A Couple of Lingering Geometric Questions...

[Andy_L]

@DPRK

That’s what I meant: start with 3-dimensional space, impose a single polynomial equation, and end up with a 2-dimensional subspace. (Over the real numbers it’s more complicated because of possibilities like x² + y² = 0, but we can stick with linear equations for the sake of this discussion.) Impose a second independent constraint and the dimension goes down to 1, etc. I just found Isilder’s phrasing ambiguous so offered a concrete example (sphere, cylinder and plane work equally well to make this point).

Ah. Got it.

[rat_avatar]

@DPRK

That’s what I meant: start with 3-dimensional space, impose a single polynomial equation, and end up with a 2-dimensional subspace. (Over the real numbers it’s more complicated because of possibilities like x² + y² = 0, but we can stick with linear equations for the sake of this discussion.) Impose a second independent constraint and the dimension goes down to 1, etc. I just found Isilder’s phrasing ambiguous so offered a concrete example (sphere, cylinder and plane work equally well to make this point).

It is a pity that complex and imaginary and real were the names given because they are far simpler in this domain but today’s XKCD is pretty funny.

Complex Numbers