Tesseroid - Notes On The 4th Spatial Dimension

Basic Concepts of the Fourth Spatial Dimension

2009-01-29T19:48:00.001-08:00

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.

Basic Hyper-Objects

2009-01-28T19:48:00.000-08:00

In this section you will learn about the most basic 4d shapes and objects, and how they are perceived both by us and by 4d entities. There are several methods that can be used to visualize 4d objects, but the one used here is the "crossections" method.

To exemplify this, imagine a 3d cube, which is bounded by 6 faces. Now, imagine that the cube is sliced into infinite squares, each one with the dimensions of a single face. The only faces bounding the cube that are important in this case are the front face (the one facing to the observer) and the back face, since they are also slices. If you rotate the cube, you are actually rotating these slices. This same method can be applied to 4d objects, except that instead of slicing the object into 2d faces, we are slicing the objects into 3d cells.

The Tesseract (Hypercube)

The tesseract is the 4d equivalent to the 3d cube, and is the more talked-about 4d object. However, many people don't seem to understand it, especially when seeing animations of a rotating hypercube. This section will demonstrate, through simple comparision with the 3d cube, how we can easily imagine a rotating hypercube, and how it is not so different than imagining a rotating 3d cube. Make sure you understand the Field of Vision section of the previous article before proceeding.

But first, let`s take a look at the 2d projection of a 3d cube. That looks just like an ordinary 3d cube drawn in a piece of paper. We (3d entities) can see up to 3 faces of the cube at same time, and we can see all the points of this projection at the same time as well. However, if this same projection was brought into a 2d world, 2d entities would only be able to see one "line" at a time: remember that they are also 2d shapes, and they would have to stay by the side of the 2d projection of the cube in order to see it (they can`t see the projection from the "front", like we can). So the only way for them to see all the points of this 2d projection at the same time, like us, would be to make this projection transparent. If the angle of the 3d cube is such that we can only see the front face, then the 2d projection will be a perfect square, and to be seen in a 2d world, it would be a transparent square.

Taking that analogy into the 4d tesseract, if the angle of the tesseract is such that 4d entities can only see the first cell (which is a 3d cube), then the 3d projection of the tesseract is merely a 3d cube. But in order for us to see all the points inside that cube, like 4d entities can, we have to make this projection transparent. So a transparent, 3d cube is the 3d projection of an opaque tesseract. Then, we can just project this transparent cube into 2d like we normally do.

But what about transparency? Well, if we look at a transparent 3d cube directly from the front, when the shape of its 2d projection is a square, we will see a big square with a small square in the middle, and four lines connecting the edges of both squares. The square in the middle is actually of the same size of the outer square, but because of depth, it looks smaller to the observer.

The same thing is true for the tesseract. When the transparent tesseract is being observed from the "front", the 3d projection will be a 3d cube with a smaller 3d cube in the center, and all their vertices will be connected. This is the most common image of the projected tesseract on the internet, but most people don't know that they are looking at a transparent tesseract, and not an opaque one. Notice that, the true 3d projection of a transparent tesseract would also be opaque, but in order for we to observe it, we make this projection transparent.

If you understood the concept described above, then imagining the rotations of a tesseract won`t be so complicated. All you have to do is take the 2d projection of a rotating cube and transform it into a 3d shape. That will be the 3d projection of a rotating tesseract.

For example, imagine that you are looking at a transparent 3d cube from the front. The 2d projection will be a square with a smaller square inside. If you start rotating it, without ever changing the orientation of the top or bottom face, the small square in the middle will start to flatten and start "moving" to the side, and the outer square will flatten and go to the other side. At some point, one of the squares will look entirelly flat, like a line. Eventually, both squares will be actually on the "sides" of the cube, and the front face is now formed by a square that was previously on the side. Notice that, while the cube is rotating, the 3d projection is not a perfect square at all times, but only when just one of the faces is on the front.

The exact same thing happens to the tesseract. The 3d projection of a transparent tesseract, when only one of the cells is brought to the "front", is a large cube with a smaller cube in its center. As the tesseract is rotated, the smaller cube will start to flatten and move to one of the sides of the cube, and the outer cube will start to flatten and move to the other side. At one point, one of the cubes will look entirelly flat, like a square. Eventually both cubes will be actually on the sides of the tesseract, and the front cell of the cube is now formed by a cube that was previously on the side. Here, the 3d projection is also constantly changing shape, in a similar fashion to the 2d projection of the cube, and only at some points it forms a perfect 3d cube. With this method, is very simple to imagine a rotating tesseract.

