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.
