Let’s, for a second, not worry about anatomy and other such mundane things.

What do we use staircases for in 3D space? To go “up”, I guess. To get from one floor to the next.

Let’s start with 2D ladders. Imagine this simple Donkey Kong-esque scenario:

The orange lines are floors, they extend all the way from -∞ to ∞ in the x direction. 2D beings can use the ladders to go from one floor to another in the y direction.

Let’s extend this into 3D space:

The orange floors extend all over the x and y directions. We can walk in every (x,y) direction and we can take a ladder to go up or down the z direction.

Intuitively, this makes sense.

Now, what if we extend this into 4D space?

Now the floors are cubes extending infinitely in the x, y and z directions. Notice that we can walk all around the cube (including the “inside”) to take a ladder to go up or down the w dimension. In 4D space, we can walk to every (x,y,z) coordinate of the cube without any magic tricks or without going through any solid material. We can walk around “on” the cube, just like we can walk around on a flat surface in 3D space.

Actually, we cheated a little bit. In the previous exercise, we didn’t really think about the shape of the ladder. Let’s not worry too much about this and just think of the ladder as a rope, extending in exactly 1 dimension.

Staircase time.

A staircase in 2D exists of stacked 2D rectangles, each a bit “taller” than the last (in the y dimension), placed next to each other:

In 3D, they’re stacked boxes, each a bit taller than the last in the z dimension:

In 4D space, they’re 4D boxes (tesseracts), each a bit taller than the last in the w direction:

(Of course, they’d need to be placed right next to each other to form a useful staircase, but for clarity I drew them apart)

Now you decide to walk up the staircase. In 2D this means the 1D sole of your shoe touches the 1D top of the stair. Every point touches every other point.

In 3D, the 2D surface of the sole of your shoe touches the 2D surface of the stair.

In 4D, the 3D volume of the sole of your shoe touches the 3D top volume of the stair. Every point of the the volume touches every other point of the volume. This would of course be impossible in 3D space, as the objects would be overlapping. In 4D space, this is perfectly possible.

All we’re really doing here is inferring properties of 4D space by looking at the relation between 2D and 3D space, both of which we are, conveniently, able to visualize.

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