As you can see, "following" one of the 3d cubes (cells) of the rotating tesseract while it rotates is just like following one of the faces of a 3d cube rotating - the outer cube eventually becames the inner cube, and then becames the outer cube again. We can see up to four cells of the tesseract at the same time, depending on the angle it's being viewed from. Also notice that, just like the faces of the cube becames "reversed" when they are not in the front, the cells of the tesseract also becames reversed when they are not in the surface of the 3d projection. That means the content inside of the inner cube of the projection (for instance) is flipped horizontally.

Another thing worth pointing out is that 4d entities can assign a "front" and a "back" side to a 3d object. Depending on the angle the 4d object is being viewed from, it might look slightly or entirelly flat to a 4d entity. When a 3d object looks completely flat, only part of the surface of the object will be visible to the entity, and the rest of the surface, as well as all the points inside the object, will be hidden behind the surface, further away in the 4d direction. If the 3d object is only partially flat, all the points of the object will be visible, but the content will look skewed. This behavior can be observed on any of the cells that form the hypercube. Remember that when a 4d entity looks at a 3d object from the "back", the content of that object will appear reversed.

The Hypersphere

The hypersphere is the 4d equivalent of the 3d sphere, and is formed by several (infinite) 3d spheres, just like a 3d sphere is formed by infinite 2d circles.

Imagine a transparent 3d sphere that has been sliced into several 2d circles. The circle closest to you is very small, or almost a point. The circle in the middle of the sphere is the largest one. That is true even if you rotate the sphere clock-wise or counter clock-wise. Now, take one of these circles, one that is between the middle of the sphere and the circle closest to you. As you rotate the sphere, this circle will appear to flatten (from your position). At some point, this circle will appear to be a single line. In fact, at this point, all the circles will appear to be a line, but since there are an infinity of them, the projected image of the sphere will still be a circle.

When a 4d entity looks at the hypersphere from the "front" (no matter what the 3d angle of the projection is), then the 3d sphere closer to the observer will be situated exactly in the center of the projected sphere. In reality, though, it would be very small, like a point. If the hyphersphere is transparent, then the 3d sphere further from the observer will also be visible, and due to dept, it will appear to be smaller than the nearest sphere.

The largest 3d sphere that forms the hyphersphere will compose the surface of the projected 3d sphere, if the hyphersphere is opaque, much like the largest circle of a 3d sphere would be its contour. If it is transparent, then all the points inside the largest sphere will be visible, as well as all the points inside all 3d spheres.

As the hyphersphere is rotated across the 4th dimension, these imaginary sphere-shaped crossections will start to "flatten" and "move" to the sides. Just imagine a 3d sphere being rotated, and the circles that compose it will have the same mentioned behavior. However, instead of flattening a circle until it becames a line, we are flattening an actual 3d sphere until it becames a circle. Notice that at all times the shape of the projection is still a sphere. Even when looking at a hypersphere from the "side", all the circles that constitute it will compose a 3d sphere.

Writing and Reading In 4 Dimensions

2009-01-27T19:49:00.000-08:00

You already know that 4d entities can draw in 3d. But what impact does that have on their everyday life? What would a 4d book look like?

To answer this, let's first take a look at our world. We can take a piece of paper and draw any 2d shape we want, as wide and tall as we want, as long as we have a big enough canvas. But when it comes to writing, what we really want is something small - simple symbols that can be written as fast as possible and are around the same size, distributed on an organized fashion across the canvas (left to right, up to down, etc...). Also, these symbols must be read in sequence to form a full sentence.

In a 4d world, things are not so different. Surely, a 4d entity can easily draw a 3d cube, and even paint anywhere inside this cube. But there are much simpler shapes they can draw. A very simple symbol in our world would be a line. When we draw a line, we can see all the points that form that line, and even rotate our canvas to look at it from another angle. In 4d, the equivalent of drawing a line would be drawing a single 3d "stick" - like a small wire. A 4d entity would then be able to see all the points outside and inside that "stick". Notice that this symbol would have to have some thickness for it to be noticed by a 4d entity.

