Tesseroid - Notes On The 4th Spatial Dimension

This section is a quick summary of concepts that are almost a pre-requisite before you venture yourself into more advanced topics. Some of the things covered here will be discussed with more detail on future articles.
Note #1: If you're new to the 4th spatial dimension, I suggest watching Carl Sagan's video first for a quick introduction. Note #2: For easier understanding, the 2d world described in this particular topic is similar to the one in Carl Sagan's explanation (it has no height - it is similar, for example, to the world in a 2d console RPG). However, for future topics, I'll refer to the 2d world as a world where it's only possible to move up, down, forward and backwards (similar to a 2d platform game).

The 2d World

Imagine a 2d world, where it's impossible to go "up" or "down".
Let's assume there's some "force" holding everything on that particular plane.
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/down (vertically) with ease.
If a 3d entity walked across the 2d world, 2d entities would only see a small 2d crossection of the 3d entity at any given time. In fact, they would only see the contour (or shape) of that 2d crossection.
In a 2d world, some or all objects might be the crossection of 3d objects.
2d entities cannot move anything up/down (without using technology) because they can't produce any horizontal action that would have a vertical reaction.

4d Entities

By comparision, imagine our 3d world, where you can move horizontally and vertically, but not in the 4d direction (let's call it 4d+ and 4d-, or "across").
Let's assume there's a force preventing 3d entities from moving in 4d+/4d-.
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.
Imagine the room you're in right now. 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, or right, 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.
If a 3d observer would wait in your room while you crossed in the 4d direction, he would simply see you disappearing, and then sudently reappearing after a while.
A 4d entity is able to see across a multitude of 3d spaces, and move 4d+/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 from us.
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 completelly passes through our "space".
In a 3d world, some or all objects might be the 3d crossection of 4d objects.
3d entities cannot move anything 4d+/4d- (without using technology) because they can't produce any spatial action that would have an "across" reaction.

4d Objects and Movement

The same way 3d entities vary in size and shape, 4d entities vary the amount of "4D size" they occupy.
The same way we can stretch our arms through a certain 3d distance, 4d entities can move part of their "bodies" across several 3d spaces, or across a certain 4d distance.
Let's imagine a 4d object. Let's say a 3d crossection of a certain part of this objects is a cube, that you can rotate and view all sides. If you walk on a 4d+ direction you would see that, as you walk, the cube might change its shape. This might look similar to animating a 3d object on the computer, and at each frame, the object can have a different shape.
But wait, let's go back to 2d: imagine that a 2d entity has a computer, and it wants to make an animation of a rotating cube. Could this entity just draw a 2d representation of a 3d cube, in the same way we draw a cube in a piece of paper, and animate it frame by frame so the cube appears to be spinning around when you press play?
No. Remember that in a 2d world, there is no height. A 2d entity would be able to draw one line at any given time. So to see a simple 2d picture of a cube, it would be necessary to display one "line" of the picture at a time. That means they would have to make an animation only to see a simple picture of a cube.
One way around this would be to make the 2d projection of this picture "transparent". Instead of looking at the cube from our position, however, a 2d entity would see it from one of the sides. This way, with a transparent projection, 2d entities can easily understand the animation of a rotating cube.
Similarly, we (3d entities) can visualize a 4d object by making a transparent 3d projection (more of it on the next section).
Notice that there's a definite start and end to any 4d object or entity, and it is NOT the same concept as "time". A piece of the 4d object that is near the initial "4D position" is in the exact same time as a piece that is near the ending "4D position" (generally).
A 4d object occupies a certain "4D size" across. To be considered small, a 4d object would have to be small not only in 3d volume but also in "4D size" in comparision to the 4d entity that is manipulating it.
A 4d entity can move 4d objects across the 4th dimension as well as through the 3d space.
If a 4d entity spins a rigid 4d object around, all 3d crossections of that object will be affected, but it can also fold a soft 4d object (like you would fold a piece of paper) through the 3rd and 4th dimensions, which would change the shape and/or position of a multitude of the 3d crossections at once, but not necessarily across the entire 4d interval.
Some of the 3d crossections can be brought together on the same 3d space of other 3d crossections of the 4d object. They can even touch each other, but can not occupy the exact same space.

