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

[Hero in a half shell]

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.

Thats my formula in a nutshell, Well done. I think I shall christen it "THE GROOVEATHON FORMULA"

 

[Lullabye]

Mhm, I thought that 4th dimension measurements were carried out with quantum computation due to there being no physically concrete(in terms of probability) way to mathematically describe its potential effects on our common (3) dimensions? Plus all those potential folded dimensions contained with themselves ....

 

[crimson5pheonix]

Here you go, a 4-dimension "cube" rotating. It's called a tesseract in the 4th dimension.

 

[Hero in a half shell]

Here you go, a 4-dimension "cube" rotating. It's called a tesseract in the 4th dimension.

I don't know what the crap that is, but I can tell you its volume with a great degree of certainty if you told me the length of one side. MUUAHAHAHAHAHAHAHA Pythagoras can keep his triangles.

 

[Eclectic Dreck]

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

The the dimension of anything in mathematics has nothing to do with measuring anything. It represents, in the simplest terms, the number of unique axis are required to represent the information. One dimensional mathematics deals with lines, two dimensions deals with planes, three with space and so on.

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

Duration can be represented any number of ways. Math does not really care what you represent with an equation. Consider modeling one of the famous and incredibly simple physics formulas, say vf = vo + a*t (that is, the final velocity of an object is equal to it's initial velocity plus the product of it's acceleration over time). If you represent this equation on a plane such that the x axis represents time and the y axis represents velocity you get a straight line. The slope of this line at any point represents the acceleration at an infinitesimal moment. The x coordinate represents time passed and the y coordinate represents the exact velocity. The area under the line and above the axis on the other hand represents the change in position with respect to time.

But that very same graph might represent something entirely different.

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.

My argument has nothing to do with what the fourth dimension might be in the physical sense but rather that, mathematically at least, there is nothing strange or weird about it. To take the simplest possible shape, consider the square. Add a dimension of equal length to any side and you have a cube. Add a new dimension and do the same and what do you have? Turns out, a hypercube. No matter what the dimension of an object, one can always project said object into a space with fewer dimensions. Thus if you have an object with a million dimensions, you can still represent it with three.

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.

The very best work I'm aware of that deals with such things is a book (and a short film) called "Flatland" which more or less addresses the problem of hyper dimensional geometry.

[/quote]

 

[Stephanos132]

Would it be 'volume' for 4 dimensions, though? I always thought volume was a 3d thing.

Also, time as a dimension is the 4th component of 'spacetime', where that distinction is made more for simplicity than anything else, and it's not a spatial dimension either (ie, you don't measure it in metres), so your 4d hybercube wouldn't bother time at all I don't think. Of course, both space and time are largely built on our admittedly limited perception of the universe we inhabit anyway, so you could define a dimension however you wanted really.

 

[Chromatic Aberration]

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.

OP, take it from someone who is currently doing his Masters in Physics - this man here speaks the truth.

And to boggle your mind a bit: In classical statistical physics you routinely work with the volume of 6N-dimensional spheres where N is of the order of 6*10^23[footnote] This number corresponds to the number of atoms in 12g of carbon. It is called Avogadros Number and sort of defines where you start looking at systems where you can invoke classical physics i.e. neglect quantum mechanical effects to a certain degree[/footnote]. Here 3N dimensions correspond to a spatial dimension i.e. the position of N particles in 3 spatial dimensions and the other 3N are momentum dimensions i.e. the momentum of N particles in 3 momentum-dimensions.

 

[Hero in a half shell]

also volume is a third dimension concept, whatever you're calculating in the fourth dimension it would not be volume.

Yes, It wouldn't be volume, but if the length of one side was 5 cm and all sides were equal, then there would be 625cm[sup]4[/sup] of the little blighters.

 

[Hero in a half shell]

OP, take it from someone who is currently doing his Masters in Physics - this man here speaks the truth.

So as a person who actually knows what he is talking about instead of us hacks, what do you think of my "grooveathon theory" (You can't criticise the name, a guy further up stated that a cube in 4d is actuallu called a 'hypercube' by mathmaticians, very Tron)

 

[thethingthatlurks]

Easy:
Find an expression of the volume of your object (that's 3D volume) in terms of w, the 4th dimensional coordinate, then take the integral of that expression over all possible values of w. Don't try to visualize these things, just follow standard mathematical procedures.

