Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 1 down vote favorite

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?

share|improve this question
3  
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 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 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 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 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 at 17:35
show 1 more comment

1 Answer

up vote 2 down vote

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 $2nL^{n-1}\ \textrm{cm}^{n-1}.$

share|improve this answer
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 at 17:45
1  
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 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 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 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 at 19:18
show 1 more comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.