two questions: 1. At 10:08, three examples are shown for the 2D manifolds of cylinder, mobius strip, and klein bottle. What would be created by altering the third manifold so blue arrows also point in opposite directions? 2. What are the factors, in terms of Cartesian products of shapes, of a sphere? I assumed circle x circle = sphere, but as explained at 15:46, clearly not.
1.) This results in what's called the real projective plane, or RP^2. It is a well-studied object, so just Googling that will give you plenty of information. 2.) There are no such factor spaces of the sphere. Showing this is true is a bit out of scope here (off hand I think we will need to invoke algebraic topology for a proof). Intuitively, one might think of spheres as more "fundamental" or "prime" topological spaces.
Epic Math Time Thanks for the explanation! I'd wondered if the sphere would be considered "prime". Might take awhile to wrap my head around RP^2 though...
Zach this video is literally amazing!!!! I have been a science freak my entire life but when I was younger I hadn't the patience for real math and I have regretted it. I don't want to show off to anyone, I just WANT TO KNOW! .....then this 19 minute video kicked me in the back of the head!! Never have I heard it explained in this way and it's like getting struck by lightning. Thank you sooooo much!!! I have so many videos to revisit now. First time comment, life time subscription!!
There's a highly interesting area in programming called computational geometry wherein topics like these are covered, this video was very helpful, it gives a nice and brief insight. Do you know any further reading (books, blogs, etc), by any chance? Thanks for sharing your knowledge.
I simply can't believe that someone with some sheets of paper, sellotape and a few basic animations can explain these topics so well. Amazing! I've learned so much. Thank you.
I was sitting here "It's a Klein Bottle it's a Klein Bottle, tell me that I'm correct" like I was sitting on hot coal. (Yes, I'm that person). Had never thought that topology was so exciting for me. I always had problems with geometry but was good in function analysis or probability.
That may also be the first time I heard a Klein bottle explained like that. I’ve wondered why they didn’t just build them with a gap around the penetration point. It’s because that intersection is an illusion, the projection of a 4D object into 3D space just like the 2D drawing of a 3D cube shows false intersections. If I looked at it from 4D there would be no intersection and the bottle wouldn’t narrow and widen.
6:15 Couldn’t you connect the top of the cylinder to the bottom of the cylinder by wrapping the top and bottom around one side on the x/y plane rather than around the y/z plane (I’m just assuming that’s how you would describe that plane)????
I’m curious about the psychology of thinking in higher dimensions. What would be different about a 4d brain that allows it to imagine in 4 dimensions? Why couldn’t a flat lander imagine 3d vision? is it simply because imagination is just a combination of images from memory and flatland guy has never seen in 3d? is it the structure of his brain? is it some fundamental truth that makes no sense to us, similar to how observing forces a quantum particle to choose a state? big hmmm
Thats a really goo remark and i think its just because we are born and live in a 3d space. There is the 4th dimension - time, but we actually cannot feel or observe it. We measure it, we live in it, we count it, but how would you imagine it? You live in it and it became such a inseparable part of our existence that it become impossible for a normal human to "think outside of the box". I bet that if you got even a glimpse of 4th space dimension, you would no longer have problem imagining it further, but in this case its like trying to imagine a color that you have not seen. It just too different from what your brain has observed up till now...
It's a really interesting question. Would a physically 4D brain hooked up from birth to a simulated 3D world like ours nonetheless somehow be capable to imagine the 4th dimension, just by virtue of physically inhabiting it? Personally I don't think so, but I imagine to have a clear answer we would have to try en experiment with a 3D brain hooked up to a 2D simulator from birth. Problem is (ethical issues aside) I don't imagine a 2D simulation would be enough for a brain to evolve complex functions like language and spatial awareness, for us to be able to ask them to imagine another dimension. Still, I believe the answer lies in the psychology of the brain and not in the physical substrate it is made of.
Eyes see pictures of the worlds, basically we see in 2D, but because we have 2 of them and the brain is actually smart, it can deduce the 3 dimension from that. I believe we instantly a Cube as a Cube simply because of experience thus (can be demonstrated with optical illusions too, it's quite easy to trick the brain with fake or impossible perspective). A flatlander would only see a line, so it would see in 1D, but with its experience, it could understand what a square is like, however there would be too much missing information for a cube. now I'm really not sure, but maybe if we found a way to have eyes that can see in 3D, we could then see objects in 4D much better. A flatlander sees 1 or 2 line from a square, but we see the whole square at once, no matter what is its rotation. Our 2D eyes see only 3 faces of a cube at most, but a 3D eye could see the entire cube at once. If you had 2 3D-eyes, maybe that would be enough to fully see in 4D? In any case, I'm confident to say it's beyond human comprehension, simply because the brain is made to interpret and compare 2D pictures, not entire 3D spaces, so even if you somehow injected 3D eyes into the brain, it probably wouldn't be able to understand anything.
