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Jan
10
comment Volume of 4-cube and n-cubes with edge length 3cm
Wait, it would look like this: upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/… Right? On a second thought: This is a rotation. The human is just moving through flatland. Hm.
Jan
10
comment Volume of 4-cube and n-cubes with edge length 3cm
(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?
Jan
10
comment Volume of 4-cube and n-cubes with edge length 3cm
+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?
Jan
10
revised Volume of 4-cube and n-cubes with edge length 3cm
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Jan
10
comment Volume of 4-cube and n-cubes with edge length 3cm
At least the first sentence. That's because of the term "volume in cm^3" in the context of a 4-cube. ;) But the rest holds still true because cubes with more than 3 dimensions have a certain number of 3-cubes. You can find the formula here: en.wikipedia.org/wiki/Hypercube#Elements
Jan
10
comment Volume of 4-cube and n-cubes with edge length 3cm
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?
Jan
10
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Jan
10
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Jan
10
comment Volume of 4-cube and n-cubes with edge length 3cm
@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.
Jan
10
comment Volume of 4-cube and n-cubes with edge length 3cm
@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?
Jan
10
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Jan
10
revised Volume of 4-cube and n-cubes with edge length 3cm
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Jan
10
asked Volume of 4-cube and n-cubes with edge length 3cm