The 4d canvas can be rotated in a similar fashion to our 3d canvas. If the 4d canvas is upside down, all the symbols projected on the 3d FOV will appear upside down. If the canvas is rotated across the 4th dimension, the symbols currently on the FOV will slightly distort or disappear completely. Also, 4d entities can only "focus" at a small point (or rather a small sphere) of the 3d FOV. That means they can only focus on a few symbols at a time, and only if these symbols are small enough, similar to the way we read.

The picture below compares our canvas with a 4d canvas:

As you can see, a single "page" of writing in our world is equal to a single "line" of text in the 4d world. Notice that the symbols used in the 4d canvas might require 3 dimensions. After all, it is just as easy for them to write horizontally as it is to write "forward". The picture also illustrate the issue of the distance between lines. In the 4d world, not only the "lines" can be spaced in different lenghts vertically, but also forwardly.

Example of a simple 3d character in the 4d world, viewed from 2 different angles. A 4d entity can easily draw this shape with 2 strokes.

But if a single "page" in the 4d world is similar to a 3d cube, what would a 4d book look like?

The answer appears to be simple: several 4d "pages" (4d canvas, very close to a 3d cube with little 4d depth) that are hold toghether. But some analysis is necessary to imagine how would they write on the back of the pages, and how are these 3d objects holded together. The first question is actually easy, and if you can understand the tesseract rotations, you probably already know the answer. Reading the front of the page would be just as described above (a 3d cube). Then, a simple rotation across the 4d axis would bring forth a different cube, with different content. Since this page would have just a small depth, at some point in the rotation the page will look like a simple flattened 3d cube (or a square), and at this point no content will be visible (only at 2 particular angles a perfect cube is visible with all the content). While reading on a 4d canvas, the cube that is on the "back" is inversed, which means that the content is flipped horizontally (much like in our world). Notice that the content will still remain distributed equally both vertically and forwardly, so a more trained 4d eye can still read it easily if the page allows seeing through it.

As for the second question, imagine a projected tesseract in the 3d FOV. Now imagine that, across the 4th dimension, the tesseract is being cut into small "slices", and each slice is equal to a very small (in the 4d axis) 4d object bounded by 3d cubes. But instead of cutting out 3d crossections of the size of each cube, a small portion of each crossection is being kept, thus linking the 4d objects toghether. Imagine, now, that a 4d entity is looking at the "cover" of a 4d book: it will look like a 3d cube (in the 3d FOV). When the 4d entity "flips" the cover, the 3d cube starts to flatten, and the next 3d cube will start to appear on the same space that was being occupied by the cover (in the 3d FOV).

As the cover flattens itself into a square, it "moves" to one of the sides of the new 3d cube. After being flattened, it starts to expand into a cube again, but now to a different side. After the cover is entirelly flipped, it will occupy a new 3d space, right next to the other 3d cube, but now it will be reversed, and the content will be different. In this moment, two 3d cubes will be visible in the 3d FOV, and they will appear to be glued together.

To conclude the visualization of the 4d book, imagine only the cover again, but right to its "left", another 3d shape will be visible. This shape represents the portion of the book that holds the pages together (the method used varies from book to book, but for this example, imagine an ordinary hardcover book). In this case, the same thing described above will occur, but after the cover is entirelly flattened, and as it starts to expand again, it will start to occupy the 3d space of the FOV that is being currently filled with this portion of the book. After a while, the volume of the 3d projection of the cover will be much bigger than the volume of the projection of the mentioned portion.

Videogames in a 4d World

2009-01-26T19:49:00.000-08:00

Assuming you have read all the previous articles, you must be well familiarized with the concept of the field of vision of 4d entities by this point. This article will go one step further, as we try to get a glimpse of what would video-games look like in a 4d world. Notice that "live games" will not be presented here, as well as whatever machines and devices are necessary to actually play 4d video-games; these are much more complex subjects and will be explored on future articles. However, this article does more than just simply imagining the counterparts of games from our own world. It will also give us a better insight of the FOV of 4d entities, and how they perceive their own universe. I'll start with a simple question: 3d entities can't perceive a 1-dimensional line, since it would be infinitely small; but can 4d entities perceive 2-dimensional shapes?

