Consider the two dimensional real place $\mathbb{R}^2$. We have a triangle (lets call it $\triangle$) with the vertices $(\frac{\sqrt{3}}{2},\frac{3}{2}),~(-\frac{\sqrt{3}}{2},\frac{3}{2})$ and $(0,-\sqrt{3})$. I need to find ALL continuous maps $f:\mathbb{R}^2\longrightarrow \mathbb{R}^2$ such that $f(\triangle)\hookrightarrow\triangle$.
I tried to convert $(x,y)\in\mathbb{R}^2$ as complex number $z=x+\imath y$ and tried to calculate the analytic ones as well. However, it is becoming too complicated at the end. Can someone please suggest some elegant method to handle the above problems (or at the least, determine all analytic $f$ with above properties). Advanced thanks for any reply.
