Space Dimension Analysis

The 4th dimension is a dense concept to really get a hold of, but we'll do our best to make sure you get the concept. Well - our researchers will. Shit if I have any clue how it works. One way or another, it's a good read. Here's our documentation on the subject, fully declassified for the your sake:

Dimensional Context

Traditionally, we are met with four essential dimensions: three spatial dimensions, and one temporal dimension. In everyday life, the spatial dimensions can be referred to as forward and backward, left and right, and up and down; mathematically, they are represented using the variables X, Y, and Z, where each axis is perpendicular to the others. Increasing in dimension, we see that the first dimension is a line, the second dimension is a plane, and the third dimension is a grid. The conflict that we run into is: in what direction can we go from there? 


When we introduce additional spatial dimensions, we are met with a concept that is absolutely un-visualizable due to our nature in a three-dimensional world. However, when you get down to the super-fine building blocks of what makes our own reality function, we can apply what we know to a higher dimension.

Understanding 3D from a 2D Perspective

For the sake of understanding what higher dimensions look like from a lower dimension’s perspective, it will be important for us to go one layer down and understand the third dimension from a two-dimensional perspective. 

Let’s flatten you into a 2D version of yourself. You want to eat a slice of cake, but there seems to be a brick wall between you and your treat. The only way for you to get to it is to go all the way around the wall, or break reality and move to the third dimension: the z-axis (normally, this should not be possible for 2D You).


Figure 1

Figure 1

Figure 2

As we traverse space in the z direction, we notice that the brick wall exists at the height z = 0 through z = 1, though it does not exist at the height z = 2. If you were to head to z = 2, you will be free to move to what would have been the other side of the wall, and then traverse back to z = 0, where the cake is.


This may feel like a rudimentary exercise in how to go over a brick wall, but remember that 2D You would have a difficult time understanding what a brick wall looks like in 3D. To them, comprehending this process would look like many 2D slices of a wall stacked in a direction that is not natural to them. From a stationary 2D perspective on the plane z = 0, you phased out of reality, and came back in on the other side of the wall.




From a 3D perspective, it’s very easy to visualize this brick wall. We can fill in the gaps and say that the brick wall exists in-between z = 0 and z = 1, but it doesn’t quite reach z = 2. Stacking up all of these small frames in-between will give us an idea of how 3D objects cast their 2D “shadows,” and thus display how you can gather information about objects in 2D to piece together how they look in 3D.



Figure 3

Figure 4

The following is a graphic that illustrates exactly that. A sphere passes through a 2D plane, and the shape to the right is how it is perceived on a 2D level. 


We will revisit these concepts once we have gone into the 4th dimension; that way we can base our understanding on these lower levels and truly grasp how the higher dimension’s continuity works.


Introduction of the W-Axis

Now that we have an understanding of dimensional variance, it’s time to pack on an additional dimension to our three. The best way to approach this will be mathematically from the very beginning:


Let’s start building higher dimensions using the smallest one: a point. We will refer to this as the “0th dimension.” A creature within this dimension would have no ability to move because it is restricted to that particular location in space and cannot leave that point. If we were to line up several 0-dimensional points, we form the first dimension that follows the x-axis. The following shows the x-axis, where there is an infinite number of 0D points between each integer.

Figure 5

Figure 6

Next (figure 6), using a line segment that’s made of 0D points, we can line up these segments perpendicularly to create the y-axis. Remember that there is an infinite number of 0D points in-between every integer for every decimal place, just as there is an infinite number of parallel lines that form the y-axis.


After that (figure 7), let’s construct the third dimension using many two-dimensional planes stacked on top of each other to make a three-dimensional grid. It’s a similar situation here as it was before; there is an infinite amount of planes in-between the z value integers. The graphic is spaced out for the sake of simplicity.

Figure 7

We begin to see a pattern: higher dimensions are constructed of many, stacked up, smaller dimensions. From this conclusion we can answer our original question of what direction you go in after you have fulfilled all three dimensions: moving in the w-axis is the only direction that we can go in, and it’s represented by a number line of individual three-dimensional grids. 


