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Imagine the 2d world, where it's impossible to go UP, or DOWN.
A 3d world implies that there are infinite planes adjacent to this 2d world.
A 3d entity can see across a multitude of 2d planes,
and move UP, and DOWN, with ease.
If a 3d entity walked across the 2d world,
2d entities would only see a small 2d crossection of the 3d entity.
By comparision, imagine our 3d world,
where you can move horizontally and vertically,
but not in the 4d direction.
A 4d world implies that there are infinite 3d spaces
adjacent to our 3d world.
We can only see one. The one we're in.
If you could move 4d+ for a while,
and go back 4d- for the same while,
you would end up in the same place.
You didn't go up, down,
left, right, forward or backwards,
you went "across".
To make it easier to understand, imagine you went across several
3d rooms similar to yours, but each of them is actually a different 3d space.
A 4d entity is able to see across a multitude of 3d spaces,
and move 4d+ and 4d- with ease.
In our example, a 4d entity could be in your room,
sitting in the exact spot you are right now,
but in a certain 4d+ distance.
This entity can see where you are just by looking across,
the same way we look at a "2d paper" that is in a certain distance.
A 4d entity has a body spread across multiple adjacent 3d spaces.
If a 4d entity would stop by your room,
you would only see a small 3d crossection of the entity.
As the entity walked across your room,
the 3d crossection would keep changing until it completely passes
through our space.
Imagine a 4d object.
Let's say a 3d crossection of this object is a cube,
that you can rotate and view all sides.
If you walked on the 4d+ direction you would see that, as you walk,
the cube might change its shape.
If a 4d entity spins a rigid 4d object around,
all 3d crossections of that object will be affected.
4d entities can see all points of a 3d object at once,
just as 3d entities are able to see all points of a 2d object at once.
A 3d entity can think in 2 dimensions.
If you imagine a cube, for example, you can only imagine it in 2d,
like a picture, even if you imagine it with depth.
A 2d entity would be able to see this projection,
as long as we make it transparent.
By analogy, anything that a 4d entity sees or imagines
can be projected into 3d.
If we make this projection transparent, we can see an entire 4d object
as viewed by 4d entities, that is,
only part of the actual 4d object.
Imagine that the "field of vision" of a 3d entity, such as human beings,
is a 2d rectangle.
Let's say that the moment you look at something,
this rectangle is initially empty.
Your eyes then scan the entire 3d space in front of you,
looking for information to fill up the rectangle.
It will first detect objects that are close to you
and take 2d snapshots of them, filling up certain areas of the rectangle.
Once these areas are filled, no more information can be drawn on them,
as there is nothing else between you and the things that were already drawn.
Gradually the entire rectangle will fill up.
Imagine, now, that the field of vision of a 4d entity is a 3d cube.
This cube is initially empty.
When a 4d entity looks across the 4th dimension,
its eyes will scan the entire 4d interval in front of it,
looking for 3d shapes to fill the cube with.
The first 3d information of 4d objects,
which are closer in the 4d interval,
will fill up certain parts of the cube first, and once this happens,
no other shape will be able to occupy those parts.
Eventually, the cube will fill up entirelly.
3d entities can paint only in 2d.
We can take a piece of paper and draw any 2d form we want.
However, we can not draw real 3d shapes, that can be rotated,
using only a pencil.
By analogy, 4d entities can paint only in 3d.
A 4d entity would draw by pressing a pen-like device against a 4d object,
or a "4d canvas".
The pen would make visible marks on the 3d spaces adjacent to the
first 3d space occupied by the 4d object. That means,
once preassure is being applied with the pen-like device,
the 4d entity might move its hands through the 3d space,
up and down, left and right, forward and backwards
to draw on top of the 4d object.
Notice that this pressure is being applied across the 4th dimension.
The analogy with our world is that, when we draw something on a piece of paper,
we actually draw above it, on the planes adjacent to it,
and not "inside" the paper,
and we are applying a force across the 3rd dimension.
It is known that 3d objects generate 2d shadows on 2d surfaces,
and that 4d objects generate 3d shadows on "3d perimeters".
A 4d device that generates light is able to spread that light not only
across the 3d space, but also across the 4d direction.
If there's a 4d object in the way of the light that is moving across
the 4th dimension, certain 4d objects that are located behind it
might not receive this light.
Therefore, when the 3d volume of the 4d entity's field of vision is
filled up with the nearest 3d information,
the brightness of the points of this 3d shape will vary in intensity.
That's how 3d shadows are formed.