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A distinction is commonly made between a ball (solid) and a sphere (the boundary of a ball). This distinction is made in other dimensions as well (e.g. circle versus disc, in 2D).

From what I've seen trying to research this online, the word "cube" is usually used to refer to the soild. Is there any standard name for just the boundary of a cube? The only thing that came to mind was "box" but that both doesn't seem to be a standard term, and might intuitively refer to the boundary of any rectangular prism.

(My question extends to squares as well, i.e. is there a standard name for just the boundary of a square? And a similar question could be asked in reverse about the torus, where that name is usual used to refer to only the boundary.)

If there is no standard name, my follow-up question would be: why do we have distinct names for spheres and balls, but not other common solids? Just historical accident? (Is the distinction made in languages other than English?)

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    $\begingroup$ For the 2D case, "lamina" seems to be a general name for the filled region, so maybe circle is to disk as square is to ... square lamina? $\endgroup$
    – user170231
    Feb 28 at 2:00
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    $\begingroup$ How about "the boundary of a cube"? $\endgroup$
    – Lee Mosher
    Feb 28 at 2:06
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    $\begingroup$ Maybe the surface of the cube? $\endgroup$ Feb 28 at 2:11
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    $\begingroup$ "surface of a cube" is pretty common and unambiguous and clear ("boundary of a cube" seems a little weird to me but ... that's just personal taste.) I'm not sure it's universally accepted that we do us sphere to specifically mean the surface. I think in a lot of cases people might (perhaps incorrectly) refer to a sphere as the solid and be understood. I think the reason we talk of spheres as surfaces is that in Topology we can have flat 2D space (a plane) and curved 2D space-- ultimately a surface of a sphere (even though we seldom step back to look at it from a 3D perspective). $\endgroup$
    – fleablood
    Feb 28 at 2:13
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    $\begingroup$ @fleablood colloquially, it's certainly common for people to mean $D^3=\{\mathbf{r}\in\mathbb{R}^3:\|\mathbf{r}\|\leq1\}$ when they say "sphere". But in the mathematical disciplines that deal with these a lot, the convention is actually very clear that $D^3$ is a ball and only $S^2 = \partial D^3$ is a sphere. $\endgroup$ Feb 28 at 14:30

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I know of no standard names for those boundaries.

Mathematicians give the sphere and ball those distinctive names because they often want to discuss both in the same context.

The boundary of the cube comes up less often. In some contexts it's the two skeleton. The one skeleton is the set of edges.

To refer to a torus together with its interior you would say "solid torus".

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  • $\begingroup$ A couple days on and seeing nobody claiming any standard name, I'll accept this answer. Out of curiosity, would you mind sharing an example of a context where both spheres and balls come up together and distinguishing them is important? $\endgroup$ Feb 29 at 15:07
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    $\begingroup$ In topology, balls have trivial homotopy but the homotopy of spheres is interesting and important: en.wikipedia.org/wiki/Homotopy_groups_of_spheres $\endgroup$ Feb 29 at 15:34
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You can refer to the faces of a cube (or other polyhedron).

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    $\begingroup$ Not really the same thing though. The boundary of the cube is the union of the faces, but "the faces" are a collection of individual squares. $\endgroup$ Feb 28 at 13:42

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