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What is the volume (I have to clarify this: not the "volume" in the context of a 4-cube but in our everyday life where it is always a cubic unit) of a $4$-cube in $\text{cm}^3$? A 4-cube contains eight $3$-cubes. So it should be $8 * 3\text{cm}^3 = 216\text{cm}^3$. But is this all there is?

And if so, what volume does a $5$-cube have? A 5-cube has besides some $4$-cubes, 40 $3$-cubes, which have $40 * 3\text{cm}^3 = 1080\text{cm}^3$ in total.

Is this the exact volume of a $5$-cube?

Also consider the following thought experiment: You have a $3$-cube and fill it with water. You pour it's content into a $5$-cube. How many times could you do this until the $5$-cube is full?

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What do you mean by a 4-cube? If you mean a 4-dimensional cube, then its volume is in cm$^4$, not cm$^3$. Or do you mean the 3-dimensional 'surface volume', analogous to the 2-dimensional surface area of a 3-dimensional cube? –  TonyK Jan 10 '13 at 17:17
    
@Tony K: I know that I don't know a lot about higher dimensions. But some sources seem contradictory. First paragraph of en.wikipedia.org/wiki/Volume#Units for example seems to tell that volume always is a cubed unit (3 as exponent). You are talking about volume in cm^4. edit: Ok, you edited the part with the surface area. Yes, I am talking about that concerning the 4-cube. So a 5-cube has a surface area of 4-cubes, which have a surface area of 3-cubes, correct? –  Leif Jan 10 '13 at 17:24
    
Well, you were the one who brought up 4-cubes and 5-cubes. What did you mean by them? And where does this problem come from anyway? –  TonyK Jan 10 '13 at 17:27
    
@Tony K: I thought, the term "n-cube" would be well defined and known, because wolframalpha knows it (wolframalpha.com/input/?i=4-cube) and wikipedia, too. If you are wondering what I mean by them, I recommend reading wikipedia. –  Leif Jan 10 '13 at 17:30
    
OK, so a 4-cube is a 4-dimensional cube, as I thought. But then the rest of your post makes no sense! –  TonyK Jan 10 '13 at 17:35
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1 Answer

Consider an analogy. A 1-cube is just a segment of length $L\ \textrm{cm}$. The 2-cube, the square, has four 1-cubes for its sides. Does this mean that the area of a square is $4L\ \textrm{cm}?$ No. The perimeter of the square is measured in centimeters, but its area is measured in square centimeters, and has nothing (directly) to do with the 1-cubes that are on its face.

In $n$ dimensions, volume has units $\textrm{cm}^n$. There is potential for confusion because we use the same word -- volume -- to refer to different quantities: just like length and area are not the same thing, 3-dimensional volume and 4-dimensional volume are also unrelated, and you don't calculate the latter by adding up the former. The $n$-cube of side length $L$ has $n$-dimensional volume $L^n\ \textrm{cm}^n$. Notice the units. Adding up the volumes of the $n-1$-cubes gives you the $n-1$-dimensional volume of the $n$-cube's boundary -- you can think of this as surface area -- which is $2nL^{n-1}\ \textrm{cm}^{n-1}.$

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Thanks, your analogy helps me to rephrase my question better. I know that the 4 1-cubes of the 2-cube have a length of 4L cm in total. But it is still true that the 8 3-cubes that delimit the 4-cube have 8 * L cm^3 of total volume, is it? So maybe we cannot easily imagine what the "volume" of an n-cube is (with n > 3) but we can tell, how many 3-cubes it has and how much volume (which is our usual understanding of "volume") these have in total. Right? –  Leif Jan 10 '13 at 17:45
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Yes. But you cannot meaningfully talk about "filling" a 5-cube with water from a 3-cube, the same way that you can't ask how many times you have to fill a square with water and pour it into a cube before the cube is full. –  user7530 Jan 10 '13 at 18:09
    
+1. We are getting close. I am trying to make up an analogy with flatlanders. If I put my hand on them, my third dimension wouldn't be visible to them, they'd only see the area of my hand touching their world. What if a 4-cube entered our 3d world? Would we only see its 3-cubes floating around? Couldn't we fill these 3-cubes with water? The problem, I am seeing there (I must be wrong) is that some of these 3-cubes seem to be overlapping, i. e. they would occupy the same space - which shouldn't be possible. Or do we only see some 3-cubes, like the flatlanders only see a part of my surface? –  Leif Jan 10 '13 at 18:20
    
Exactly. You would see a cross-section of the 4-cube, which (assuming the cube is properly aligned) would look like a three-cube, just like a cross-section of a 3-cube looks like a square. –  user7530 Jan 10 '13 at 18:50
    
(You can only edit for 5 minutes, argh) Let's continue this analogy. I am annoying, right? Sorry. :) Let's assume a human would fly through the surface of flatland. The parts appearing in flatland would look like this: worldsstrangest.com/neatorama/full-body-mri With this as an example, I can hardly imagine how a 4-cube flying through our world would look. Would his 3d cross section change shape like the 2d cross section in the image above? How would this look like? –  Leif Jan 10 '13 at 19:18
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