hi chat

hi @Semiclassical

i'm trying to decide about the best way to word something

Usually linguistically is the best way to word something

suppose I have a 1-periodic vector function $\vec{r}(t)$. In general, that'll be a closed loop in $\mathbb{R}^3$. But if $\vec{r}(t)$ lies in some plane then that's a closed loop in $\mathbb{R}^2\subset\mathbb{R}^3$.

in that case, I can pick orthogonal coordinates $(x,y)$ on this plane. What I'm trying to figure out is the best way to say the following: I can identify $(x,y)\in\mathbb{R}^2$ with $z=x+i y\in\mathbb{C}$.

Does it not go without saying? I was under the impression that every $\Bbb{R}^2$ has 2 bijections in domain $\Bbb{C}$ ($x+iy$ and $y+ix$)

to me it does, but i want to avoid any confusion

@MickLH — way more than that ...

does anyone know about my question?

I don't see why all the points M_i form a regular n-gon

@TedShifrin psssh only infinitely more, not that many

any $a x+ b y$ with $a,b\in \mathbb{C}\setminus \{0\}$ and $a/b$ not real should work, I think?

though that really amounts to to a different choice of initial coordinates $(x,y)$ (possibly obligue)

hi @TedShifrin @BalarkaSen

anyone know for my question?

hi @Ali

@usukidoll did you figure out your proof?

I got nice intuition regarding to homology @BalarkaSen

let me know when you come