We can put together many dots to get a line, we can put together many lines to get a plane and we can put together many planes to get ..., what's the term for what we get?
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I guess it's called hyperplane. Can someone confirm this? – timtam Jul 18 '22 at 10:42
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Not quite: a hyperplane is often used to mean a dimension one less than the overall space. You might consider whether volume has the 3-dimensional meaning you seem to be aiming for – Henry Jul 18 '22 at 10:56
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But wouldn't volume be the equivalent to length in 1d or area in 2d? A hyperplane in 4d would be a flat affine 3 dimensional subspace, no? And a hyperplane in 3d, would simply be plane, no? And a hyperplane in 2d would be a line, no? So I think hyperplane is the term I am looking for – timtam Jul 18 '22 at 11:05
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That would be a space. – John Douma Jul 18 '22 at 11:26
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I am actually not aware of a term for this. "Hyperplane" means something slightly different; you could say "$3$-plane" or more formally "$3$-dimensional affine subspace" but it's slightly awkward. – Qiaochu Yuan Jul 18 '22 at 11:57
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For dimensions 0 to 3, we use "point, line, plane, space". This is enough for everyday life were we exist in 3 dimensions. Above this, we usually use "$n$-plane" or "$n$-space" where $n$ is the dimension of the object (which depends on personal preference). So a point is a $0$-plane, a line is a $1$-plane, space is a $3$-plane, etc. If the particular dimension is not important, then "affine subspace" is probably the most common description. – Paul Sinclair Jul 19 '22 at 18:27
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FYI - A set of points is "affine" if for any two distinct points in the set, the entire line through those points is also in the set. If the set consists of a single point, then there are not two distinct points in the set, so it vacuously satisfies the condition to be affine, – Paul Sinclair Jul 19 '22 at 18:33