Figure 8

The Perpendicular Approach

If raising a dimension means going in a direction perpendicular to the original, then that means that moving along the new axis can be independent of the original axis; in other words, going from z = 0 to z = 2 does not need to produce a change in the x and y values. In this case, when we move along the w-axis, the x, y, and z values do not have to change. This serves as a type of mathematical proof and a separate way to look at reaching the 4th spatial dimension conceptually. 

A Retrospective Application


With the w-axis fully set up now, we’re ready to apply our perspective of 3D from a 2D perspective onto 4D with a 3D perspective. We can repeat the “brick wall experiment” by claiming that there is an incredibly tall and incredibly wide wall in w = -1 and w = 0, but it doesn’t exist at w = 1. Traversing the w-axis will allow you to get around where the wall would have been in three-dimensional space and return to w = -1.

Figure 9

This is another scenario where, from the stationary perspective of w = -1, you disappeared from reality and appeared again on the other side of the wall after traversing the w-axis, just like 2D You did from the stationary perspective of z = 0.

An Applied Understanding


Refer back to the 2D brick wall example from earlier. From the 2D perspective, each “slice” in the z-axis looked relatively similar to those around it: z = 0 featured yourself, a brick wall, and cake. At z = 1, we found that the brick wall remained, though you and the cake are not there. In z = 2, there was no more wall. 2D You would view this as a jumbled mess of constants and variables, with each 2D slice feeling like an alternate version of the last.


As we apply this to 4D, we see a similar case where w = -1 holds you, the wall, and the cake. Value w = 0 has just the brick wall, and w = 1 is empty space to be roamed. A way to think about this is almost like “alternate universes,” where each w value has slightly differing contents, but you can tell that it’s still the same x, y, and z values. 


Just as the 3D wall was able to be seen as many different slices of a 2D wall stacked together, the 4D wall is the same way: many different walls stacked in a direction in which we can’t see. 


If you remember the animation from earlier that showed a sphere passing through a 2D plane, we can apply this kind of thinking to an abstract 4D situation now. Just as the 3D sphere casts a 2D “shadow” of a growing and shrinking circle as it passes through, the 4D sphere (called a “hypersphere”) will cast a 3D shadow (or “projection”) as it passes through a certain w value. Let’s go on to understand what this truly means in realistic terms. 

Figure 10

The Slanted Brick Wall - 2D and 3D Projections


In order to warm up the idea of how bizarre the 4th dimension can look through a 3D lens, we will backtrack and see how bizarre the 3rd dimension can look through a 2D lens. The following is a portrayal of how 2D You perceives a brick wall that has been slanted. On the right is the 3D movement of the brick wall, and on the right is the wall’s 2D projection pattern.


Physically, we know that this is a 3D object passing through a plane; nothing is reality-breaking about it whatsoever. It’s this context that allows us to see that it’s the limitations on our perspective that makes it appear as though the impossible occurs: a brick wall phases in, moves, and phases out of reality before our very eyes. Being able to see the situation from a higher dimension gives the context of continuity that you just don’t get with lower dimensions. We will come back to this concept when string theory is addressed.


Relating this subject back to the case of the 4th dimension, we saw the hypersphere phase in and out of 3D space gradually. A 4D wall will also slip in and out of 3D reality gradually, but how it does depends on how it’s constructed along the 4th dimension, comparable to the slanted brick wall’s 2D projection pattern. The way that it moves in and out is influenced by its physical 4D shape (such as the hypersphere) in addition to the viewing orientation it’s observed in.






Figure 11

Figure 12

Just by switching the orientation of the way the wall is passed through the 2D plane, we can get a dramatically different projection pattern. Every single orientation will have its own pattern, with some ability to produce repeats. 


Misleading Projections


Keep in mind that it is possible for different shapes to create the same projection onto lower dimensions. This can be somewhat confusing when we see 3D projections of 4D objects; we need more information to properly get a picture of what the entire object resembles. 


