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Parallel lines are 1-dimensional objects in 2+ dimensional space. Parallel planes are 2+dimensional objects in 3+ dimensional space.

Is there a notion of e.g. parallel 3D volumes in 4D space? If not, why not?

The question is inspired by some really well-made videos on YouTube that explain what a hypothetical 4D object would look like to us as it passes in or out of our 3D "slice".

RuslanD
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  • I guess I "discovered" hyperplanes for myself – RuslanD Apr 06 '24 at 02:33
  • There are also parallel $k$-dimensional affine subspaces in an affine $n$-dimensional space. – Moishe Kohan Apr 06 '24 at 02:53
  • @MoisheKohan would it be correct to talk about 3D space as we perceive it as a 3D hyperplane in putative 4D space? Ignoring for a second the complications of "curved" space near large masses? Or are those unrelated/incompatible things? – RuslanD Apr 06 '24 at 03:12
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    Yes, of course, you can embed the Euclidean 3-dimensional space as a hyperplane in the Euclidean 4-dimensional space. – Moishe Kohan Apr 06 '24 at 03:14
  • For a simple specific example, the graphs of $u=0$ and $u=1$ in (ordinary Euclidean) $(x,y,z,u)$-space are parallel 3D "planes", in the same way that the graphs of $y=0$ and $y=1$ in $(x,y)$-space are parallel 1D "planes" and the graphs of $z=0$ and $z=1$ in $(x,y,z)$-space are parallel 2D planes. @Red Five: Since you teach high school (I've done so in two different schools, for a total of 4 years), I especially recommend some of the expository publications I give in this MSE answer. – Dave L. Renfro Apr 06 '24 at 06:40

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If two lines in $n$-dimensional space are parallel then any line perpendicular to line $L_{1}$ will also be perpendicular to $L_{2}$.

For planes, it is a similar argument. If $\Pi_{1}$ and $\Pi_{2}$ are parallel then any line perpendicular to $\Pi_{1}$ will also be perpendicular to $\Pi_{2}$.

Generalising this to hyperplanes or any object up to $n-1$ dimensions in $n$ dimensional space should therefore follow the same logic, should it not?

Lines in $n$ dimensional space can be written using $n$-dimensional vectors and the dot-product is defined in $n$-dimensions, so establishing that two geometric objects are perpendicular should be possible always.

This does assume that $n\geq 2$ of course.

Or have I over-simplified the situation and missed something important?

Red Five
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  • You have not oversimplified the situation - I just didn't have enough mathematical background to answer my own question with certainty. Your answer makes sense to me. I'm curious - as a thought experiment, if there were a fourth spatial dimension (that we can't perceive), how would we define the 3D hypersurface corresponding to the 3D world as we perceive it? Or is that an ill-posed question? – RuslanD Apr 06 '24 at 05:45
  • @RuslanD that is a really good question and totally beyond my field of expertise. My days working at a university are long gone - I just teach high schoolers now. I'm sure there are others on this site who would be able to assist though, so wait and see what other responses come through. – Red Five Apr 06 '24 at 05:48