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We know that, in $1$-D, we just have a straight line where all the things happen. In $2$-D Cartesian co-ordinates, we have $2$ axes which in turn give rise to a plane, commonly called $xy$-plane, where all the work is done and in case of $3$-D Cartesian co-ordinates, we have the $x,y,z$ axes forming $3$ distinct planes which divide the $3$-dimensional space into $8$ octants.

My question is, what happens where we get into $n$ dimensional Cartesian space where $n \geq 4$? Do we have $n$ perpendicular axes forming $n$ planes? How do these axes look like?

I thought that, perhaps for $n = 4$, we will have $4$ axes criss-crossing to form $4$ planes giving rise to "God Knows What". But geometrically, I could not visualise $4$ perpendicular axes acting as basis and forming planes. Perhaps they form some different structures and they have some cumbersome geometry. It is difficult to ascertain what kind of geometric structures the $n$-dimensional Cartesian axes form and what these axes look like. Well, I have heard that they call it "Cartesian Hyperspace".

So, can anybody help? And mind it, I am strictly talking about Cartesian and not the general curvilinear co-ordinates.

SchrodingersCat
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  • Are you looking for a visualization? Because it is basically impossible to accurately visualize an n-dimensional space in 3D. We can distort it to force it into 3D, but those distortions will make it unrecognizable for larger n – dimpol Nov 03 '16 at 13:48
  • @dimpol No need for a visualisation. Can you at least tell me how they look like? Any geometric description? – SchrodingersCat Nov 03 '16 at 13:49
  • I tried, see my answer – dimpol Nov 03 '16 at 14:07
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    This might interest you: http://math.stackexchange.com/questions/389729/imagining-four-or-higher-dimensions-and-the-difference-to-imagining-three-dimens – David K Nov 04 '16 at 02:04

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You get $n$ axes, and ${n \choose 2} = \frac{n(n-1)}{2\cdot 1}$ coordinate planes, and $n \choose 3$ coordinate "solids", and so on.

John Hughes
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For an n-dimensional space, we have n perpendicular axis. This is hard to vizualize, because the automatic idea is to imagine just another axis a 3d-space. However to take 4d as the easiest example, the 4th axis is perpendicular to all 3 other axis. Maybe one of the easiest ways to imagine this is to take the 4th axis to be time. This is of course wrong, because we are talking about a space with 4 spacial axis. However you can see that the 4th axis is completely perpendicular to the 3 other axis. Consider saying: "see you same place tomorrow".
Just considering 3d-space, this is a baffling statement. Are we meeting more to the left or right? Are we meeting more up or down? Are we meeting more forwards or backwards? The answer to all those questions is no. Yet we are not meeting 'here'. We are meeting 24 'hours' away.
If I want to 'walk' those 24 hours, what direction do I go? Well, walking along the 3 spacial axis isn't going to do you any good. Even worse, it is going to bring you farther away from the 'place' we are meeting.

Consider the strangeness of adding an extra axis, an extra dimension and the consider the step from 4 to 5 dimensions is equally strange. And from 5 to 6 as well, and from 6 to 7. All we can hope to do for those higher dimensions is to describe their properties in the language of mathematics. English largely failed me in describing the 4th dimension above, trying to even describe the 17th dimension is a useless exercise. Let alone something like the 1234567th dimension.

dimpol
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The parallel coordinate transformation provides usable 2-d transforms from N-D which I have found very helpful in trying to understand the strangeness of N-D.

lottalarfs
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