Why axes should be 90 degrees to each other?

Question

So why we can't (or we actually can?) draw coordinate system like this:

We can add more than 3 dimensions!

Are there any problems in this coordinate system with describing n-dimentional bodies?

PS: I've heard something about affine coordinates and parallel coordinates but I can't put it all together and clarify those ideas.

Answer 1

There is no problem drawing coordinate systems with more than 2 axes (which of course can no more be pairwise perpendicular on the drawing).

You are actually performing a parallel projection from $nD$ space to $2D$ and this is conveniently done with a $2\times n$ projection matrix.

Notice anyway that doing this you lose the ability to see right angles and measure distances, and a single $2D$ point corresponds to infinitely many overlaid $nD$ points.

A very common case is that of the axonometric projection ($3D$ to $2D$) used in technical drawing. The angles are normalized.

Below, a $2D$ representation of a $4D$ cube (crossed-eyes stereoscopy), also known as a tesseract.

Answer 2

Because the usual system guarantees at the same time the uniqueness of the representation of points in space with the coordinates and the orthogonality of the direction of axis, which is important for example to calculate the distance between two fixed points.

Answer 3

IMO, this is kind of a confused question, since obviously the title of it does not match its description. The title asks about the $90^{\circ}$ angle between axes, while the description asks whether the proposed picture has represented an n-dimensional space. Here, I want to disprove the latter case clearly:

1. Regarding the axes you've shown in your picture, you also have to make sure the basis vectors in those directions will be linearly independent(You can't, read $2$ below).

2. You won't be able to do $1$. Because it can be proved that every possible Basis for a vector space will have the same cardinality as the other ones. I.e., Since we know that x, y, and z dimensions can cover all the 3D space, so no matter if you add more axes like the picture in your question, some of them will definitely be superfluous(wherever they could go, could be covered with x, y and z only as well), i.e., you have not added more "dimensions" in your picture, at all.