Tomáš Musil why 4th dimension of space? Space cannot be has only 3 dimension and 4the or 5th beside time , there might be other dimension but not space. Something we cannot understand
I feel enlightened. For the first time, I understand the what that Klein bottle and how that is related to 4th dimension. It's just mindblowing but unfortunately no one can ever witness the higher dimensions.
Love this video! great way to explain this concept! I've been binging this stuff on youtube for years and this helped me put a lot of concepts together. Thannk you so much! great work!
Old games here: King's Quest 1 is a Torus (like Asteroids). The map cycles around in all 4 directions. King's Quest 2, 3, 4 (Most of them anyway, depending on map) are cylinders like the first new example. Very cool vid; great perspective!
The way you explain the things is far far far much better. Rather amazing. I used to watch pbs videos earlier. But he somehow failed to explain the things clearly. I would request you to please explain those videos in your way. I am sure everyone would understand the things in perfect way.
@Zach: Now do: Portal - multiply connected space-time. (Also, Splitgate) Braid - closed time-like curves Miegakure - 4D world. Antichamber - 4D (? ) Escher-like world. (Also, Manifold garden) Any game with scale-based maps that the player transitions between (eg Star Control II, No Man's Sky) Monument Valley - world that can be manipulated by perspective. So is it a projection of a higher dimensional space onto a lower dimension? Fez Contrast - game where the player can move between 3D world and shadows projected by light sources.
i am so happy that i recognized the möbius strip surface and the klein bottle almost instantly but SOMEHOW did not instantly recognize the cylinder because did not understand what the heck were those zebra stripes on the corners
The video is really well done , definitely one of the best ones out there ... to the viewers who want to move into the 4 th dimension, it’s just not possible for us to do that in the physical human form. We lack the extra spatial dimension to move into it . It’s like a shadow wanting to be in the 3rd dimension, it has height and width but no thickness ... you however can consider yourselves to be the projections of your 4 th dimensional persona. Something like the shadow is the 2D representation of a 3D body. The shadow has no awareness of its 3D higher self but it exists when light is obstructed by its 3D self before casting the shadow on the 2 dimensional floor it wall on which the 2 D shadow is cast.
I’ve played a game called “Manifold Garden” before. It’s a first person puzzle game (it’s 3D). In that game if you fall you land back on the level floating in space. You can rotate yourself in any direction and fall back into the level itself like before. I now understand the title and see every level is a 3 torus.
I’ve had this question on my mind for days now. my question is what’s the topology of a world map? Going left to right and right to left is easy to imagine, but what about going up? You don’t end up on the bottom. Think about this one, it’s tricky. Here’s an example of one of my broken theories: Imagine when going up, your x coordinate becomes its opposite (- to +, and + to -), and you get flipped around, facing South. This works, except that when walking strait into the arctic in the middle of the map, you would see the same terrain when after being turned around, walking back toward the equator (the terrain should be different!).
I just had a eureka moment. If you had 2d slices of a Klein bottle (like the animation) and moved them through time there would be no self intersection, but if you put it all at the same time, there is self intersection. We live in 4 dimensions and perceive the fourth axis as time.
Thank you so much, Zach, this is the clearest explanation of this I have ever seen. Now I can follow Slavoj Zizek a little better when he talks about how topology is related to psychoanalysis!
Okay... so for 2D manifolds, you have: >Square >Cyllinder >Mobius Strip >Flat Torus >Klein Bottle But what about where both pairs of edges are pointing in opposite directions? Like... top and bottom are opposite, and so are left and right? Does that form something?
Hey there! I saw the answer to this in a thread nearby, and someone said that it was called the Real Projective Plane (represented as RP^2). There's a few good Wikipedia articles on the topic, so here they are: REAL PROJECTIVE PLANE - en.wikipedia.org/wiki/Real_projective_plane REAL PROJECTIVE SPACE - en.wikipedia.org/wiki/Real_projective_space
Wow... you've represented the "hairy" topic of topology in such an intuitive, easy to understand way... even up to the fourth dimension! Amazing! You've just won yourself a subscriber, good sir!
