Understanding the 4th Dimension

Quoting Mp202020

you could/would instantly find yourself inside of a solid object of some sort and instantly die?

No, not quite. You'd still bump into things. But I suppose it depends on how the fundamental forces would work in 4 spatial dimensions.

The game simulates being able to perceive a 4th dimension as a 3-dimensional observer, essentially perceiving a 4th dimension, one 3-dimensional plane at a time.

A 4-dimensional observer would perceive everything contained within the 4 dimensions simultaneously.


Though, I'm not sure whether simulating 4-dimensional space is the most intuitive way to understand the problem, since space in our universe seems fundamentally 3-dimensional? Maybe theories concerning the existence of dark matter changes things, but it's certainly counter-intuitive.


Perhaps a more intuitive way to imagine four dimensions is as follows:
1. Imagine observing a town. (Normal 3-dimensionality)

2. Imagine observeing the town throughout time. (Observing 4-dimensionality as a 3-dimensional observer, like the game). Note that people already do this.

3. Imagine being able to observe the town throughout time in its totality, instead of one 3-dimensional snapshot at a time.

Note also that the breadth of the time window one would be able to perceive simultaneously would be limited by whatever sense organ one would use to perceive it.

Finally, note that it's impossible to "bump into" time (as per your question), because its nature is different from space.

Three spatial dimensions, one time dimension. Spacetime. Don't try to make time into a fictitious part of space. But who really cares?

Quoting Mp202020

isn’t it extremely likely, no, inevitable, that you could/would instantly find yourself inside of a solid object of some sort and instantly die?

No. The interface in the video always shows the user rotating in place, so where he is does not change, only things at a distance as the cross section rotates through different things. The rotation never changes the user's coordinates, only walking does, and one does not walk into solid things.
The 2D frog cannot rotate into the interior of a 3D object since it stays put when only rotating.

The same way objects appear in thin air can happen right where you are standing

Again, no. Anywhere else, but not where you're standing. You're always at the axis of (actually plane of) rotation, so rotating does not put you somewhere else (into a solid object say).

You can literally be standing in the same place another object in a separate 3rd dimensional cross section is currently standing at the same time but you are separated by the different 4D location you are at.

The 'other object' is at a different location per your description. At least one of its four coordinates is different than the one where you are.

This is the concept of “parallel” dimensions

Not really what most are talking about when speaking of 'parallel dimensions',

2- I do not understand why burrowing underground in your current 3D cross section would protect you any more than the 3D four-wall structure you built?

I see no need for protection at all.


Quoting Tzeentch

The game simulates being able to perceive a 4th dimension as a 3-dimensional observer, essentially perceiving a 4th dimension, one 3-dimensional plane at a time.

Just so, yes. Wonderful implementation done too. His (very capable) computer seems to have a rough time trying to keep up. Mine (not so capable) does even for a 3D world, and I have to turn the resolution and rendering distance down to keep the frame rate reasonable.

Quoting jorndoe

You'd still bump into things.

I bump into things now in three dimensions.

Quoting Tzeentch

2. Imagine observeing the town throughout time

There is a difference between the 4-space of Euclidean space,, and the 4-space of relativity. Once you include time, the metric changes. The analogy of viewing time as a fourth Euclidean dimension is often used to explain time as the fourth dimension, but this is not how relativity thinks of it.

If @Mp202020 is talking about mathematical Euclidian 4-space, it's characterized by the Euclidean metric giving the distance between two four-vectors [math](x_1, y_1, z_1, w_1)[/math] and [math](x_2, y_2, z_2, w_2)[/math] as

[math]\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2 + (w-1 - w_2)^2}[/math]

This is the familiar Euclidean distance formula we learned in high school. It can be generalized to any finite number of dimensions.

But once you start talking about time as the fourth dimension, the Euclidean metric is no longer used. Rather, the Minkowski metric is used,

[math]\sqrt{ - (t_1 - t_2)^2 + (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}[/math]

There is a minus sign in front of the coordinate corresponding to time.

This is often expressed different ways, sometimes with a plus sign for time and a minus sign for the three spatial dimensions, or with or without the square root, or with the speed of light multiplied by the time, or the speed of light normalized to 1 as I've done here.

Point being that 4-space in physics does not use the familiar Euclidean metric. Relativity is not just doing standard Euclidean math while calling one coordinate [math]t[/math]. The way you calculate distances changes.

Just wanted to note this dismbiguation for clarity.

@Mp202020, are you talking about trying to understand Euclidean 4-space? Or relativistic 4-space as used by the physicists?

Quoting jgill

Three spatial dimensions, one time dimension. Spacetime. Don't try to make time into a fictitious part of space. But who really cares?

The physicists care a lot.

Minkowski himself said, From henceforth, space by itself, and time by itself, have vanished into the merest shadows and only a kind of blend of the two exists in its own right.

Reply to noAxioms as much as I appreciate your insight, I am perplexed at the unnecessarily demeaning tone with which many on this site choose to provide their responses with. It to me posits a deep lack of basic social skills, throughout my schooling in computer sciences I’ve met many man-children who were incredibly smart but held themselves back in life due to their unnecessary behavior and attitude such as this. A sign of being on the autistic spectrum.

Rather than simply communicate to me your explanations, you communicated to me your explanations like one of these man-children needlessly. Instead of respect and appreciation, which would have been the natural consequence if you spoke normally, I instead think negatively of you. I’m sure you don’t care about my opinion of you per se, but I’d bet you aren’t a very likable person in real life and I’d wager you don’t have healthy relationships.

