Paradox of the Möbius Strip and Klein Bottle - A 4D Visualization

How does your brain process  what you see on the screen? To you, there could be 6 triangles  arranged to form a hexagon. Or you could see it as a hexagon whose outer  vertices are all connected to the central vertex.

Both ways of seeing are  correct, but how about now? Now, your brain is thinking: there's no hexagon at all! It's a cube!

Suddenly your mind went from 2D mode to 3D even  though you're still looking at a flat-screen. You understand that some  edges have to intersect on the screen you're watching to represent the cube. Despite the intersections, you know  none of a cube's edges actually overlap.

The way we look at the world in 2D mode is  totally different from how we see it in 3D. But will humans be ever capable of  comprehending 4 dimensions and beyond? For the sake of clarity, let's define a few terms.

Topology. It's a branch of mathematics concerned  with the properties of a geometric object that are preserved  under continuous deformations. For example: the number of holes that a doughnut has No matter how much we stretch, twist, crumple, or bend it, it will always have one hole.

Manifold. In simple terms, it's anything  that can exist in any dimension. For example, one-dimensional  manifolds include lines and circles; two-dimensional manifolds include  open surfaces such as a plane, topological versions of the plane, a disk, and an annulus, and closed surfaces such as a sphere, and a torus.

Three-dimensional manifolds include  a ball, also known as a solid sphere, a solid torus, or commonly a doughnut, and a tetrahedron. On the other hand, a boundary, for our purposes, refers to an abrupt stop to a surface. For example, the boundary of this  plane is the four edges surrounding it, and the boundary of this annulus  is the two concentric circles that define its shape.

Applying topological transformations  does not remove or add new boundaries. A sphere or a torus, on the other hand, doesn't  have a boundary since they are closed surfaces. Now, consider an open 2D manifold.

It's easy to see it has two sides, one on the top and the other on the bottom. If an ant is on one side, it must cross  the boundary to reach the other side. Here's a riddle: Can you come up with a 2D manifold that has only one side?

That is, where an ant on the  top side can reach the bottom side without crossing a boundary? And is that even possible? Pause the video now if you want to think about it.

A particular way to do this is to get a  strip and join one edge to its opposite. But before doing so, give the strip  a half-twist, then connect them. The resulting one-sided surface  is called a Möbius strip.

To prove its one-sidedness, notice that an ant crawling along one side of  the strip will eventually reach the other side. In other words, the top and bottom sides  are actually the same side. Zooming in, we can see the  boundaries on either side appear to run relatively parallel along  the whole length of the Mobius strip.

And as we know, parallel lines never intersect. But taking a step back, we see that the strip only has 1 continuous boundary. Isn't that trippy?

Another cool thing you can do to Mobius strips is cut them in half. We have this notion that cutting  anything in half, well, will result in two separate parts of the thing we're cutting. But take a look at what happens when we try to cut the Mobius strip in half.

It doesn't split into two! But how? Keep in mind there is only  one continuous boundary.

Since the cut in the middle  never gets to the boundary, the strip doesn't cut into separate pieces. Another way to intuitively visualize this is to first cut the original strip in half, bend so the edges can meet, give a half-twist, and then finally join the opposite edges. If you think that's cool, consider cutting one-third along  the width of the Mobius strip.

Cutting along it, we find it takes 2 trips around the Mobius strip for the  cut to form a closed loop. Separating the parts, we get this. Now that's real trippy.

But we can visualize this by first doing the cut, the twist, and then joining the edges. Now in a seemingly different direction, ever heard of the grandfather paradox? The grandfather paradox is usually  used to disprove the existence of time travel.

The idea is that if you were to travel  back in time to kill your grandfather before he had your mother, then it would be impossible  for your mom to have been born. So you wouldn't have been born to travel back in  time and kill your grandfather in the first place. Thus, it's a paradox!

And some people would assert  that because of this potential logical inconsistency, time travel cannot exist. But, there's supposedly a  clever way to resolve this. And to do that, you guess it, we use a Möbius strip.

Let's represent a timeline on  the surface of a Möbius strip. Suppose, in 2021, you gain access to  a time machine for unknown reasons, and you decide to travel back in time  to kill your grandfather when he was still a child—still for unknown murderous reasons. So now you're back in 1940,  and you kill your grandfather.

If your grandfather is dead as a kid in 1940, in 1970 your mother can't have been born, and in the year 2000 you can't have been born, so in 2021, there was no you to travel back in time. Now here's where it's different. We just simply move past this  paradoxical point and see what happens.

If there's no you to travel  back in time in the first place, then there's no one to kill  your grandfather in 1940, so he would be alive to have your mother in 1970, and your mom is present to give  birth to you in the year 2000. And by 2021, you use a time machine  to travel back in time to kill your grandfather, and the cycle repeats forever. Now we see that the whole surface of the Mobius strip is written over.