The drawn cube has 1-D intersections, the Klein bottle has 2-D intersections. That’s made higher dimensions as clear as a mill pond. I had seen the Klein bottle many times but didn’t realise that it was a shadow of a 4-D object. Every time I had tried to visual a 4-D object my brain always plays back the clip from Terminator 2 where T1000 is in the molten lava and he kind comes out of his own mouth incidentally this looks a lot like the intersection on a Klein bottle.
First part I didn't think to create donut by using the top bottom connection outside, which is in a way easy, like the first time I thought to twist towards each other towards cylinder insider and then imagined expanding cylinder's side walls going outward or bulging
I clicked a video about the probability of dice, now here I am 4 videos later stretching my mind in 4D trying to envision the progression by which I can imagine a Klein bottle without an intersection.
One particular thing that I believe is worth mentioning, is that a lot of people (myself included) tend to think of the circle as a 2D shape, when in actuality it is the embedding of the topological circle into 2D space. This little subtlety, if not recognized, might make you wonder: if a circle is a 2D object, then why is the product of two circles not a a 4D object? Actually, the resulting object is the so-called Clifford torus, and its stereographic projection is the standard 3D torus.
Can you talk about tensors? I never understood what they are. How do they relate to vectors? and how can we represent them? (for example vectors are arrows and tensors are ... ??)
Zack, I think I've found your future career. I haven't heard you mention it but I feel you should go into teaching. You're great at it and, have a lot of stage presence. Which is important to keep students engaged. I know I've learned a lot from watching your videos in my free time. I think the world could benefit from it too. P.S. I love the new channel as well. Not what I expected, very much on my level, and nice to see a different, relatable human side. Thanks for doing what you do. I enjoy it and want to see more.
Antichamber and Manifold Garden are two 3D puzzle games that make you think outside the box, kinda like topological problems. I suggest you should give them a try if you haven't spoiled the fun by watching a playthrough.
I started watching this channel for the fun little mind puzzles and now I’m trying to conceptualize the fourth dimension through numbers and variables at 4am, wtf happened
Want a Klein bottle?: stemerch.com/collections/maths-toys/products/klein-bottle-1
sorry dude but above time there isnt dimension anymore. Dimension means: borders, boundaries, mathematical dependency and can be explained by formula (only from a higher dimension into a lower one)
every try to explain higher dimension from a lower is a CONCEPT and isnt true but only approaches the truth eg klein bottle. Concept: a lower dimension thinking about a higher one
Definition: a higher dimension observing a lower one.
a mobius strip is the two dimensional infinity and can only exist by staying in a higher dimension!
6:15 - is what we see as a donut shape. The torus. The klein bottle is the three dimensional infinity existing in a higher dimension (=something like a proof there is at least one higher dimension above 3D, what i called CONCEPT above)
en.wikipedia.org/wiki/Curse_of_dimensionality
more views means a better idea of the concept of the dimension that is above 3D (4D i think is the time dimensionality)
How do you make a cone this way?
Want a Mobius strip? I'll sell them for half price of the Kline bottle. Some assemble will be required because the Mobius strip is delivered as a post card to you and you have to tape the two ends of the postcard together yourself. The instructions are on the post card. In fact, the instructions appear on both sides of the post card after you have taped the ends together to form the Mobius strip.
Drink from that Klein bottle!
ooeorr rueuvoo ruruv kvooki u dudue
How many people associate a klein bottle with wild gray hair, exuberant arm movements and one very excited guy?
YEAH!!
Numberphile reference
Yeah boiii man that guy is great
Ah, yes
Lmao. Everyone
Edit: Actually I thought more about Matt Parker of Stand-up Maths at first.
I've never felt the need of a fourth dimension in my life so much before
You won't , ask them who are data scientist..😂
Me to
If we were 2D people . You would have said the same that you wouldn't need 3 Dimension 🤣🤣
same
Need? It's a prerequisite.
For some reason, nobody has ever explained to me a cube on 2d intersecting itself is like a Klein bottle on 3d intersecting itself. Idk why. I got the concept a while ago, but never had it explained like that.
Mathematically, this is a visual of the failure of a certain property of functions..
For an object to embed in n-dimensional space, there must be a function that brings the object into n-dimensional space that is, among other requirements, one-to-one.
The object intersecting itself is a manifestation of the function's failure to be one-to-one. Points that were previously distinct on the object are brought to the same point.
