Length is to one dimension as Volume is to three dimensions as what is to four dimensions?

Question

If you have a one dimensional object, it would just have length. If you had a two dimensional object, it would have area, and it's edges would have length. If you had a three dimensional object, it would have volume, it's faces would have area, and it's edges would have length. If I had a four dimensional object, it would have what, and it's what would have volume, and it's faces... etc.

Additionally, in English, are there prefixes and roots for going even further for these two terms. I googled this, and I wasn't able to hit the right terms to lead me to the awnser.

Answer 1

Mathematicians are apparently not as clever as you think. Rather than make up new words, they instead use "volume" to mean all of these things. Length is $1$-dimensional volume, area is two dimensional, etc. There may be a special word for four dimensional volume somewhere, but it's not in common use.

To generalize what you mean when you say "faces," you could call it the boundary.

Answer 2

The notion of volume generalises to $n$-dimensions: $$V_{2k}(R)=\frac {\pi^k}{k!} R^{2k}$$ is the volume of the $2k$-ball of radius $R$. In odd dimensions it's $$V_{2k+1}(R)=\frac {2^{k+1}\cdot \pi^k}{(2k+1)!!}R^{2k+1}$$

*Note that the expressions go to zero as $n$ goes to $\infty $.

For volumes of arbitrary sets see volume element , which is used to integrate.

Furthermore the boundary $\partial M$ of an $n$-dimensional manifold $M^n$ is generally $n-1$ dimensional.