In a $1$ dimensional object, the name for the region enclosed by it is the length of the object. In a $2$ dimensional object, the name for the region is the area of the object. In a $3$ dimensional object, the name is the volume of the object. What is the name in a $4$ dimensional object? Hypervolume? In general, what can we call the name of this region in a $n$ dimensional object? ("the region enclosed by this $n$ dimensional object" seems too long and wordy).

  • $\begingroup$ I've seen the term "bulk" used for polychorons, but the general term is indeed "hypervolume". $\endgroup$ – J. M. is a poor mathematician Jun 1 '13 at 6:11

I would say that just volume is actually a fine choice for any dimension $\geq 3$, but if you really want to emphasize the relevant dimension, you could say $n$-volume or $n$-hypervolume. Take a look at the Wikipedia page on Lebesgue measure.

I think good catch-all terms (i.e., for dimensions $1$ and $2$ as well) would be measure or content.

  • $\begingroup$ $4$-$volume$ sounds a little strange when said aloud, but I like it. $\endgroup$ – Justin Jun 1 '13 at 6:24
  • $\begingroup$ Glad I could help :) By the way, to get formatting in text, I suggest using Markdown (see here), so that to get $4$-volume, you would write $4$-*volume*, not $4$-$volume$. $\endgroup$ – Zev Chonoles Jun 1 '13 at 6:29
  • $\begingroup$ Thank you, I forgot about that (but I did use it in the question itself). Also, no wonder why I recognized your username; you were one of the old moderators! $\endgroup$ – Justin Jun 1 '13 at 6:30
  • $\begingroup$ Ah, I should have noticed, sorry! $\endgroup$ – Zev Chonoles Jun 1 '13 at 6:31

How about $n$-volume? Many objects carry the name from 3 dimensions as they are generalized to higher dimensions: $n$-cube, $n$-ball whose boundary is an $(n - 1)$-sphere come to mind.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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