This was the best explanation of the concept behind the Klein Bottle I've ever heard. The comparison was great and really helped me understand
Tgon Mwort you don’t need to see it. no one can imagine 4 dimensions, and a taurus is embedded in 4d. but you can imagine a cube, and you can imagine a drawing of a cube, and you can see how certain point have to intersect on the paper to represent it. despite these apparent intersections, you know that none of the vertices of a cube actually overlap, and you can compare that to a drawing of a klein bottle. remember, you don’t need to imagine it. just understand the comparison.
Tgon Mwort no one can. just the way the brain works.
@Tgon Mwort wait how can you see into the future
That feeling when you see a square with arrows that you drew in five minutes for a Wikipedia article back in 2006 show up on a RUclips thumbnail in 2020…
R u the author or
Bruh!
That 5 minutes might be your biggest contribution to humanity considering the reach. Unless you are a teacher of some sort.
@@aqaridot en.m.wikipedia.org/wiki/File:TorusAsSquare.svg
here it says "Drawn in en:Inkscape by Ilmari Karonen"
That's awesome.
There is something about the thorus that took me quite some time to understand which is the difference between it and a sphere. I took a while to understand why the universe of the asteroids game couldn't be a sphere, since it is also true that in the sphere when you travel all the way to the left you come out on the right and the same is valid for up and down (and in the same orientation). The answer I found is that two perpendicular lines on the thorus only intersect once, whereas on the sphere they intersect twice. So if two spaceships would start at the same point in the universe of astroids and one travels up and the other travels to one side, let's say right, then they would meet again in the same point, the first comming from below and the second comming from the left (assuming equal lengths). However if the universe was a sphere, they would have to meet in another point, which would be the diametrically opposite to the one they started, whic is not the case. I just thought this would be something nice to point out
This was really well explained. Thank you.
Yeah, the video also had me think "what about spheres?" at several points.
The thing about spheres is you can't topologically embed them in a plane, so it's hard to compare. If we use something like Mercator projection (how globe maps are usually drawn), yes if you go west you come out on the east end, but the same is not true for north-south. As you move towards a north-south edge of a map, once you're there, you come back down from the same edge but in opposite direction (on the map) at a different point that isn't even connected to the previous one. That's because essentially, if you embed a sphere in a plane, you have to turn the poles from points into edges (the north and south edge of the map).
So basically, if such a 2D game were to have spherical-ish topology, if you go through the left border you'd come out on the right, but if you move through the bottom border (going down) you come out on the bottom border going up but exactly half a screen length displaced.
Interesting thoughts. I did think asteroids is a sphere. The issue I had with a torus was, I thought if your spaceship was on the inner ring, and pointing along the mid plane of the torus, if you press up then your spaceship would move in a circle on the screen (or perhaps run along a single vector line and back again without going off screen). Ultimately, the entire surface of a sphere cannot be laid flat in 2D to make a SQUARE, but with a torus it can... I think... The bend radius would have to be very small. Ok now I need some paper and scissors! It would have been good if this was mentioned in the video.
@@gmweb1304 I don't know if this is relevant, but the position of the observer is different in the two examples. When mapping a sphere we do so from a position above the flat surface to be mapped. With the game of Asteroids it appears as though the point of observation is from below what could be described as the flat surface of their environment.
The sphere can be pictured with a rectangle. Just look at a world map. If you travel to the top (or bottom) you still come out to the top (or bottom) just at a different point - left or right.
Finding a connection between games and (currently) intangible science is hard to believe. Cheers for this one man!
I doubt it will ever be tangible to us, sadly. Unless, we give our brain a 4D update lol
@@angadsingh9314 Lmao xD
@@arunavaghatak8614 Nobody wants to see that shit.
Game Theory X Evolution is typing...
How many Topology videos can you make?
Zach Star: Yes.
What shape is Asteroids?
To answer this, we need to talk about parallel universes
**SM64 file select music starts playing**
@@moversti92 Step 1: build up speed for 12 hours
circle
The asteroid shape is torus
Or is it?
*Music intensified
W H E R E A R E T H E T U R T L E S
I remember when I was in grade 8 my math teacher spent a class talking about infinity and topology. That was the first time I felt math is interesting and I wanted to be a mathematician. 11 years later, unfortunately, pi is no longer pi to me. pi=e=3
π=3 but π^2=10
@@bayekofsiwa7035
no, it's g
but g is 10...
ok nevermind
Really sad, but yeah thats the truth...lesser ppl are getting into true math by the time.