The short answer is no. Take a look at the 3d FOV once again, which in our analogy, is similar to a 3d cube. Imagine, now, that there is a 2d picture in the center of the FOV. In our world, assuming we do the same with a transparent cube, we would be able to see that 2d picture from almost any angle, because that shape would occupy a large portion of our own 2d FOV. But if we look at that picture from the "side", it will look like a small line to us, and occupy a small portion of our FOV. Therefore, it would be almost unperceptible. A real 1-dimensional line would be impossible for us to perceive, but we can perceive a real 2-dimensional shape as long as we look at it from a certain angle.

Analogously, a real 2d shape would be impossible for a 4d entity to perceive. Once you place a factual 2d shape inside the 3d FOV, it will look infinitely small. Unlike our transparent cube in the previous example, there is no empty space on the 3d FOV: it is a solid cube, filled with the nearest 3d information. The 2d shape would be hidden in the center of the cube, next to more perceptible 3d volumes. Since 4d entities can only focus on particular 3d volumes of the FOV, this 2d shape would only take a infinitely small portion of the information currently on focus. Therefore, 4d entities can't perceive 2d or flat 3d shapes.

This lead us to assume that, even the simplest of the games in the 4d world would make use of 3d geometry. In the following sections, we are going to retrace the evolution of video-games in our own world, and by analogy, imagine how would video-games evolve in a ficional 4d world, from the simplest forms to the most advanced.

Basic 3d Games

Look at the pictures to the right. Even though they are both screenshots of 2d games, you should notice that each game belongs to a different niche: the first game is a "pure" 2d game, while the second one is a "layered" 2d game.

What that means is that the second game, while still being 2d, attempts to simulate our reality by using background images and layers. The first game, however, creates a reality of its own, without trying to portrait ours. That's important in our analysis because the 2d world mentioned in previous articles can be compared to the first game. Any 2d game with more than one layer of image doesn't depict a 2d world accurately.

With that in mind, we can assume that in a 4d world, there are also two kinds of 3d games: pure and layered. Pure 3d games can be used to portrait the reality of a 3-dimensional world. That means a 3d screenshot of such game would, in a way, appear to be a 3d model that we can easily understand, as long as you exclude the "black space" from this model. The black space I'm refering to is similar to the black background on the first game above: it is just there to fill in the gaps, but in a 2d world, that would be only space.

Let's try to imagine a simple tetris game in the 4d world: the equivalent of a simple, unlayered 2d tetris in our world. The rules are the same: blocks come falling from above and the player must control their position and orientation before they hit the bottom. Since 4d entities have a 3d FOV, the space through which these blocks fall is a cubic shape, and not a rectangle, and the pieces are also 3d shapes. The game would be presented in a way that it would only take a small portion of the 3d FOV. In our version of the game, once a row is entirely filled up, the parts of the pieces that are in that row (which are small 2d squares) disappear. Similarly, in the 4d version, once a grid of the bounding cube is entirelly filled (ie, an entire grid is filled with small 3d cubes) those cubes disappear. Notice that if we tried to play such game, we wouldn't be able to see 3d blocks that are "under" other blocks, unless, of course, these blocks were transparent (although that would still be confusing). But 4d entities can see all points of the 3d FOV, just as we can see all points of our 2d tetris game. Notice, also, that most of the space of the 3d FOV would not be blank space, but would be filled with "black background" (or black pixels; remember that the display screen of their TV set is a cube, and not just a surface like in our world). Note: I'll refer to the entire image formed by the pixels generated in the 3d screen as an hologram.

We can also try to imagine very basic 3d layered games. A game similar to our tetris game with a background image would be similar to the one described above, but instead of the black pixels, there would be a different 3d hologram that would be the equivalent to our 2d photos. So let's imagine a simple tetris game in the 4d world with a tesseract as the background image, and let's exclude the black background information that would make up the rest of the background. In the beggining of the game, the bounding cube would still be appearing in the center of the 3d FOV, but now, in its center, you would be able to see another 3d cube, which is the 3d projection of the tesseract (as seen in previous articles). Even though it is a 3d shape, the 3d blocks can fall "through" it, and occupy the same space as the tesseract projection. It's important to understand this concept, because that's how 4d entities actually perceive their world: they see a cubic 3d shape that keeps changing constantly, with closer 3d projections occupying the space of projections of 4d objects that are further away. In this case, however, the tesseract is not further away from the observer: the tesseract, the bounding cube and the 3d blocks are all part of the same hologram that changes with time.