Just as the side of a 3D cylinder can be projected as a 2D rectangle, it’s possible for a projected 3D cube to be part of a 4D cylinder.


You may have noticed an additional misconception: notice that the 3D brick walls not only had shape distortions as they passed through, but they also had an additional trajectory. In reality, the 3D wall is only moving along the Z-axis in the negative direction (down), but when this movement is translated to the 2D grid, it appears as positive movement along the X-axis. Unfortunately, this is not where the misconceptions end.



Figure 13

The Restricted Perspective


We’ve spoken so much about how things would project onto 2D from a 3D world, but we haven’t properly considered what the 2nd dimension looks like through the eyes of someone in 2D. 


While a true 2D perspective would be flattened to have no height whatsoever (which will be discussed in the section regarding Logistical Complications), we can instead visualize it as a small strip of our 3D view, as seen in Figure 14. This view has something particularly interesting about it: it is, effectively, a 1D projection: a line segment. Of course, the variance in visible materials gives context to how the line segment fluctuates, making it appear 2D; in other words, it can be seen where the sphere ends and the air behind it begins. Going up a dimension, we notice that this exact same thing happens with our own eyes: a 2D image is projected onto the backs of our eyes, which is then transmitted to our brain. The difference in materials is what begins to create depth perception - the illusion of a 3D space allows us to extract 3D information from it. 


Just for the sake of fun, try to imagine what the world would look like from a 4D being’s perspective; a 3D projection onto their eyes that gives the illusion of 4D. 

Figure 14

Figure 15

On the left of Figure 15 is a top-down representation of constant angular velocity. This is a perspective only possible in 3D and above, so if we were in a 2D world we could see perspectives such as the two on the right. The top-down 2D perspective doesn’t offer a whole lot of the big picture, but notice a glaring restriction that’s found in the side view: its velocity appears to be speeding up and slowing down, even changing direction entirely. It’s a phenomenon similar to the Ames Window illusion, which features a spinning trapezoid that appears to flip direction and change velocity. The truth is, we actually need the top-down 2D perspective to get the full picture - but how does a 2D being move in a direction that is unnatural to it? This is another logistical complication that will be brought up later.

The purpose of bringing up rudimentary illusions in a mathematical context is that when we start working with such abstract concepts as the 4th dimension, it’s easy for visualizations and arithmetics to trick our brain due to the lack of several perspectives. When we see things that make no spatial sense from our perspective, we can gather information to build the whole picture by changing the point of view in the higher dimension.


The Cracked Brick Wall - Continuity in Space


We’re back at our brick wall again, but this time let’s put a crack in it. We will say that  z = 1 has the deepest part of the fracture, and at z = 0 and 2, the wall is unbroken. As 2D You will see, going up and down the wall will display changes in the wall’s state; going from z = 0 to z = 1, you will see different bricks and a crack that wasn’t there before. This is a dramatic change. However, going from z = 1 to z = 0.9 will show a significantly less intense change: the crack will be more shallow, and the bumps on the surface of the bricks may have changed even though it’s still the same brick. As we take smaller slices of Z-values, we see smaller changes through that 2D lens, and it looks more similar to the original slice’s reference point. From 2D You’s perspective, it seems like a parallel reality where things are slightly as they used to be, but not entirely the same.


The main point is continuity. Everywhere you go in the universe in time and space, there is a natural flow from point A to point B; dust doesn’t just suddenly appear on a surface, it falls from the air and builds up over time. Hard cut surfaces such as the top of a brick wall may seem like a sudden change in material between brick and air, though if we take a small enough 2D slice, we can still see the ever-so gradual change in the bumps on this surface.



Figure 16

Temporal Misunderstanding


You may have heard the “4th dimension” previously referred to as time, and to some degree it’s an accurate comparison, especially if we consider the continuity aspect of it all. Obviously, we’ve been referring to it as the 4th spatial dimension though, which bears some discrepancy.

