I was curious when someone suggested a rotation of $\pi/4$ and a rescaling by $\sqrt{2}$ gives a nice isometry between the taxicab and max metric on $\mathbb{R}^2$. Thus, let $$f: (x,y) \mapsto (\frac{1}{2}x-\frac{1}{2}y,\frac{1}{2}x + \frac{1}{2}y).$$
I'm just starting out with an good intro book on metric geometry, and this problem seems to be giving me troubles. Unlike motions on the real line, I don't seem to quite see the intuition behind this type of mapping. Is it that the distances in the $x$ and $y$ directions generally form right angles, so a $\pi/4$ rotation straightens these segments out? And the rescaling by $\sqrt{2}$ preserves the vector component's magnitudes?
For example, $(6,8) \mapsto (-1,7)$, so $\max(|6|,|8|) = |-1| + |7|$.