Calculating Volume in the 4th dimension: A mathematical curiosity,

[Hero in a half shell]

I was watching a stupid educational video about calculating volume, and noticed something curious: warning, possibly dodgy theoretical maths ahead.

Say you have a cube, and you know it to be a cube, its dimensions are 5x5x5 centimetres.

In the first dimension all you will percieve is a straight line, 5 cm long. so you could say:
1d=5 [where d is the dimension]

In the second dimension you will percieve the cube to be a square, 5cm long, 5cm high, to get the volume you multiply the edges by themselves, so:
2d=5x5 [or 5 [sup]2[/sup]]
so 2d=25

In the third dimension you have the whole cube, 5 long, 5 high and 5 wide. Volume is 5x5x5 so:
3d=5x5x5 [or 5 [sup]3[/sup]]
so 3d=125

Now we get to the interesting part, does this mean that in the forth dimension, whatever the heck that may be, that the volume of the cube is
4d =5x5x5x5 [5 [sup]4[/sup]]
following the formulae that we have, the cube in the forth dimension is actually 625cm[sup]4[/sup]

If this formulae is right then we are able to calculate the volume of an object in dimensions that we can't even imagine. Now I know that the forth dimension is touted to be time, but that has never sat well with me, as the other dimensions are each physical plains of geometry, so could this formulae disprove the supposed forth dimension of time? And did you know that the volume of a 5cm cube in the 10th dimension is over 48 million cm[sup]10[/sup]

EDIT: It is now called the Grooveathon formula (by me)

 

[DefunctTheory]

Try this.

EDIT: The fourth dimension is time, not more space.

 

[Vakz]

Incredibly clever clip I must say, though I do feel I'll have to watch it another five times or so before I actually get it

 

[Joehova]

The fourth dimension isn't necessarily time. It could be space and we just can't perceive it.

 

[Owlslayer]

I agree with Vakz. That was quite educational, though i still get the dimensions mixed up. And at some points it was a bit difficult to imagine what the video was talking about.

 

[venatus]

Try this.

EDIT: The fourth dimension is time, not more space.

not quite, truth is there is no official fourth dimension, much in the same way there's no official first dimension or second, time is a fourth dimension we live in but scientists are also pretty certain there exist more spacial dimensions as well, but even mathematically it's hard for us to go beyond 4 spacial dimensions.

 

[MiketheBassMan]

There's a fundamental problem with just increasing the exponent. Cube implies a special rectangular prism with equal side lengths, which is why from 1 to 2 to 3 dimensions, the logical volume's (area and length in 2 and 1d) exponent increases by one for each dimension. To say that it's fourth-dimensional volume would be x^4 would be to say that some well-defined fourth dimension exists that is said to have some units of magnitude equal to the side lengths of the cube in the third dimension. I don't think 'cube' implies that at all.

 

[Eclectic Dreck]

EDIT: The fourth dimension is time, not more space.

It doesn't matter what the fourth dimension is in the slightest. One can calculate the length of a line for example without caring what the line represents just as readily as one can calculate an area of a plane or the volume of a space. Adding a new dimension adds difficulty only when it comes to representing the information visually. One can still perform all the calculations in the same basic way regardless of dimension or what is being represented.

 

[Hero in a half shell]

Try this.

EDIT: The fourth dimension is time, not more space.

not quite, truth is there is no official fourth dimension, much in the same way there's no official first dimension or second, time is a fourth dimension we live in but scientists are also pretty certain there exist more spacial dimensions as well, but even mathematically it's hard for us to go beyond 4 spacial dimensions.

Exactly, so could my formulae show that there is in fact a further geometric plain before time, or that time strictly speaking is not a dimension at all, since there is no point that time will fit into the formulae, dimensions are therefore striclty geometric based, time is an assumption that must be made to allow any dimension to exist?

 

[Akiraacecombat]

in mathematics and linear algebra n-number dimensions are consider,the problem is that they cannot be represented with graphics,the is defined in matrices as R dimensions,which can be mathematically correct,but is impossible to represent,some people say that the 4th dimension is time,and that might be true but mathematically there is n-number of dimensions,read wikipedia,the chapter about linear algebra and matrices and you will see what I mean.

 

[DefunctTheory]

EDIT: The fourth dimension is time, not more space.

It doesn't matter what the fourth dimension is in the slightest. One can calculate the length of a line for example without caring what the line represents just as readily as one can calculate an area of a plane or the volume of a space. Adding a new dimension adds difficulty only when it comes to representing the information visually. One can still perform all the calculations in the same basic way regardless of dimension or what is being represented.

I never said one couldn't measure it, only that, if there is a 4th dimension, its 'duration,' from what I've read.

And 'duration' does not have volume, so it wouldn't increase the size of anything.

But, like so many others have brought up, it COULD be space, or time, or the bacon dimension for all we know. It's hard to be sure about something that you cannot, under any circumstances, perceive.

I only brought up that video because I stumbled it yesterday, and I thought it was a neat concept, and visualized what I've been brought to understand.

Then again, I blow at physics, and therefore math, so what the fuck do I know.

EDIT: Just noticed I have a 117 score. Not too shabby for my first MM.

 

[Hlain]

EDIT: The fourth dimension is time, not more space.

Lol!

We observe the universe as 4 dimensional space-time, meaning we see 3 space dimensions and 1 time dimension. That doesn't stop a hypothetical cube of having 4 space dimensions.
The dimensions aren't numbered, with nr 1 being this, nr 2 that etc.

