Understanding Hypercubes: Exploring the Concept of 4th Dimension

[Nugso]

Hello everyone once again. If I remember correctly, hypercubes are said to have 4th dimension. But as far as I know, the 4th dimension is time. So, when we're describing the 4th dimension I guess we can say, " Let's meet at my house at 12.20 pm". When I say "at my house" I'm talking about the space(3 dimension) and by saying 12.20 pm I'm talking about time(4th dimension). My question is, how can Hypercubes have 4 dimension?

 

[Mandelbroth]

Hello everyone once again. If I remember correctly, hypercubes are said to have 4th dimension. But as far as I know, the 4th dimension is time. So, when we're describing the 4th dimension I guess we can say, " Let's meet at my house at 12.20 pm". When I say "at my house" I'm talking about the space(3 dimension) and by saying 12.20 pm I'm talking about time(4th dimension). My question is, how can Hypercubes have 4 dimension?

A hypercube is a generalization of the idea of a cube to any arbitrary dimension $n$.

Time is not "the" fourth dimension. If we wish to describe an object's position in space and a point in time, then we can say that time is one of the 4 dimensions given. For all intents and purposes, time could be the first dimension here as well.

Informally, an object's dimension can be thought of as the minimum number of coordinates needed to specify a point on the object.

 

[Nugso]

A hypercube is a generalization of the idea of a cube to any arbitrary dimension $n$.

Time is not "the" fourth dimension. If we wish to describe an object's position in space and a point in time, then we can say that time is one of the 4 dimensions given. For all intents and purposes, time could be the first dimension here as well.

Informally, an object's dimension can be thought of as the minimum number of coordinates needed to specify a point on the object.

So wen can safely say 5-dimensional hybercube. Right? Thanks by the way!

 

[micromass]

So wen can safely say 5-dimensional hybercube. Right?

Sure, that exists.

Dimensions are well-studied in mathematics in various contexts. In particular, it's true that time isn't necessarily the fourth dimension. But it could be a fourth dimension.

What would a hypercube look to us if time were the fourth dimension? Well, we would perceive it as an ordinary cube appearing out of nowhere at a time. It remains stationary for a certain time until it suddenly disappears. There are various other ways in which hypercubes can appear to us if time is the fourth dimension.

If you haven't, read the book flatland by Abbott. It's a very good book and freely available on the internet.

This also looks good: http://www.felderbooks.com/papers/4dplots.html

 

[Nugso]

Sure, that exists.

Dimensions are well-studied in mathematics in various contexts. In particular, it's true that time isn't necessarily the fourth dimension. But it could be a fourth dimension.

What would a hypercube look to us if time were the fourth dimension? Well, we would perceive it as an ordinary cube appearing out of nowhere at a time. It remains stationary for a certain time until it suddenly disappears. There are various other ways in which hypercubes can appear to us if time is the fourth dimension.

If you haven't, read the book flatland by Abbott. It's a very good book and freely available on the internet.

This also looks good: http://www.felderbooks.com/papers/4dplots.html

I've yet to read a book that's about 5-dimensional thingummies and stuff. Thanks both for the link and for the explanation.

I got another question by the by, what else can we describe as 4th dimension besides time? Density?

 

[micromass]

I've yet to read a book that's about 5-dimensional thingummies and stuff. Thanks both for the link and for the explanation.

I got another question by the by, what else can we describe as 4th dimension besides time? Density?

Anything can be a fourth dimension really. Very loosely speaking, a dimension corresponds to labeling an axis on a graph. For example, in this graph:

we have a 2-dimensional situations. The first dimension is temperature and the second dimension is pressure. This is the best way to think of dimensions.

Of course, you're speaking of "physical" dimensions (whatever that means). So the first three dimensions are given by length, width and height, and you want to extend this. But there is no reason why it should be time. The theory of relativity might say it's time because it's useful there. Other theories might say other things. In this case, saying that time is the fourth dimension is just a mathematical convenience. It gives a theory that works. But it's only that, mathematics. It's not actually real.