An asteroid strikes Earth, and all frames of this event are lined up in a timeline. We have t = 0, which is before the impact, t = 1,  being the height of the impact itself, and   t = 2, where the impact has concluded. Using our continuity mindset from the previous section, we know that t = 0.9 will resemble slightly less impact than t = 1. This is an appropriate way to consider the variable dimensionality of the situation, but the element that it lacks is freedom. This may be a somewhat difficult way to wrap your head around the concept, but just as we envisioned the brick wall from the 2D perspective, it’s appropriate to envision this “slice” of Earth from a 3D perspective as we apply it to 4D. Its continuity is what brings it all together and creates a type of “multiverse” effect - where one version of Earth is very slightly different than the one next to it along the W-axis. 


With our 3D world, we’re able to move freely within our space, but we’re constantly propelled down the slope of time. This is why it’s inaccurate to consider the 4th spatial dimension to be time, though you can see the parallels between the two concepts. After all, you’ve heard of a timeline, just as you have now heard of a 4D line.


There’s another element to this, which is the question of what it means to TRULY return to a point in space, but for our purposes we will brush past it.

Beyond the W-Axis


By now we have established that the 4th spatial dimension is made up of many three-dimensional grids, which are made of many two-dimensional planes, which are made of many one-dimensional lines, which are made of many zero-dimensional points. We can apply everything that we have learned thus far to making a 5th spatial dimension: the v-axis, which is a plane of four-dimensional lines. From there, adding a 6th, u-axis, would make it a grid of five-dimensional planes.


Once we have gone up and down, left and right, and forward to backward, we are, yet again, faced with the question of where to go from here. The pattern is, simply, that the process will repeat itself in increments of three. The 7th dimension will be a line of six-dimensional grids just as the 4th dimension is a line of three-dimensional grids. We can even revisit the 0th dimension, as it displays this exact pattern as well: the 1st dimension is a line made of 0D points. Essentially, we’re just substituting “points” for entire grids once you have gone up three dimensions.

Figure 17

Real World Theoretical Applications


Today as we delve deeper into quantum and particle physics, we are met with concepts that don’t seem to make any spatial sense; for example the dense intricacies of what holds up string theory.


String theory is an ever-changing concept about how every particle at its core is made up of strings - be it lines or loops - that vibrate at certain frequencies to produce different types of overall particles. While it is incredibly difficult to prove with our current technology, it has consistently held up mathematically. These mathematics could point to 10 total spatial dimensions, plus one for time. The issue is: Is it truly possible for these dimensions to have any effect on one another?

Logistical Complications


There is a conflicting nature about additional spatial dimensions, being that on paper there should be no way for three-dimensionally bound entities to interact with other dimensions. The same rule applies to every other dimension. 


Imagine the thinnest thing in the world; whatever it is, the second dimension must be thinner than this, as it has absolutely no thickness whatsoever. By three-dimensional standards, it does not exist. If we have a cube of side lengths 2x2x0, it simply has no volume and does not exist. Even if we just consider the nature of the atom, it requires three dimensions to function properly, so reducing it to two would be insufficient. Whatever type of atom is present in a 2D reality is missing a 3D component that would make it function in 3D, and it’s reasonable to say that a 3D atom is missing a 4D component to make it function in 4D.


Despite these glaring issues, what keeps it all tethered in reality is this apparent teleportation in circumstances such as string theory. When we encounter mathematics that point to the possibility of a break in continuity such as a sudden jump in space, it’s worth investigating the possibility of a higher spatial dimension at play, and it could hold the keys to discovering more about this bewildering reality.

Key Term Glossary


Axis - The path that a dimension takes - typically used in a mathematical context

Continuity / Continuous - connected; no dramatic cuts or immediate changes in trajectory

Dimension - a direction in which something can be measured

Hypersphere - the raised dimensional equivalent to a sphere

Integer - a whole number; not a decimal

Orientation - the perspective from which an object is viewed - typically rotation-based

Perpendicular - in opposing direction to

Spatial - relating to space

Temporal - relating to time

..... Yeah, like I said, I've got no idea what any of that shit means.

But hey, that's not my wheelhouse. Come by any time if you need a refresher.