How can we cut a 3D Cube using 3D Cube?

Question

Cutting a 1D Line:

We can cut a 1D line via 0D point.
if we want to cut a 1D line via a 1D line.
we have to move 1 dimension up that is in 2D

(Image for 1D Line cut by 0D pt.)

(Image for 1D Line cut by 1D line)

Cutting a 2d Plane:

We can cut a 2D plane via 1D Line.
if we want to cut a 2D plane by a 2D plane
we have to move 1 dimension up that is in 2D

(Image for 2D plane cut by 1D line)

(Image for 2D plane cut by 2D plane)

Cutting of a 3D Cube:

We can cut a 3D Cube by a 2D Plane
So the question arises that
Can we cut a 3D cube by a 3D cube?
And if yes then how ?
For it we may require 1 more dimension (4th dimension).

(Image for 3D Cube cut by 2D Plane)

So how can we do that!

In all the cases sizes doesn't matter:

Size of a Line doesn't matter that if we cut a Line by another Line. And Size of a Plane doesn't matter that if we cut a Plane by another Plane.

Is it same for Cubes? Does Cubes size also doesn't matter. Can we cut a Cube by another Cube(Like a Plane cuts a Cube) and that another Cube's size can by 10x, 100x, 1000x etc ?

I think understanding how a 3D Cube is cut by a 3D cube help me to understand (or say visualize) what 4th Dimension is?

Answer 1

The short answer to your question, in your terms, is that the 3D cube cut by a 3D cube will be a be a 2D thing in the fourth dimension.

Your deeper question

I think understanding how a 3D Cube is cut by a 3D cube help me to understand (or say visualize) what 4th Dimension is?

is a hard one. Mathematicians study the geometry of $n$ dimensions by using algebra. When $n$ is $1$, $2$ and $3$ you have the number line, the plane with points $(x,y)$ and 3D space with points $(x,y,z)$. There you can match algebraic arguments with pictures. For four dimensions, work with points $(x,y,z,w)$ - but there and in higher dimensions all you have is the algebra.

The algebraic answer to your "cutting" question is that in general, if you have subspaces (what you call "cubes") of dimensions $k$ and $l$ living in $n$ dimensional space they will cut each other in a subspace of dimension $k+l-n$. That pretty much covers all your examples.

But visualizing this is not something humans are naturally equipped to do. My thesis advisor, Andy Gleason, once told me he would give a lot for one good look at the fourth dimension.

Answer 2

@EthanBolker already gave a great general answer. This answer gives one way to visualize a 3D cube cutting a 3D cube in 4D. I wouldnt say this way is generally useful, but it might be useful in the case of cubes.

Imagine the 4th dimension as "time" $t$. The first cube is $\{(x,y,z,0): -1 \le x, y, z, \le 1\}$. So this cube "doesnt exist" except at one moment $t=0$, and at that moment it exists as a regular 3D cube $[-1,1]^3$ in the $(x,y,z)$-space.

The second cube is $\{(x,y,0,t): -1 \le x, y, t \le 1\}$. First you have to convince yourself that this is a 3D cube in 4D. :) Then if you go along the time dimension, the cube "doesnt exist" except in the time range $t \in [-1, 1]$, and at any moment during that time, it exists as a regular 2D square $[-1,1]^2$ on the $(x,y)$-plane (where $z=0$).

The intersection of the two is $\{(x,y,0,0): -1 \le x, y \le 1\}$ which is a 2D square that "exists" only for one moment $t=0$, but at that moment the 1st cube also "exists" and this square bisects the 1st cube. (This square also bisects the 2nd cube.)

Like I said, making a dimension special (calling it time) may not be a generally helpful method, but perhaps it helps in this case?