We hear it all the time, that our world is three-dimensional.
It makes sense after all,
we can only define up to three directions each at a right angle to each other,
they don’t interact with each other.
No matter how far left or right you go,
you won’t move up at all.
But the moment we try to add another dimension into the next,
it all breaks down.
No matter how hard you try, in our three-dimensional universe,
you can’t lay out four or more directions that won’t interact.
Now I know what some people are thinking:
“Can we consider time is the dimension?”
In simple terms, it does make sense.
We can measure time and tell one point in time from another in a linear sense.
Four hours later is always four hours later,
just like how four feet to the left is always four feet to the left.
Hey, you could even make time act like spacial dimension.
Take a flipbook, each frame of the animation is a two-dimensional image,
but by traveling through the 3D stack of these two-dimensional frames at a consistent rate,
we can make time act like a spacial dimension,
where height in the stack of 2D frames is the same as time between two events.
In other sense, this 2D animation just became 3D, right?
换句话说 二维动画变成了三维的 对吗？
Just because time can match up with the spacial dimension doesn’t mean that it is one.
Notice how in the animation time only moves one way,
from the bottom of the stack to the top.
But in regular 3D space,
one can move either way: up or down.
This doesn’t mean it’s wrong to imagine time as a spacial dimension
since it clearly can be done,
but rather it’s wrong to assume a spacial dimension and time are exactly the same.
So now that’s cleared up:
what would four-dimensional space be like?
The easiest way to answer that question is
actually to do the opposite.
Rather than add a dimension in,
let’s take a dimension out.
Try to picture what it would be like to live on the surface of a sheet of paper.
You can no longer jump up or down off the paper,
because up and down don’t exist to you anymore.
You can’t look up off the paper either.
All you can do is look forward across the paper.
When you do look in front of you, you’ll see a horizontal line.
And in that line, you’ll see the edges of other 3D figures.
Imagine a circle in a box,
clearly to us this is three-dimensional beings.
We can see the circle in the box.
But to a two-dimensional being,
the size of the box would prevent anyone from seeing inside.
Also consider that whenever something casts the shadow onto, say a wall,
the shadow forms a line on the wall.
Of course to you, all the walls will be lines anyways,
so you wouldn’t really notice.
Let’s bring that same situation into three dimensions.
As you look around, all you can see are two-dimensional objects,
just like how even a box in two dimensions look like nothing more than a line.
Since you can only view it from one side at a time,
the front of the box on three dimensions only looks like a square.
Also, suppose the ball was still in the box.
Now, sphere inside a cubic box.
We still can’t see it now because all the sides are still covered.
The shadow the box casts on the wall looks different.
Instead of just being a line like before,
now it’s a flat 2D shape.
If you wanted, you could even trace it on the paper
and get a 2D drawing out of it.
Here’s where things get tricky.
Let’s try to add a new dimension that we haven’t ever seen before.
Let’s just dip our toe into the four dimensional water,
and use the differences between 2D and 3D space
to determine what would happen in 4D space.
One of the first things you would notice when you looked in a 4D space,
is that you would see the world is a 3D shape.
In a similar way to how you view a 3D world and can put it onto a 2D picture.
But there’s one catch: you can see all the sides of a shape.
Remember how if you were to put a circle on a 2D box,
you could still see it looking down from above in 3D?
while now in 4D,
you can see every side of the 3D box when looking down on it,
including seeing the ball inside.
Are you freaked out yet?
After you get over the bit of wonder
you experience being able to see inside of the box,
you might also notice that another 4D box is casting a shadow on the wall.
And the shadow, it’s a 3D shape.
Yes, just like how 2D space has linear shadows,
3D space has 2D shadows,
here in 4D space, we have three-dimensional shadows.
But what would the shadows look like?
Here’s an example.
This is called the tesseract,
which is the shadow you would get from casting light through a 4D cube.
Notice how it’s made of two cubes connected at the vertices,
just like how you can draw a cube on paper from two squares
and connecting the corners.
Now the tesseract looks weird enough sitting still,
but watch what happens the moment you start rotating it.
I’ll give you a moment to clean up you mind,
which is most likely at this point exploded.
Yes, that’s what the shadow of a 4D cube looks like
when they keep rotates.
Why does it do this?
Where the middle cube stretches out to become the outer cube?
Let’s take a look at the shadow of a 3D cube.
First, note the two squares,
one is smaller than the other because of perspective,
making further objects appear smaller.
If you follow the small square,
you’ll see that it moves to the other side.
before eventually moving to the front of the cube, and becoming the big square.
Now, watch the tesseract again.
The little cube because of 4D perspective
when rotated to be closer becomes bigger.
where the big cube rotates to be smaller.
So just like in a 3D cube where the big and little squares switched places,
in a 4D cube,
the big and little cubes will switch places.
All in all, four-dimensional space is not all that bad.
In fact, with the proper understanding of what will happen,
one can devise some clever shapes
that do things we can’t even imagine doing
in the 3D space.
Ever hear of a Mobius Strip?
It’s a plane that thanks to being bent around in 3D space,
has only one side.
Because what would normally be the top and bottom sides are now connected,
making them the same.
Things to a little 4D trickery, a figure called a Klein Bottle can exist,
where the inside and outside are the same.
Ever notice how you can mere writing
on the page by flipping it over?
In 4D space,
you can mirror a whole 3D object just by flipping it over four dimensionally.
4D space shows us what is arguably the best part about mathematics,
it works regardless of your bias.
In a previous video,
I showed how we could use two digits to represent any value that 10 digits came.
And in a similar way,
just like how we can do any mathematical function
possible in base 10, in base 36,
we can have geometry that functions consistently,
whether it has three dimensions? Four dimensions? or possibly even more.
不管是在三维 四维 甚至更多维空间