Exploring the 4th Dimension: Questions and Answers

[L.Newton]

Hey guys here is my question. 1D = Line
2D is a plane. 3D is space. So shouldn't a fourth dimension be something else? Is there really such a thing as fourth dimensional space. There is no such thing as a 3 dimensional plane. Though you can have a 2D plane in 3D space.

Next question leads to this. Why can't the fourth dimension be time itself. Could the four fundamental forces be themselves dimensions. These questions may be pretty dumb. Some helpful links could go a long way. Thanks!

 

[micromass]

Hey guys here is my question. 1D = Line
2D is a plane. 3D is space. So shouldn't a fourth dimension be something else? Is there really such a thing as fourth dimensional space. There is no such thing as a 3 dimensional plane. Though you can have a 2D plane in 3D space.

In mathematics, you can certainly work with dimensions beyond $3$. You seem to have a quite geometrical way of thinking of dimensions, and this makes it difficult to consider higher dimensions. In mathematics, we define higher dimensions very algebraically.

Perhaps you know how every point in 3D can be denoted by three coordinates $(x,y,z)$. Well, the four dimensional space that mathematicians look at is just the set of four-tuples $(x,y,z,t)$. There might not be a point associated to these coordinates in the classical sense, but we act like there is anyway. In this rather algebraic way, we can work with 4 dimensions very easily. And we can even work with "3-dimensional planes in 4D space", they are called hyperplanes.

A course in linear algebra should teach you all this and more.

Next question leads to this. Why can't the fourth dimension be time itself.

In some parts of physics, time is indeed a fourth dimension. But there is a big difference between "spatial dimensions" (which are the three dimensions we know right now) and other dimensions.

 

[L.Newton]

You see that's my question. I'm currently taking linear algebra and cal 3. Both work heavily, especially cal 3, in three dimensions. So I could not help but ask myself the question of "is there such a thing as 4 dimensional space". Are there any applications of four dimensional space? I understand M theory requires 10 and it makes use of the yang-mills theory on spatial dimensions, which I don't fully understand. Perhaps we are losing something by thinking in terms of space.

 

[micromass]

You see that's my question. I'm currently taking linear algebra and cal 3. Both work heavily, especially cal 3, in three dimensions. So I could not help but ask myself the question of "is there such a thing as 4 dimensional space". Are there any applications of four dimensional space? I understand M theory requires 10 and it makes use of the yang-mills theory on spatial dimensions, which I don't fully understand. Perhaps we are losing something by thinking in terms of space.

In mathematics, 4-dimensional spaces and higher certainly exist. And you bet there are applications of them. The applications range from physics to computer science and engineering.

The harder question is whether these 4-dimensional spaces exist in nature. That's a question of physics, and not of math. But 4-dimensional spaces (whether they exist in nature or not) have a ton of applications already.

 

[L.Newton]

Sorry I didn't read your first reply.

 

[L.Newton]

Right now were discussing linear transforms from R^2 -> R^3. Are we far off from this type of topic?

 

[L.Newton]

The professor tends to just go through examples and say what's in the textbook. He doesn't seem to have a lot of patience for these type of questions

 

[Mark44]

Right now were discussing linear transforms from R^2 -> R^3. Are we far off from this type of topic?

Probably not. You might take a peek ahead in your book at the parts that talk about vector spaces. R² and R³ are examples of relatively low-dimension vector spaces. There will likely be some examples and problems that involve 4, 5, or higher-dimensions vector spaces. Although difficult to imagine geometrically, most of the concepts in two or three dimensions extend pretty naturally to higher-dimension spaces.

 

[L.Newton]

I will do. Thank you very much

 

[Nugatory]

So I could not help but ask myself the question of "is there such a thing as 4 dimensional space". Are there any applications of four dimensional space?

All the math in Einstein's general relativity and in the modern formulations of special relativity is based on a 4-dimensional space; it takes four coordinates - for example, $(x,y,z,t)$ - to completely identify a point in that space.

Interestingly, this four dimensional space has different properties than the four dimensional space that you get if you just take the next logical step in the progression: zero-dimensional point; one-dimensional line; two-dimensional surface; three-dimensional solid; is there a four-dimensional something? At least mathematically, yes: google for "tessaract" and "hypersphere". That line of thinking leads you to the four-dimensional space in which the distance $\Delta{s}$ between two points is given by $\Delta{s}^2=\Delta{x}^2+\Delta{y}^2+\Delta{z}^2+\Delta{t}^2$ ($t$ here has nothing to do with time; it's just that there wasn't a letter beyond $z$ that I could use). That space is mathematically fascinating and has led to a few fun science fiction stories (Heinlein's "He Built a Crooked House", for example) but I am no aware of no serious applications of it.