@@englishmotherfucker1058 no i meant that in engineering they take π^2=10
@Hassan Akhtar its a joke yes, but once in my physics exam the teacher said that we have to take π^2=10
Great Job Zach
Papa flamy!
@@rainbow-cl4rk
A rare instance of a mathmatician complementing an engineer
Multiplying shapes is trippy. I need to re-watch this part more than once.
When you do enough math, you'll see that you can have a product of a lot of different things, provided that you can define the product.
Russians living in 4D be like: R Я
N И
Ñ Й
@@rainbowbloom575 X X :P
@@MsSonali1980 q р
@Rainbow Bloom d b
A classic game of Snecc?
>snecc
everytime the answer was torus, i really thought it was gonna be a sphere 😣
I think it was not a sphere because you can make a sphere using only one kind of circles that continues to go in one direction anywhere. That creates a sphere successfully. The answer was torus because you cant only do it with one sphere going one direction and you need two so that the hole of the torus can be made. And i know i explained that very terribly but im tired.
@@alexumitsuu3237 its cool, thanks! hope you get some rest
@@randaljr.8581 ssssssssaaaaaaaaaaaaaaaaaaame on the oc lol
me too
Np
My real thought process for circle x circle:
Well, what's a circle of circles?
*thinks of spheres oriented in a circle*
Oh I know, it’s a necklace!
Honestly, yeah, that is sorta what a torus is, a loop traced along another loop. Although I suppose the bead and necklace idea that I associate with that idea does make think of those oops as separate from each other I'd more readily picture them as dangling from each other, but I'm pretty sure that's just a "me" thing. It's probably why I'd try to describe the loops of a torus as embedded into each other, but I'm pretty sure I'm just being pedantic.
All kidding aside, this is blowing my mind. I'm getting old so I don't like to talk about myself too much, but the cartesian product as you're describing here is the way I stumbled upon for describing a simple 4D object as a teenage stoner. Of "stacking" spheres upon one another in the same way you've described, to "extend" the sphere out into 4-space.
How was this made yesterday?
Obama i think it's that patreons can see the video earlier than anyone else
@@barni3045 wtfff. No way.
Trump 2020
m.ruclips.net/video/FAvFFYu1RGM/видео.html
@@angadsingh9314 way
Awesome! Two questions:
1. At 10:08, three examples are shown for the 2D manifolds of cylinder, mobius strip, and klein bottle. What would be created by altering the third manifold so blue arrows also point in opposite directions?
2. What are the factors, in terms of Cartesian products of shapes, of a sphere? I assumed circle x circle = sphere, but as explained at 15:46, clearly not.
A sphere is the same as RxR but you need to add an extra point at infinity.
You ask the same question as me😂
Are you in my head?😂😂
I thought the same thing about the sphere
1.) This results in what's called the real projective plane, or RP^2. It is a well-studied object, so just Googling that will give you plenty of information.
2.) There are no such factor spaces of the sphere. Showing this is true is a bit out of scope here (off hand I think we will need to invoke algebraic topology for a proof). Intuitively, one might think of spheres as more "fundamental" or "prime" topological spaces.
Epic Math Time Thanks for the explanation! I'd wondered if the sphere would be considered "prime". Might take awhile to wrap my head around RP^2 though...
Understanding a 4D structure(T3) using a 3D manifold on a 2D screen using a 1D brain.
"It's evolving, just backwards"
sorry, but real envolve doesn't think of dimension as dimensions but realtime
sciencephile the AI references here
0:47 As someone who answered rectangle, I said "what?" exactly the moment the name torus came out.
My god, when I realized the Klein Bottle is a 3d representation of a 4d object, I literally got chills. I love math.
I was imagining that Three-Torus as a hollow donut, where the inside face of the donut is connected to the outside face.
a klein-donut
You could do 2 of the connections in 3 dimensions, it's that last connection that gets you into trouble.
Me too!!!
This made the difference between Topological and Geometrical SO clear.
Thank you.
Zach this video is literally amazing!!!! I have been a science freak my entire life but when I was younger I hadn't the patience for real math and I have regretted it. I don't want to show off to anyone, I just WANT TO KNOW!
.....then this 19 minute video kicked me in the back of the head!! Never have I heard it explained in this way and it's like getting struck by lightning. Thank you sooooo much!!! I have so many videos to revisit now. First time comment, life time subscription!!
