Wikipedia:Reference desk/Archives/Mathematics/2020 December 12

= December 12 =

Difference between a sphere and a cube
Go to:

http://mathforum.org/library/drmath/view/54696.html

and scroll down to a post made by "Dr. James" it's just after the one made by "Dr. Jerry". Read the first 3 paragraphs in Dr. James's post. It's saying:

''That's a very tricky question, and I can see exactly where your confusion lies.

''In one sense, we'd like to think that a sphere is a three-dimensional object, since we can take a ruler and measure its length, width, and height.

But in another sense, it can't be a three-dimensional object because it has no volume (remember, a sphere is only the "skin of a ball," not the inside of the ball too), just like a flat plane has no volume.

Now, suppose we replace "sphere" with "cube" and "ball" in the parenthetical phrase in the third paragraph with "box". Any reason it isn't valid any more?? (The important thing is that this appears to be a valid statement about the surface of any 3-dimensional object the same way it is with a sphere.) Any reason this trick question isn't valid with cubes?? Georgia guy (talk) 22:29, 12 December 2020 (UTC)
 * In strict mathematical usage, there's a very clear distinction between a sphere (which is a 2-dimensional manifold) and a spherical ball (which contains a nonempty neighborhood of 3-dimensional space). Admittedly it is not always strictly observed; you'll find all sorts of references to the "volume of a sphere", which really mean the volume of a spherical ball, as the volume of a sphere, strictly speaking, is zero.
 * Whether the corresponding distinction is standardly made for cubes and other polyhedra I really don't know. You'd have to ask a geometer.  Our articles are a bit confusing on the point (for example polyhedron says that polyhedra are made up of vertices, edges, and faces, without mentioning the interior, but then says that a convex polyhedron is the convex hull of finitely many points, which isn't true unless you include the interior).
 * In any case, this is a question about terminology, not really about mathematics. The answers are obvious once you agree what you're talking about. --Trovatore (talk) 22:38, 12 December 2020 (UTC)
 * Generally, a sphere is the surface of a ball while a cube is not a surface but the whole volume, corresponding to a ball. The surface of a cube does not have a name. PrimeHunter (talk) 22:43, 12 December 2020 (UTC)
 * Go one dimension down and you'll find a similar ambiguity with the Area of a circle. A circle is a curve, a one-dimensional object, and as such it has length but no area (or rather it has a zero area). When we talk about the area of a circle, it is, as the linked article says, an area enclosed by the circle. (For more strict talk we sometimes use a name disc for that figure.) In a case of cubes and other polyhedra, we ususally consider a polyhedron a 3D figure, so we talk about a volume of a polyhedron and an area of a surface of the polyhedron. --CiaPan (talk) 23:13, 12 December 2020 (UTC)


 * (ec) The ambiguity between the filled object and its surface was why I used the term "solid cube" in a response to an earlier question. In non-mathematical use, a cube (as in "a cube of sugar") is a solid thing. In mathematical use, it can be a much more abstract concept than the surface of a solid cube; the term is sometimes even used to mean a particular graph (discrete mathematics) of eight vertices and twelve edges, an abstract combinatorial entity not embedded in a geometric space. Likewise, when it is said that gravity pulls a planet into the shape of a sphere, this is not meant to suggest planets are hollow. Topologists use the term "sphere" for a particular two-dimensional manifold, which is topologically indistinguishable from the (idealized) boundary forming the surface of a smooth potato. These are just some of the different senses in which these terms are used. There are also multiple senses for terms like "square", "line", and "curve". --Lambiam 23:39, 12 December 2020 (UTC)