But here's where things start to get interesting. An unlayered 3d game (in a 4d world) could be used to portray our own 3d reality. That means, one of the first games created in a 4d world, when the games industry would still be on its infancy, could look like a full-fledged 3d game to us! Take a 3d game like Zelda OOT, for instance. Theoretically, all 3d models of characters, enemies, weapons, etc, could be used in the 4d world with minor adjustments (the textures would require some thickness to be visible). The model of Link would look exactly the same (even though 4d entities would be able to look "inside" Link, remember that we are talking about "unlayered" 3d games here, which means the insides of Link could be filled with black pixels, and 4d entities would only focus on its contour). However, since 4d entities have a 3d FOV, you might be wondering how exactly do they see all the information that we see: for example, where, in the 3d FOV, would the "sky" appear? How would 4d entities see very large, outdoor landscapes in their FOV?

They wouldn't. This is where their version of the game would differ from ours. Imagine, for a second, an unlayered 2d game in our world that tries to simulate the reality of a 2d world. In reality, we are simulating that reality from our point of view. That means, in a true unlayered 2d game, we can only see a small portion of that 2d world at a time, and we can not see the sky or anything that is too far from the main character. Notice that a 2d entity has a 1d FOV (a line), and this entity could look up to see the sky, which would then fill up its FOV. But we are not playing the game from the point of view of 2d entities.

So, in the 4d version of Zelda: OOT (or rather the unlayered 3d version in the 4d world), the 4d entities would only see a small portion of the 3d scenery at a time, and the camera angle would most likely to be fixed and locked on Link, since for them, there is no difference between seeing a 3d scene from an angle or another. The space that is not filled with the ground, the characters, enemies or other objects would be filled up with empty (black) pixels, and not with the sky, or any other information. That seems very restrictive for the 4d entities, and it is, which is why they would eventually evolve their games into layered 3d games.

Layered 3d Games and True 4d Games

Let's imagine, once again, a layered 2d game that tries to simulate our 3d world. Notice that, even if you ignore the background and foreground layers, it would be impossible for the 2d characters of these games to live in a true 2d world. For example, a 2d character might be holding a weapon on its right hand. In a 2d world, that would be like holding a weapon "inside" itself. That means the information on layered 2d games just can't be used to imagine a 2d world, they are more like 2d projections of a 3d world.

By analogy, we can say that layered 3d games are made up from 3d projections of the 4d world. There is no 3d game in our world that can be compared to a layered 3d game. However, we can still speculate about it to some extent. Looking again at our 2d layered game, we can say that a 2d entity would perceive it as something similar to their own world, but with solid information filling up what would otherwise be empty space (ie, the background and foreground images). This information would change constantly. Also, the characters portraited in this game would be perceived as 2d entities that have a morphing ability. Remember that, for 2d entities, only the contour of 2d shapes is perceived.

Finally, the mechanism of the game would be similar to that of unlayered games. That means the mechanism of a 3d layered game in the 4d world would be similar to the example given above (of the Zelda game). In essency, the 3d FOV would still display a small 3d scenery with a fixed camera angle (locked on the main character). But now, the empty 3d space would be filled with different colors and shapes, and not just black pixels. The model of Link would also be different. It would have the same overall shape (assuming we are only porting the same game to a layered style), but the insides of the 3d model would also be textured accordingly to simulate a 4d entity. The shape of the character would also change constantly, just like the tesseract appears to change shape when it rotates.

As you can see, a layered 3d game would make full use of the 3d FOV of 4d entities, but the game would still be very restrictive for them. This lead us to the final stage of evolution of games in the 4d world: true 4d games. This type of game would simulate a 4d world, which means the hologram that fills up the 3d FOV would be constantly changing, and the FOV would not act like an ordinary 3d camera anymore. At least not always - a true 4d game, if viewed from a specific and fixed angle, might look very similar to our own world. This simple comparision might help us understand the point of view of a 4d entity.