[OscarCunningham]

The convention is to name things for their inherent dimensionalitly, not the space they live in.

A sphere in three dimensional space is a two dimensional object, in the sense that it's a surface on which the points can be described by two coordinates (say longitude and latitude).

Similarly a point on a circle can be described by a single coordinate (distance around the circle).

So the 1-sphere is a circle, and normally lives in 2 dimensional space, while the ordinary sphere is called a 2-sphere, and lives in three dimensional space.

[algorias]

If you talk about the "volume of a sphere" (like the article does), then a sphere embedded in 3 dimensions should be 3 dimensional. This would usually be called a ball and not a sphere, though.

Note that analogously, a cube in 3 dimensional space is always considered to be 3 dimensional.

[OscarCunningham]

Right, the 2-sphere is the surface of the 3-ball. I think "volume of a sphere" is okay though. It just means the volume enclosed.

>a cube in 3 dimensional space is always considered to be 3 dimensional.

So "cubes" are taken to include their interior. That makes sense I suppose. I wonder what the surface of a cube is called. A "box" maybe?

[HelloNurse]

Things that are named for their inherent dimensionality regardless of the dimensionality of the space they are contained in are others, such as points (0), lines (1), planes (2), general manifolds, vector spaces, etc. Spheres (more properly balls) and cubes (or hypercubes) are understood to have the same number of dimensions as the space they are contained in.