 

[someonehairy-ish]

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.

I think that's basically what is meant by 'spacetime'

The cube thing up there is awesome btw, I'm thinking 'Tesseract' would be a good name for some kinda prog band xD

 

[Hungry Donner]

We exist in a world with three spacial dimensions and one temporal dimension, so in terms of general human perception time is "the" forth dimension.

However as others have pointed out there are theoretical spacial dimensions beyond the third. No numerical dimension is officially tied to anything, in fact an animated picture would have three dimensions: two would be space and the third in this case would be time.

While it's difficult to visualize higher dimensional space it isn't impossible. My dad, who had the benefit of visual aids, explained it to me this way:

Let's start with a 0 dimensional object, a point. (A point occupies no space at all)

Now let's grab that point and stretch it over to another area. We now have a line, a 1 dimensional object.

Drag a line and we end up with a square, a two dimensional object.

Drag a square and we end up with a cube, a three dimensional object.

Grab that cube and drag it and you end up with a hyper-cube, or tesseract. We cannot see forth dimensional space but just as we can approximate a cube when drawing on a piece of paper we can approximate a tesseract with a 3D model. (It is also possible to approximate a tesseract with a 2D drawing but it is very messy. Three dimensions, whether as a 3D spacial model or an animated 2D image, makes it much clearer).

 

[Outright Villainy]

Here you go, a 4-dimension "cube" rotating. It's called a tesseract in the 4th dimension.

fun fact, that's supposedly stationary in the fourth dimension and the only way for us to perceive it is that animation.


also volume is a third dimension concept, whatever you're calculating in the fourth dimension it would not be volume.

The problem with trying to show someone a tesseract and make them understand it is it's far easier to do so with a 3D model, which would be a projection of a 4th dimensional object in 3D dimensions, much in the same way that this:

Is the projection of a cube in 2 dimensions. All the lines are perpendicular, but in lower D projection that's skewed. I think it gets way too cluttered to make any sense in a 2D gif, but maybe that's just me. I thought it helps to imagine it as a smaller cube within a big cube with lines joining the corners of the inner cube to the corresponding corners of the other, and then imagine those lines are perpendicular as well, but that's the skewed projection, as well as the inner cube being smaller of course, it'd be the same size in it's true shape.

 

[Grimfoxw00t]

I think that you are getting things mixed up, as Dajosch pointed out. Following the formula might give you the calculation for spacial dimensions in a cube that would contain more and more spacial dimensions. But speaking of the fourth dimension in terms of physics, as time, has nothing to do with that im afraid.

 

[Chromatic Aberration]

OP, take it from someone who is currently doing his Masters in Physics - this man here speaks the truth.

So as a person who actually knows what he is talking about instead of us hacks, what do you think of my "grooveathon theory" (You can't criticise the name, a guy further up stated that a cube in 4d is actuallu called a 'hypercube' by mathmaticians, very Tron)

Well not much I'am afraid - except for the name of course
I bet (and unfortunately I do not know that for certain) a n-dimensional cube is defined in such a way that it will yield the volume as you said. But to conclude anything about time or whatever from that would be nonsense: As Wyes said, a dimension is - first and foremost - only a purely mathematical thing which is independent of any units or any physical theories whatsoever.

And I would also be very careful in just saying that the fourth dimension is time even when you are speaking about the natural world: what you can show in physics is that you can have wonderful coherent mathematical theories about the behaviour of nature if you consider so called 4-vectors [http://en.wikipedia.org/wiki/4_vector] that include time along with the three spatial components. Thus, what we have is still only a mathematical construct and to derive any absolute claim as "The fourth dimension is time!" from that is to be considered very carefully.

 

[HitoriNoOkami]

Try this.

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

Yeah, so the same guy who narrated that ^ video
made this one (notice the song at the end):

http://www.youtube.com/watch?v=V-Y4xseftgQ

The point I want to make is that we can't really empirically measure anything beyond the 3rd (maybe 4th if you want to count time) dimension and that once you start trying to extrapolate higher dimensions, you're bound to hit the limits of your imagination or just become crazy to have enough imagination to think about it.