And a nifty thing about using the  Mobius strip is that opposite events like "your mother is born"  and "your mother is not born" are precisely on the locally  opposite sides of the Mobius strip. In 2000, you were born and not born at the same time. In 2021, you travel back in time and do  not travel back in time simultaneously.

In 1940, you kill your grandfather and not  kill your grandfather at the same time. Each event and its opposite is happening at  the same time along the length of the strip. It looks much more of a paradox than  the paradox it is trying to resolve.

Still, when you restrict your thinking along  the looping events happening at the surface, the logic behind the causes and effects seems to blend smoothly. It's become a causal loop. So the reason why you had to kill your grandfather in the very first place?

You did that so you could live. Isn't that amazing? Let's take this to a higher level.

Consider a sphere. Like a rectangle, it is a 2D manifold, but it is a closed surface  because it has no boundary. Moreover, we can also take it that it has 2 sides: one on the outside and the other one you can't  see unless we open it up, the inside.

Here's another riddle: Can you think up a closed surface 2D  manifold where an ant on its outside can just walk freely to get to the inside  without passing through the manifold? And is that even possible? Pause the video now if you want to think about it.

It sounds impossible. But is it? Well, yes and no.

Yes, it is possible, but no, since  it can only exist in 4 dimensions. But still, we can try to see what it  looks like to our 3-dimensional brains. Are you ready?

This 2D manifold is called a Klein bottle. It is a closed surface with no boundary. If an ant were walking along its surface,  it would be able to reach the inside and outside without ever encountering  a boundary or falling over an edge.

That's because, as mentioned, the  outside and inside of the Klein bottle are one and the same side despite  it being a closed surface. Notice that there's a part where the Klein bottle appears to intersect itself. This is only a manifestation of  the limits of 3D visualization.

Here's an analogy: At the beginning of the video, we saw a 3D cube. We understood how certain points have to  intersect on the screen to represent it. Yet despite these apparent intersections, we  know none of the cube's edges actually intersect, and we can precisely apply the same idea here.

In addition, realize that you are looking at a 2D image of a 3D representation of an object that can only be embedded in 4 dimensions. Trying to explain the 4th dimension is like  describing colors to a person blind since birth. There's no way to actually visualize it correctly.

But we can try. So far, we've only talked about dimensions that are spatial, that is,  dimensions that relate to space. But there's another dimension we can understand: the temporal dimension, also known as time.

Imagine a 1-dimensional line named Linus trying to comprehend the 2nd dimension. How could he tell apart a disk from a triangle? Well, one way is for these 2D shapes to  slowly cross Linus's plane of existence, or in his case, his 'line of existence'.

As a disk crosses this line,  Linus would grasp it, at first, appearing as a single point that gets longer and longer until it gradually stops. Then gets shorter and shorter until it disappears. For the triangle, on the other hand, first, Linus would comprehend it  appearing suddenly as a long line that gets shorter and shorter until it disappears.

Basically, we just cut these 2D shapes into lines  that Linus can comprehend one slice at a time. TIME. Since Linus can only understand  up to 1 spatial dimension, he needed the passage of time or  a temporal dimension to compensate for his lack of comprehension for a 2nd dimension.

In other words, instead of space, he  is using time for the 2nd dimension. Next, imagine a 2D square named Squirrel trying to comprehend the 3rd dimension. How could she distinguish  a ball from a tetrahedron?

Well, she follows the same process. As the ball slowly crosses into  Squirrel's plane of existence, she would grasp it, at first,  appearing as a single point, turning into a disk that gets bigger and bigger until it gradually stops, then becomes smaller and smaller where it then disappears. On the other hand, as the tetrahedron crosses her plane of existence, she would comprehend it  appearing suddenly as a triangle that gets smaller and smaller until it disappears.

In the same manner as before, what we basically did was slice up these 3D objects into flat 2D  shapes that Squirrel can comprehend one slice at a time. Do you see where this is going? Going back to the Mobius strip, it's a 2D manifold that can only be  embedded in at least 3 dimensions.

If we were to restrict it to two dimensions, basically to squash it flat  and leave only its boundary, it is clear this curve intersects itself right about here. If we lift some part of it into the 3rd dimension, it will no longer cross itself! It appears that adding an extra dimension  will remove the self-intersection.

When squirrel hangs around nearby, she can see the boundary of the Möbius strip one slice at a time. To her, points that supposedly intersect, that is, occupying the same space, now occupy different points in time. So she’s able to understand that the  curve in 3D space does not self-intersect even if its 2D representation does.

Finally, we are ready to see the Klein bottle in its full glory. Like the Mobius strip, the Klein bottle is a 2D manifold but  can only exist in at least 4 dimensions. What you see here is only its 3D representation.

And just like what we just saw, we can remove the apparent  self-intersection of the Klein bottle by lifting a part of it to the 4th dimension. And to visualize the 4th dimension, we cut up the Klein bottle to see one 3D slice at a time with the help of the passage of time.