12:58 i probably feel WAY prouder of myself for understanding this than i should lol
Cliff Stoll: *heavy breathing*
There's a highly interesting area in programming called computational geometry wherein topics like these are covered, this video was very helpful, it gives a nice and brief insight. Do you know any further reading (books, blogs, etc), by any chance? Thanks for sharing your knowledge.
So I finally get a little basic knowledge about this pretty interesting topic!🤩
Thanks Zach!☺️😉
I simply can't believe that someone with some sheets of paper, sellotape and a few basic animations can explain these topics so well. Amazing! I've learned so much. Thank you.
This video explain klein bottle and mobius strip so good that I want to restart my life
I was sitting here "It's a Klein Bottle it's a Klein Bottle, tell me that I'm correct" like I was sitting on hot coal. (Yes, I'm that person). Had never thought that topology was so exciting for me. I always had problems with geometry but was good in function analysis or probability.
That may also be the first time I heard a Klein bottle explained like that. I’ve wondered why they didn’t just build them with a gap around the penetration point. It’s because that intersection is an illusion, the projection of a 4D object into 3D space just like the 2D drawing of a 3D cube shows false intersections. If I looked at it from 4D there would be no intersection and the bottle wouldn’t narrow and widen.
just found your channel, absolutely love it. the whiteboard flip @8:50 earned my subscription
While watching i realized I’m looking at a 2D image of a 3D representation of a 4D object...😳🤯
with yo 1D brain
6:15
Couldn’t you connect the top of the cylinder to the bottom of the cylinder by wrapping the top and bottom around one side on the x/y plane rather than around the y/z plane (I’m just assuming that’s how you would describe that plane)????
I’m curious about the psychology of thinking in higher dimensions. What would be different about a 4d brain that allows it to imagine in 4 dimensions? Why couldn’t a flat lander imagine 3d vision? is it simply because imagination is just a combination of images from memory and flatland guy has never seen in 3d? is it the structure of his brain? is it some fundamental truth that makes no sense to us, similar to how observing forces a quantum particle to choose a state?
big hmmm
Thats a really goo remark and i think its just because we are born and live in a 3d space. There is the 4th dimension - time, but we actually cannot feel or observe it. We measure it, we live in it, we count it, but how would you imagine it? You live in it and it became such a inseparable part of our existence that it become impossible for a normal human to "think outside of the box". I bet that if you got even a glimpse of 4th space dimension, you would no longer have problem imagining it further, but in this case its like trying to imagine a color that you have not seen. It just too different from what your brain has observed up till now...
It's a really interesting question. Would a physically 4D brain hooked up from birth to a simulated 3D world like ours nonetheless somehow be capable to imagine the 4th dimension, just by virtue of physically inhabiting it?
Personally I don't think so, but I imagine to have a clear answer we would have to try en experiment with a 3D brain hooked up to a 2D simulator from birth. Problem is (ethical issues aside) I don't imagine a 2D simulation would be enough for a brain to evolve complex functions like language and spatial awareness, for us to be able to ask them to imagine another dimension.
Still, I believe the answer lies in the psychology of the brain and not in the physical substrate it is made of.
Eyes see pictures of the worlds, basically we see in 2D, but because we have 2 of them and the brain is actually smart, it can deduce the 3 dimension from that. I believe we instantly a Cube as a Cube simply because of experience thus (can be demonstrated with optical illusions too, it's quite easy to trick the brain with fake or impossible perspective). A flatlander would only see a line, so it would see in 1D, but with its experience, it could understand what a square is like, however there would be too much missing information for a cube.
now I'm really not sure, but maybe if we found a way to have eyes that can see in 3D, we could then see objects in 4D much better. A flatlander sees 1 or 2 line from a square, but we see the whole square at once, no matter what is its rotation. Our 2D eyes see only 3 faces of a cube at most, but a 3D eye could see the entire cube at once. If you had 2 3D-eyes, maybe that would be enough to fully see in 4D? In any case, I'm confident to say it's beyond human comprehension, simply because the brain is made to interpret and compare 2D pictures, not entire 3D spaces, so even if you somehow injected 3D eyes into the brain, it probably wouldn't be able to understand anything.
Tomáš Musil why 4th dimension of space? Space cannot be has only 3 dimension and 4the or 5th beside time , there might be other dimension but not space. Something we cannot understand
I got to say this is probably the best video out on RUclips to explain about basic topology!!
when ZACH asked what is CIRCLE * LINE, I answered PATREON..😜😉😎
that slide of hand you used on the moebius strip demonstration was awesome
"what is a circle times a line segment?"