 

[Jaime_Wolf]

"The fourth dimension" is a purely hypothetical construct. Saying that "the fourth dimension is time" makes as little sense as when people just dictate "the third dimension is depth!". Nature doesn't conveniently number dimensions for you. Saying anything about a fourth (spatial) dimension is just talking about four-dimensional space, it doesn't make any sense to actually say which dimension is which. And given the context, it's very obvious that the poster is talking about an additional spatial dimension, so it's especially dumb to try to break out your popular physics books and condescendingly lecture him (also, again, you're wrong, or at best incomprehensible).

I'm not a geometer, but I imagine the formula does generalise such that the volume of a tesseract (a four-dimensional cube) is indeed sidelength^4. On the other hand, I'm suspicious of what "volume" even means in four dimensions. For instance, there is no identical concept of volume when you have only two dimensions, so I'm not sure if it's really fair to call the corresponding thing in four dimensions "volume".

 

[Hlain]

For instance, there is no identical concept of volume when you have only two dimensions, so I'm not sure if it's really fair to call the corresponding thing in four dimensions "volume".

Well, in 2 dimensions, we have area. In 3 volume.
So there must be something corresponding to that in 4 space dimensions, call it whatever you will. Volume is as good a placeholder as any.

 

[Jaime_Wolf]

For instance, there is no identical concept of volume when you have only two dimensions, so I'm not sure if it's really fair to call the corresponding thing in four dimensions "volume".

Well, in 2 dimensions, we have area. In 3 volume.
So there must be something corresponding to that in 4 space dimensions, call it whatever you will. Volume is as good a placeholder as any.

Meh, area and volume are SORT of the same thing, but not really. I'm just saying that it's effectively impossible to imagine what "four-dimensional volume" would even be like, so it's hard to have intuitions about appropriate formula.

Now this thread is just reminding me of Flatland.

 

[The Seldom Seen Kid]

The 4th dimension doesn't actually exist a way that makes sense to our human eyes, it's purely theoretical. From what I know (which really isn't much), it's technically there, but we can't see it.

The problem with you equation is that to be able to calculate x^4, you'd need a clearly defined dimension of the cube that can be accurately measure, not imaginary mumbo-jumbo, as the case may be.

 

[The Diabolical Biz]

Try this.

EDIT: The fourth dimension is time, not more space.

Very good clip!

OT: I'm not sure if that's how it works, it seems a bit like a 'Too easy' scenario, but then what do I know

 

[Wyes]

Try this.

EDIT: The fourth dimension is time, not more space.

not quite, truth is there is no official fourth dimension, much in the same way there's no official first dimension or second, time is a fourth dimension we live in but scientists are also pretty certain there exist more spacial dimensions as well, but even mathematically it's hard for us to go beyond 4 spacial dimensions.

Exactly, so could my formulae show that there is in fact a further geometric plain before time, or that time strictly speaking is not a dimension at all, since there is no point that time will fit into the formulae, dimensions are therefore striclty geometric based, time is an assumption that must be made to allow any dimension to exist?

Mathematically speaking, we can have as many dimensions as we please. In fact, when you start doing some more advanced algebra (as in, after high school), you start learning about everything in n dimensions, where n is any positive integer (i.e. 1, 2, 3,...,n). The object you're describing is known as a hypercube (and objects of that type are known as hyper-volumes), and I'm fairly certain that there's much maths devoted to the subject, though I don't know it.
Physically speaking, there are only four dimensions we're directly aware of; the three spatial dimensions, and time. There are theories, such as string theory, which predict many more dimensions, though by their nature we're unable to observe them in every day life.

It is not meaningful to suggest that there's another geometric dimension before time, because dimensions are not 'fixed', for lack of a better term. What I mean is that if I want to, I can have displacement (i.e. length) on one axis, and time on the other. Or I can have length vs breadth, or I could have volume on one angle and time on the other, and there's no reason whatsoever I can't do that. We get to pick our coordinate system (and in fact there are many different types of coordinate systems beyond Cartesian/rectangular coordinates, such as polar, spherical, curvilinear etc.) It really just depends on how you look at it. It is meaningful to talk about there being another geometric dimension, and the fact that we can have such objects in maths means if we do discover more dimensions, we will be prepared to deal with them analytically, if maybe not intuitively for most of us.

Also crap, that wasn't meant to be such a long post.

 

[venatus]

Try this.

EDIT: The fourth dimension is time, not more space.

not quite, truth is there is no official fourth dimension, much in the same way there's no official first dimension or second, time is a fourth dimension we live in but scientists are also pretty certain there exist more spacial dimensions as well, but even mathematically it's hard for us to go beyond 4 spacial dimensions.

Exactly, so could my formulae show that there is in fact a further geometric plain before time, or that time strictly speaking is not a dimension at all, since there is no point that time will fit into the formulae, dimensions are therefore striclty geometric based, time is an assumption that must be made to allow any dimension to exist?

this is way beyond me but I don't think a calcul;ation for volume would preclude time as a dimension

 

[Hlain]

OT: I'm not sure if that's how it works, it seems a bit like a 'Too easy' scenario, but then what do I know

Just think about it like this. First we have a line of length, or size, x.
We put a bunch together to form a plane, with the length x on every side, size x^2.
Same with next dimension, put planes together, get a cube, size x^3.

The mathematical 4th space dimension follows the same rules as the lower. There's no reason the rules suddenly stop applying. We overthink it, because we can't imagine a 4-dimensional body.
The size is just x^4.