Point of View of a 4d entity

4d entities can see all points of a 3d object at once, just as 3d entities are able to see all the points of a 2d object at once (since there is no such thing as the "back" of a real 2d object).
Think about it this way. A 2d entity can only imagine in 1 dimension. If a 2d entity sees a circular shape in front of it, it will still only see the countour of that shape, which is a line.
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. To imagine all sides of a cube, you would have to imagine a cube spinning: that is, in reality, several 2d images in sequence.
You can only see "part" of a cube at any given time (let's call it the "first 2d information" you see).
Let's say you are thinking about a non-moving cube: that's a 3d object being projected into 2d. A 2d entity would be able to see this projection, as long as we make it transparent. This 2d entity is seeing the entire cube as we (3d entities) "imagine" it.
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 (or what is on their field of vision, that is, only part of the actual 4d object).
Remember that 3d entities can't see all of the 3d objects at once: only the first 2d information. A 4d entity also can't see all of the 4d object at once, but only the first 3d information. For example, if the 3d crossection of a 4d object is a cube, and all the consecutive crossections are shapes of equal or smaller volume located on the same spatial position, the only information that will be interpreted by the 4d entity on that particular volume of its 3d field of vision is the first 3d crossection (the 3d cube).
Here is an example that will sum this section up in a way that is easy to understand. Imagine that the FOV (field of vision) of a 3d entity (you) 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 FOV of a 4d entity is a 3d cube. This cube is initially empty. When the 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 that 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.
With the above reasoning, imagine that a 4d entity is simply observing a "grass field" and the sky. At first, the 3d FOV is empty. Then, the 3d space closest to the entity fills up a small portion of the FOV: imagine that the bottom of the "cube" that represents the 3d FOV is filled up with part of the ground (with a small height). Now, suppose the next 3d space is similar to the first one: since this crossection is a little further away from the observer, it will be formed just a little above the previous crossection. Repeating it until the last possible 3d space, the half bottom of the FOV will now be filled entirelly. Since there is nothing now between the observer and the sky, the rest of the FOV will be filled with it.

4d Boundaries / Walls

If a 4d entity can move across 4d+/4d-, how would you "trap" a 4d entity? What would a small "prision cell" look like?
Let's take a look at the 2d world again. To trap a 2d entity, you simply need several 2-dimensional walls to surround it. If you would do the same to a 3d entity, however, it would simply walk "above" the walls, even if these would had some "height" (3-dimensional walls).
You can only trap a 3d creature by closing the entire space, and not leaving any hole on any horizontal or vertical direction.
The same reasoning applies to 4d entities. Let's say you have a small 3d room, without any windows, and only 1 door. If you place a 4d entity in the room and close the door, it would simply move across 4d+ and escape. Even if the walls would have some "4D size", the entity would simply look on the 4d direction and "see" where the walls "ends".
So, in order to trap a 4d entity, you would first need to determine the "4D size" of the room, as well as the spatial size. Then, you would have to place a solid block, with the exact volume of the room (or bigger), on the "4D edges" of the room. The "4D size" of this solid block corresponds to the thickness of a 3d wall.
This solid block also can't have any "hole" that is bigger than a 4d entity. A 4d entity is then able to "see" the block, but not pass through it.
Let's say one of the walls, at some point in that 4d interval, is smaller than necessary to block the room from the outside world. If the interval is too small, in relation to the aprisioned 4d entity, this entity might not be able to pass through that hole.

Drawing in a 4d World

As you already know, a 2d entity can only draw in 1 dimension. It can simply draw dots or lines (with no height).
Let's look at our world. We can take a piece of paper and draw any 2d form we want. However, we can't draw "real" 3d shapes, that can be rotated, using only a pencil. We can construct any 3d object and paint its surfaces, but that would not be a drawing anymore.
We also can't draw a line with absolutelly no height, it would always have a height, no matter how small. That means 3d entities can paint "only" in 2d.
By analogy, 4d entities can paint only in 3d. Even if a 4d entity attempted to draw in 2d, a small 3d information would always be present. A 4d entity can't draw in 4d, only construct in 4d.
A perfect 3d canvas would be a 3d object that mimics an unfolded 2d plane with a big enough area (a piece of paper, for instance).
A perfect 4d canvas would be a 4d object that mimics an unfolded 3d realm with a big enough volume (something close to a 3d cube, occupying a small 4d interval).
A 4d entity would draw by pressing a pen-like device against a 4d object. The "pen" would make visible marks on the 3d spaces adjacent to the first 3d space occupied by the 4d object. That means, once pressure is being applied with the pen-like device, the 4d entity might move its "hands" through a 3d space, up/down, left/right, forward/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 in a piece of paper, we actually draw "above" the paper, on the planes adjacent to it, and not "inside" the paper, and we are applying a force across the 3rd dimension.

Light and Shadows in a 4d World

It is known that 3d objects generate 2d shadows on 2d surfaces, and that 4d objects generate 3d shadows on 3d perimeters. But how would a 3d shadow be formed? This is actually easy to understand if you grasped the concept of the 4d entity point of view.
A 4d device that generate light is able to spread that light not only across the 3d space but also across the 4d direction. That light would shine very bright on adjacent 3d spaces and loose power as you move across the 4th dimension.
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 FOV is filled up with the nearest 3d information, the brightness of the points of this 3d shape will vary in intensity.
To have a perfect 2d shadow of a 3d object, you need a straight 2d surface with area big enough to acomodate that shadow. This shadow would have almost the same shape as a 2d snapshot of the 3d object.
To have a perfect 3d shadow of a 4d object, you need a solid 3d perimeter with volume big enough to acomodate that shadow. If the volume is not enough, the shadow will be spread across multiple 3d spaces, thus looking distorted.
A 3d entity might not be able to see a shadow of a 3d object when this object is between the observer and the shadow. This is also true for a 4d entity. Depending on the position of the 4d entity, the shadow of the 4d object might not be visible on any part of the 3d FOV of the entity, which means it can not be observed from that position.

Notes On The 4th Spatial Dimension

Basic Concepts of the Fourth Spatial Dimension