_brain.exe has stopped working. windows is searching for a solution to the problem._
U describe these topic in such a way that I'll watch these a
Whole day
E
E
I correctly guessed the mobius loop and the Klein bottle and was increasingly proud of myself
I wish all math was taught like this; with ideas and visuals over just numbers.
Please do some more of those multiplication. I always had trouble with those. Your explantions help a lot to understand these better.
That explanation about the cube intersections blew my mind
Holy hell man.. this Topology and higher dimension stuff is really interesting, i have a whole newfound view of everything..
Please do more videos like this. This video was great and really interesting.
I feel enlightened. For the first time, I understand the what that Klein bottle and how that is related to 4th dimension. It's just mindblowing but unfortunately no one can ever witness the higher dimensions.
My man went from being a students counsellor(i.e major prep)to a mathematician(Zach Star) in just few years. Keep the work dude.
Question at 12:45 ... Who says {A} is only on the x axis?
Shouldn't this product give a second set of dots, with {A} on the y axis, {B} on the x axis?
Your videos are great. I like how you make sure to connect the abstract to the everyday.
This video is amazing, cleared so many of my questions about topology.
Love this video! great way to explain this concept! I've been binging this stuff on youtube for years and this helped me put a lot of concepts together. Thannk you so much! great work!
This video is one of your best vedio.
Higher dimensions always exites me.
A brilliant way of representing manifolds. I immediately predicted the Möbius Strip and Klein Bottle just from the polygon representation.
Old games here:
King's Quest 1 is a Torus (like Asteroids). The map cycles around in all 4 directions.
King's Quest 2, 3, 4 (Most of them anyway, depending on map) are cylinders like the first new example.
Very cool vid; great perspective!
The way you explain the things is far far far much better. Rather amazing.
I used to watch pbs videos earlier. But he somehow failed to explain the things clearly. I would request you to please explain those videos in your way. I am sure everyone would understand the things in perfect way.
@Zach: Now do:
Portal - multiply connected space-time. (Also, Splitgate)
Braid - closed time-like curves
Miegakure - 4D world.
Antichamber - 4D (? ) Escher-like world. (Also, Manifold garden)
Any game with scale-based maps that the player transitions between (eg Star Control II, No Man's Sky)
Monument Valley - world that can be manipulated by perspective. So is it a projection of a higher dimensional space onto a lower dimension?
Fez
Contrast - game where the player can move between 3D world and shadows projected by light sources.
Excellent video. Very interesting and informative, and a must see for every student interested in topology.
I literally saw the title and thumbnail in my feed and clicked and subbed instantly. Never made a better bet in my life
i am so happy that i recognized the möbius strip surface and the klein bottle almost instantly but SOMEHOW did not instantly recognize the cylinder because did not understand what the heck were those zebra stripes on the corners
After watching dozens of videos related to topology, this one finally gave the required intuition and fundamentals.... Thank you.. 😌
The video is really well done , definitely one of the best ones out there ... to the viewers who want to move into the 4 th dimension, it’s just not possible for us to do that in the physical human form. We lack the extra spatial dimension to move into it . It’s like a shadow wanting to be in the 3rd dimension, it has height and width but no thickness ... you however can consider yourselves to be the projections of your 4 th dimensional persona. Something like the shadow is the 2D representation of a 3D body. The shadow has no awareness of its 3D higher self but it exists when light is obstructed by its 3D self before casting the shadow on the 2 dimensional floor it wall on which the 2 D shadow is cast.
This is awesome! Very illuminating for a topic I found hard to understand at a cursory glance
I’ve played a game called “Manifold Garden” before. It’s a first person puzzle game (it’s 3D). In that game if you fall you land back on the level floating in space. You can rotate yourself in any direction and fall back into the level itself like before. I now understand the title and see every level is a 3 torus.
I’ve had this question on my mind for days now. my question is what’s the topology of a world map? Going left to right and right to left is easy to imagine, but what about going up? You don’t end up on the bottom. Think about this one, it’s tricky. Here’s an example of one of my broken theories: Imagine when going up, your x coordinate becomes its opposite (- to +, and + to -), and you get flipped around, facing South. This works, except that when walking strait into the arctic in the middle of the map, you would see the same terrain when after being turned around, walking back toward the equator (the terrain should be different!).
I just had a eureka moment. If you had 2d slices of a Klein bottle (like the animation) and moved them through time there would be no self intersection, but if you put it all at the same time, there is self intersection. We live in 4 dimensions and perceive the fourth axis as time.