 

[Nugso]

Anything can be a fourth dimension really. Very loosely speaking, a dimension corresponds to labeling an axis on a graph. For example, in this graph:

we have a 2-dimensional situations. The first dimension is temperature and the second dimension is pressure. This is the best way to think of dimensions.

Of course, you're speaking of "physical" dimensions (whatever that means). So the first three dimensions are given by length, width and height, and you want to extend this. But there is no reason why it should be time. The theory of relativity might say it's time because it's useful there. Other theories might say other things. In this case, saying that time is the fourth dimension is just a mathematical convenience. It gives a theory that works. But it's only that, mathematics. It's not actually real.

Thanks micromass, I guess I see now. However, once again, I'd like to ask one more thing. A friend of mine has just told me that hypercubes' 4th dimensions are "physical" dimensions such as the coordinates x, y, z. I have a hard time understanding it. Is he right?

 

[micromass]

Thanks micromass, I guess I see now. However, once again, I'd like to ask one more thing. A friend of mine has just told me that hypercubes' 4th dimensions are "physical" dimensions such as the coordinates x, y, z. I have a hard time understanding it. Is he right?

No, I don't think he's right. The extra fourth dimensions are not spatial dimensions. So really anything can be a fourth dimension, it's just that time is the best one. So I wouldn't call the 4th dimension "physical". But this is of course philosophy instead of mathematics. This means that there is not really a "right" answer to your question, and many people will disagree with me and will have very valid reasons for it.
The best thing you could do is to think about it yourself, read up on the math and decide for yourself whether you consider the extra dimension to be physical. Any answer is right, really, depending on how you argue it.

 

[Nugso]

No, I don't think he's right. The extra fourth dimensions are not spatial dimensions. So really anything can be a fourth dimension, it's just that time is the best one. So I wouldn't call the 4th dimension "physical". But this is of course philosophy instead of mathematics. This means that there is not really a "right" answer to your question, and many people will disagree with me and will have very valid reasons for it.
The best thing you could do is to think about it yourself, read up on the math and decide for yourself whether you consider the extra dimension to be physical. Any answer is right, really, depending on how you argue it.

Thanks but I mean, according to the definition of hypercubes, it looks like it's a physical dimension. Not sure though.

Not: I'm very confused. :(

 

[micromass]

Thanks but I mean, according to the definition of hypercubes, it looks like it's a physical dimension. Not sure though.

Not: I'm very confused. :(

OK, but what do you mean when you say "physical"? And why does the definition make you think it's physical?

 

[Nugso]

4th coordinate like x, y and z. Tesseracts', 4th dimensional hybercubes'(am I right?), picture make me thing it's physical.

http://en.wikipedia.org/wiki/File:Schlegel_wireframe_8-cell.png

It doesn't look like it's got something to do with time. :/

 

[micromass]

4th coordinate like x, y and z. Tesseracts', 4th dimensional hybercubes'(am I right?), picture make me thing it's physical.

http://en.wikipedia.org/wiki/File:Schlegel_wireframe_8-cell.png

It doesn't look like it's got something to do with time. :/

It's not because you can draw a picture of something that it actually exists.
The question what you mean with physical was actually more rhetorical. The answer whether the 4th dimension is physical, depends on that answer.

I might define physical as "it actually exists". Then time might make a good 4th dimension. But I can make up a lot of dimensions in mathematics that don't exist in reality. So they are not physical.

 

[Nugso]

It's not because you can draw a picture of something that it actually exists.
The question what you mean with physical was actually more rhetorical. The answer whether the 4th dimension is physical, depends on that answer.

I might define physical as "it actually exists". Then time might make a good 4th dimension. But I can make up a lot of dimensions in mathematics that don't exist in reality. So they are not physical.

Yea, sorry for explaning it very badly. What I meant was, things that have 4th dimensions(the 4th dimension here is like x, y, z. Length, width etc that is) are called tesseracts(hybercupes).

 

[micromass]

Yea, sorry for explaning it very badly. What I meant was, things that have 4th dimensions(the 4th dimension here is like x, y, z. Length, width etc that is) are called tesseracts(hybercupes).