However, the four-dimensional space of relativity, which definitely does have practical applications all over modern physics, has the property that the distance between two points is given by $\Delta{s}^2=\Delta{x}^2+\Delta{y}^2+\Delta{z}^2-\Delta{t}^2$ - note the minus sign, and here $t$ really is time, and the minus sign captures the common-sense observation that time is somehow different than any of three spatial dimensions.

 

[micromass]

Interestingly, this four dimensional space has different properties than the four dimensional space that you get if you just take the next logical step in the progression: zero-dimensional point; one-dimensional lie;, two-dimensional surface; three-dimensional solid; is there a four-dimensional something? At least mathematically, yes: google for "tessaract" and "hypersphere". That line of thinking leads you to the four-dimensional space in which the distance $\Delta{s}$ between two points is given by $\Delta{s}^2=\Delta{x}^2+\Delta{y}^2+\Delta{z}^2+\Delta{t}^2$ ($t$ here has nothing to do with time; it's just that there wasn't a letter beyond $z$ that I could use). That space is mathematically fascinating and has led to a few fun science fiction stories (Heinlein's "He Built a Crooked House", for example) but I am no aware of no serious applications of it.

There are applications enough! For example, if you do optimization or regression with 4 variables, then you work in $\mathbb{R}^4$ with that distance function.

 

[L.Newton]

Very interesting. Iv actually been wanting to get into science fiction. Would you recommend any that have a heavy focus on mathematics or physics?

 

[micromass]

Very interesting. Iv actually been wanting to get into science fiction. Would you recommend any that have a heavy focus on mathematics or physics?

Not really science fiction, but be sure to read flatland. Wonderful little book. Here's a free version: http://www.gutenberg.org/ebooks/201

 

[L.Newton]

There are applications enough! For example, if you do optimization or regression with 4 variables, then you work in $\mathbb{R}^4$ with that distance function.

I meant application of 4 spatial dimensions as opposed to 4 dimensions. We've done work in linear algebra as high as R^5 in chemistry but my main wonder was 4 spatial dimensions

 

[micromass]

I meant application of 4 spatial dimensions as opposed to 4 dimensions. We've done work in linear algebra as high as R^5 in chemistry but my main wonder was 4 spatial dimensions

Not even relativity has 4 spatial dimensions. Time doesn't count as a spatial dimension, but as a temporal one. The only physics with more than 3 spatial dimensions seem to be string theory and the like.

To be honest, more than 3 spatial dimensions might be cool, but the real fun starts when you have more than 1 temporal dimension :tongue:

 

[L.Newton]

Not really science fiction, but be sure to read flatland. Wonderful little book. Here's a free version: http://www.gutenberg.org/ebooks/201

Oh yeah I'm familiar with that one but iv never read it. Thanks for the link

 

[micromass]

For some books about dimensions in literature:
http://en.wikipedia.org/wiki/Dimension_(mathematics_and_physics)#Literature
http://en.wikipedia.org/wiki/Fourth_dimension_in_literature

 

[Nugatory]

There are applications enough! For example, if you do optimization or regression with 4 variables, then you work in $\mathbb{R}^4$ with that distance function.

D'oh -of course, and thank you.

 

[L.Newton]

For some books about dimensions in literature:

http://en.wikipedia.org/wiki/Dimension_(mathematics_and_physics)#Literature

http://en.wikipedia.org/wiki/Fourth_dimension_in_literature

Great stuff. Thanks for the feedback. A second temporal dimension sounds pretty cool. Would it run counter to the current arrow of time? Or am I confusing two different topics

 

[SteveL27]

Search engines use n-dimensional vector spaces to determine the degree of similarity of two text documents.

http://en.wikipedia.org/wiki/Vector_space_model

 

[micromass]

Great stuff. Thanks for the feedback. A second temporal dimension sounds pretty cool. Would it run counter to the current arrow of time? Or am I confusing two different topics

I think you should ask this stuff in the physics forum. It are great questions, but they're not really math.

Search engines use n-dimensional vector spaces to determine the degree of similarity of two text documents.

http://en.wikipedia.org/wiki/Vector_space_model

Wow, this is good stuff!

 

[L.Newton]

Gonna move the temporal dimension over to the physics section.