I'd have never even thought of these questions. Excellent work Zach.
your channel is amazing Zach
Thank you so much, Zach, this is the clearest explanation of this I have ever seen. Now I can follow Slavoj Zizek a little better when he talks about how topology is related to psychoanalysis!
Wow this stuff really blew my mind! and the explanations were also really awesome and easy to wrap your head around!
Okay... so for 2D manifolds, you have:
>Square
>Cyllinder
>Mobius Strip
>Flat Torus
>Klein Bottle
But what about where both pairs of edges are pointing in opposite directions? Like... top and bottom are opposite, and so are left and right? Does that form something?
Hey there! I saw the answer to this in a thread nearby, and someone said that it was called the Real Projective Plane (represented as RP^2). There's a few good Wikipedia articles on the topic, so here they are:
REAL PROJECTIVE PLANE - en.wikipedia.org/wiki/Real_projective_plane
REAL PROJECTIVE SPACE - en.wikipedia.org/wiki/Real_projective_space
Wow... you've represented the "hairy" topic of topology in such an intuitive, easy to understand way... even up to the fourth dimension! Amazing!
You've just won yourself a subscriber, good sir!
I am not usually a fan of topology stuff, but I thought that this was well done.
The drawn cube has 1-D intersections, the Klein bottle has 2-D intersections. That’s made higher dimensions as clear as a mill pond. I had seen the Klein bottle many times but didn’t realise that it was a shadow of a 4-D object. Every time I had tried to visual a 4-D object my brain always plays back the clip from Terminator 2 where T1000 is in the molten lava and he kind comes out of his own mouth incidentally this looks a lot like the intersection on a Klein bottle.
i fucking love your simplistic but caring way of explaining things.
First part I didn't think to create donut by using the top bottom connection outside, which is in a way easy, like the first time I thought to twist towards each other towards cylinder insider and then imagined expanding cylinder's side walls going outward or bulging
I clicked a video about the probability of dice, now here I am 4 videos later stretching my mind in 4D trying to envision the progression by which I can imagine a Klein bottle without an intersection.
Awesomely explained! Cleared so many things
Nice video. I’m studying topology in university at the moment and this shit is mind-bending and cool.
One particular thing that I believe is worth mentioning, is that a lot of people (myself included) tend to think of the circle as a 2D shape, when in actuality it is the embedding of the topological circle into 2D space. This little subtlety, if not recognized, might make you wonder: if a circle is a 2D object, then why is the product of two circles not a a 4D object? Actually, the resulting object is the so-called Clifford torus, and its stereographic projection is the standard 3D torus.
9:00
HOLY SHIT THAT BLEW MY MIND!!!!
This video is excellent and clears many things up. Thank you
Mobio boy: A game where a boy named Mobio travels across strange lands, flipping gravity and orientation by just wrapping around places.
Me: ”It’s time to actually dive into topology”
Zach: ”I’ll make a video for you yesterday”
Didn't even notice that it is already sponsor time. Nice content as always
Can you talk about tensors? I never understood what they are. How do they relate to vectors? and how can we represent them? (for example vectors are arrows and tensors are ... ??)
I was finally able to get Sheldon's "why did the chicken cross the mobius strip? To get to the same side" joke. Thank you!
Cliff Stoll is an expert on Klein Bottles. He has been featured on Numberphile.
Zack, I think I've found your future career. I haven't heard you mention it but I feel you should go into teaching. You're great at it and, have a lot of stage presence. Which is important to keep students engaged. I know I've learned a lot from watching your videos in my free time. I think the world could benefit from it too.
P.S. I love the new channel as well. Not what I expected, very much on my level, and nice to see a different, relatable human side. Thanks for doing what you do. I enjoy it and want to see more.
This is the best introduction to topology ever.
Antichamber and Manifold Garden are two 3D puzzle games that make you think outside the box, kinda like topological problems. I suggest you should give them a try if you haven't spoiled the fun by watching a playthrough.
What about projective plane? Both pairs of edges are anti-parallel
Amazing video!! Looking forward to next one
Awesome visual presentations!
Love your content !! Keep it up!!
Explanation is so good
I thought topology is a very complex subject. Now I feel that it is damn easy. Thanks man.
now after seeing your videos it clear to me the cyclic nature of time and space explained in Vedas.
I started watching this channel for the fun little mind puzzles and now I’m trying to conceptualize the fourth dimension through numbers and variables at 4am, wtf happened