Thank you for your reply however, I hope you one day learn how to interact with others.

Reply to fishfry wow man, I am in no way qualified to even engage with your response, I’m a mere layperson with some questions that popped up pertaining to the attached video, I cannot understand much of what you said. I’ll try to read over a few more times.

That being said I deeply appreciate the thought and effort you’ve chosen to spend on my questions. It’s really a beautiful work of art, thank you so much. The least I can do to show my appreciation is give it a very true attempt to understand everything you’ve laid out here. Thanks again, will get back to you after I chew on this.

Quoting Mp202020

wow man, I am in no way qualified to even engage with your response, I’m a mere layperson with some questions that popped up pertaining to the attached video, I cannot understand much of what you said. I’ll try to read over a few more times.

That being said I deeply appreciate the thought and effort you’ve chosen to spend on my questions. It’s really a beautiful work of art, thank you so much. The least I can do to show my appreciation is give it a very true attempt to understand everything you’ve laid out here. Thanks again, will get back to you after I chew on this.

Thank you for the kind words. I was not trying to impress anyone with my erudition, believe me. I am a total physics ignoramus. I just told you literally everything I know about the subject. Apologies if I misunderstood the level or the intent of the question. I was just trying to be helpful by disambiguating math 4D from physics 4D.

When people say, "Time is the fourth dimension," that's physics. In math, the fourth dimension is just the fourth dimension, and there are fifth, sixth, etc. dimensions. It's simpler than you can imagine. If you saw the Cartesian x-y plane in high school, Euclidean n-space is just that on steroids.

But when physicists use time as a dimension, they do NOT just swap in the variable "t" for one of the coordinates, then go on like usual. On the contrary, they change the very definition of what it means for two points in spacetime to be a certain "distance" from each other.

So when people say that "Time is the fourth dimension," that is actually misleading. Because in physics the three spacial dimensions are related to the time dimension in a way that doesn't happen in the Euclidean case.

If anything I wrote is of interest, that's good, and if not, that's good too. Apologies again if I missed the mark. I did want to get my thoughts out there in a discussion of the fourth dimension.

You would enjoy a book called Flatland, A Romance of Many Dimensions

Feel free to ask any questions about the fourth dimension and I'll do my best.

Reply to fishfry That is very interesting.

To be honest, I understand very little about the mathematical underpinnings, but I think I understand your point. Time and distance behave differently so cannot be substituted - something that intuitively appears agreeable.

Do you know why they behave differently?

When writing down the example I realized that we cannot move back and forth in time as one could with distance, but there might be more to it?

Quoting Tzeentch

To be honest, I understand very little about the mathematical underpinnings, but I think I understand your point. Time and distance behave differently so cannot be substituted - something that intuitively appears agreeable.

Yes, I wanted to make that distinction. But at the same time your example is good too as a visualization. Meet me at ground level at Third and Main at Noon. Four dimensions.

But it's worth noting that physicists aren't just substituting in "t" for a spacial coordinate. Time and space are intertwined.

Quoting Tzeentch

Do you know why they behave differently?

I literally am a physics ignoramus. It's quite shameful. I'm at my physics limit.

I think I have a sense of what might be going on just looking at the distance formula. I've seen other versions of this where the quantities being squared are the differentials at each point, meaning how fast time is going. So if the time term is small, meaning time is going slowly, the [math]-(t_1 - t_2)^2[/math] is negligible, and the three spatial coordinates predominate.

But if time is going fast, then [math]-t_1[/math] and [math]-t_1[/math] are far apart, and [math]-(t_1 - t_2)^2[/math] is a large negative number that can cancel out the contributions of the three spatial coordinates. So the overall distance in spacetime could be very close to zero, even if the two points were far apart spatially ... as long as you're going fast enough.

In other words, as you go faster, distances shorten.

That's how it seems to me at this moment. If anyone took freshman physics I hope they can straighten me out on this. I couldn't find a clear explanation online. But that's what I think is going on. The minus sign on the time term of the distance formula represents the shortening of spatial distances as your velocity increases.

Quoting Tzeentch

When writing down the example I realized that we cannot move back and forth in time as one could with distance, but there might be more to it?

Yes, time is special that way. We can travel in any spatial dimension, but time only goes in one direction.

That brings up an entirely separate topic, but one that is of great philosophical interest. Why does time only go one way? Physicists call this the "arrow of time," and it's a big mystery. The equations of physics are symmetric with respect to time, so there's no fundamental reason time can't go backward. But we don't experience that. I think I've seen a Sean Carroll video about this.

We may quantify space/distance with meters and time/duration with seconds, which aren't quite the same. Meters per second isn't just a (unit-less) factor, but what you might get a speeding ticket for.

Running with spatial 4d (so 5 when including time), I suppose we'd have to clarify whether
• we're 4d beings (spatially), or
• we're 3d beings in a larger 4d world,
while just perceiving 3 either way, in our thought experiment.

Implicitly, the geometry of the 4d is similar to our familiar 3d, not like Kaluza-Klein for example. This is what the neat Minecraft thing suggests.

Not the same as "parallel worlds", more like the "extra" spatial dimension (or direction) gives a continuum of 3ds.

There'd be some implications for the physics I think, whereas the mathematics itself can model either fine.

To hike further along, what about an extra temporal dimension...? :)