Why do we extrapolate 4d based on 3D volume, and not surface area? For 2D->3D we use area. Why would we then use volume in 3D->4D when surface area is an option?
The space that is enclosed by a unit sphere in dimension 1 is the interval (-1, 1). In physics this is measured in meters. This is referred to as length. The space enclosed by the unit sphere in dimension 2 is measured in meters squared. This is referred to as area. In dimension 3 we call it volume and measure it in meters cubed.
In dimensions 4 through infinity we quickly come to a problem. Do we come up with unique names to refer to amount of space enclosed by a unit sphere? Mathematicians have decided to just use the word volume. We rely on context to make it clear what the dimension is. Similarly we use the word n-sphere to refer to the collection of points in n-dimensional space that are exactly 1 unit from the origin.
In my applied mathematics education the highest-dimensional unit of measurement of a given dimensionality was referred to generally as ‘content’: length is the content of 1D objects, area is the content of 2D objects, volume is the content of 3D objects, hypervolume is the content of 4D objects, and henceforth you just start numbering them...
We're using volume the whole time. Area is just the two-dimensional equivalent of volume. The two-dimensional equivalent of surface area would be perimeter, which is relatively rarely used.
The more general term for ’area’ and ’volume’ in higher dimensions is ’content’: area is the content of a 2D item, volume is the content of a 3D item, hypervolume is the content of a 4D item, et cetera.
Same reason we don't extrapolate 2D->3D based on circumference. You need all the information of the (n)D thing, plus a new length, to get to the (n+1)D thing.
