eli5: what is 4D space?


I just dont know what Is 4D space and i can’t imagine it…

In: 5

4D space isn’t something that we can intuitively imagine because it’s not a part of our sensory perception.

There are various ways to think about it though. For example, imagine you need to graph something in 4 dimensions. If you know what a 3D plot looks like, imagine you add color to the data being plotted. The color is the 4th dimension.

Another way to visualize it in your mind is by imagining a cube. The cube is 3D. Now imagine there are infinitely many more cubes to its left and right (like a number line, but made of cubes instead of numbers), that’s 4D space.

If the 4th dimension were time, you can imagine each 3D cube as one specific moment in time.

Well, 4D doesn’t work well with our minds unfortunately. The best way to imagine it is via its 3D representation.

Take a 3D cube. It’s got three dimensions. They’re height, depth and width. It’s easy to imagine what would happen if we changed any of those dimensions of the cube, so I won’t go into that.

Now let’s say we have a 4D cube. Just like the 3D cube it’s got width, depth and height. But it’s got one more, because we said it did. This could be anything though. 4D space doesn’t exist as far as we know, so there’s no reason to assume that our 4th dimension is a spatial dimension.

Time, for example, is a perfectly valid 4th dimension. Changing the cube’s time dimension might mean it looks pristine at t=0, and old and crumbly at t=10000. That’s one 4D cube.

Or the 4th dimension could be sphericalness, in which case varying the 4th dimension could change how spherical it seems.

Or the 4th dimension could be *volume*, independent of outer dimensions. A big 4th dimension would mean that somehow it’s huge on the inside.

I like to think of black holes as hypothetical examples of 4D space. Due to gravitational spacetime warping, it’s not unreasonable to expect a black hole to have a much greater volume inside of it than the outer 3 dimensions would allow.

Think how differently life would be in only 2 dimensions, and how easily you could see everything going on if you were a 3rd party observer.

Now bring that up one level and you can sort of imagine what it might be like, even though we can’t actually imagine or perceive it.

Personally I do feel like time might be a dimension, as 3D worlds moves through it (albeit only one way) – in fact I’ve always thought of time as the fourth dimension in my minds eye.

But time could be the 5th or 11th dimension too, and we can’t just perceive or visualize the ones in between.

In a 3D space, you need three values to identify any particular point in it: `(x, y, z)`, like `(1, 2, 3)`. In 2D space you only need `(x, y)` coordinates. In 4D space, you simply need one more value, like `(1, 2, 3, 4)`. You don’t necessarily need to visualize it as physical axes, that additional value could be anything. For example time. Or if you imagine any `(x, y, z)` point to be a cube itself (picture a warehouse of cubes), then the forth value could denote the side of the cube. Or you don’t imagine it like a “space” at all, and just take it as a filing system in a library: “1st floor, 2nd quadrant, 3rd shelf, 4th section”.

If you stop trying to picture *n*D coordinates as physical axes at all and just take it as “you need *n* values to identify any one location in this space”, it might make more sense.

Nobody can we are built for 3D thinking.

So what do we mean by dimension?

The number o basis vectors required to reach every point in your space is the dimension number. (You can define dimensions in other ways but this’ll do.)

So what are basis vectors?

Imagine a line. Each point on the line can be reached with a vector pointing to it from the origin. So lets say we define what 1 unit is. Lets say the basis vector is 1 unit in lenght. It points from 0 to 1. Lets call this v1. Every point now can be defined as some number c times that vector v1. So two would be 2v1, -6 would be -6v1.

When we have more dimensions it means that we need v1 and v2. Now v2 has to be linearly independent from v1. This means that v2 isn’t a multiple of v1. Make them have an angle between them and this will hold true. Like make them perpenticular. Now each point on the plain can be reached with the linear combination of the two. This is basically: any point = c1v1+c2v2. This is almost like coordinates but we can add and multiply in comfortable ways. So a point (6,2) is 6×v1+2×v2.

We can add a third independent vector v3 which cannot be reached with a linear combination of v1 and v2. So it points out of the plain. Together they stretch out the 3D space.

Mathematically there is nothing weird about adding a 4th basis vector each point will have 4 coordinates, and we can reach any point by doing c1v1+c2v2+c3v3+c4v4. You can continue this as long as you want.

So that is a 4D space its streached by 4 basis vectors.

In physics you can treat time as a 4th dimensions. You have 3 spacial directions and 1 temporal. Since a lot of motion happens in a plain like a planet orbiting a star we can drop 1 spacial dimension and look at the orbit in the x,y plain and add a 3rd dimensions as time. So in this 3D diagram the orbit looks like a spiral. So that 3D diagram is the path of your object through space and time. This is called a worldline.

Now why you might want to do this is because it allows you to use different math. For moving objects you got dynamics we deal with that if we need to figure out the path of an object. It involves solving usually difficult equations. But lets say we know the path and we want to calculate different things, like time dilation in special reactivity. In general the whole path through space and time gives you a 4D diagram, but since time is one directional it behaves differently than spacial dimensions we often call a diagram like that a 3+1 spacetime diagram. But once you do that your framework is static geometry, things dont change, nothing moves its just lines, curves and coordinate transformations.

So using a 4D or higher space can be helpful in physics. Phase diagrams for example plot position and velocity. Its a good way to visualise what a system is doing. Like a pendulum will oscillate around its stabil point, and that appears as ellipsis or circles on the phase diagram. Think it through the pendulum swings left and stops. It has 0 velocity and maximum negative x direction so its in -x then it moves back to 0 gaining velocity in 0 it has +v velocity then it slows down at +x with 0 velocity and moves back with -v velocity. It draws a circle around 0. Of course if we can let that pendulum swing in any direction our phase diagram got x,y,z and v_x,v_y,v_z. Giving us a 6D phase diagram. And again looking at the curves we can geometrically understand how the system behaves. Identify stabil and instabil equilibrium states.

So higher dimensional vector spaces are like most things mathematical tools we can use.