But that's a mathematical description. It doesn't need to be real or physical in any way. I can describe a lot of mathematical things which don't actually exist.

 

[Nugso]

But that's a mathematical description. It doesn't need to be real or physical in any way. I can describe a lot of mathematical things which don't actually exist.

Alright once again. I'd like to make it clearer once again, I meant spatial dimension when I said physical dimension.

Thanks!

 

[micromass]

Alright once again. I'd like to make it clearer once again, I meant spatial dimension when I said physical dimension.

Thanks!

Then no, there are only three spatial dimensions (that we know of). Not even time is a spatial dimension.

 

[Nugso]

Then no, there are only three spatial dimensions (that we know of). Not even time is a spatial dimension.

Okay, guess it's clearer now. Thanks for englihtening. :D

 

[The17YearOld]

Nugso, I believe I understand what you mean by physical dimensions. Correct me if I'm wrong but I believe you are talking about the dimensions at work in our own world. Not necessarily the dimensions on paper or numbers used to represent them, but the known dimensions like the fact that the Earth is a 3d solid and space is 4 dimensional. What I am trying to get at is how the 4th dimension relates to time because I believe I have a fairly good way of explaining how time and the 4th dimension are linked. If you are interested please respond and tell me if I am somewhat understanding what you mean by the physical dimensions.

 

[Theorem.]

Nugso, I believe I understand what you mean by physical dimensions. Correct me if I'm wrong but I believe you are talking about the dimensions at work in our own world. Not necessarily the dimensions on paper or numbers used to represent them, but the known dimensions like the fact that the Earth is a 3d solid and space is 4 dimensional. What I am trying to get at is how the 4th dimension relates to time because I believe I have a fairly good way of explaining how time and the 4th dimension are linked. If you are interested please respond and tell me if I am somewhat understanding what you mean by the physical dimensions.

I think you might be confusing a mathematical object with a physical model? A hypercube is a mathematical construct, independent of a physical world. the physical dimension corresponding to time is completely unrelated to the concept of a hypercube with dimension 4.

 

[Nugso]

Nugso, I believe I understand what you mean by physical dimensions. Correct me if I'm wrong but I believe you are talking about the dimensions at work in our own world. Not necessarily the dimensions on paper or numbers used to represent them, but the known dimensions like the fact that the Earth is a 3d solid and space is 4 dimensional. What I am trying to get at is how the 4th dimension relates to time because I believe I have a fairly good way of explaining how time and the 4th dimension are linked. If you are interested please respond and tell me if I am somewhat understanding what you mean by the physical dimensions.

Yes, but it looks like things which's 4th dimensionns are spatial(There're no 4 spatial dimensions though, only imaginary) are called hypercubes.

 

[WannabeNewton]

5 dimensional Kaluza-Klein theory has 4 spatial dimensions. It's really not clear what you are asking; I can't tell if you're talking about physics or math.

 

[PhanthomJay]

Tesseract hyper cube: 4D projection onto 3D...requires 3-D glases old red and green style lenses..available free...http://dogfeathers.com/java/hypercube2-nogl.html

 

[Nugso]

5 dimensional Kaluza-Klein theory has 4 spatial dimensions. It's really not clear what you are asking; I can't tell if you're talking about physics or math.

Didn't know its name before. I meant "spatial". I guess it's about math.

@PhantomJay

Don't have a 3d glasses :(

 

[WannabeNewton]

If it's about math then the phrase "physical dimensions" don't really have any meaning. Mathematics doesn't prescribe physical meaning to dimensions of abstract geometric objects; I can call them whatever I want. A hypercone satisfies the equation $x^2 + y^2 + z^2 - t^2 = 0$ and as far as mathematics is concerned, the coordinates $(t,x,y,z)$ are just arbitrary labels constrained by the above equation; there is no physical meaning ascribed to any of them. I can then choose to call the $t$ coordinate as "time" and the $x,y,z$ coordinates as "spatial" and we basically end up with the light cones from special relativity.