# Isometry from Manhattan plane to Euclidean plane?

Does there exist an isometry from a Manhattan plane $A$ to a Euclidean plane $B$? I.e. a function $\varphi:A \to B$ that suffices $\|\varphi(a)\|_B = \|a\|_A$ for all $a \in A$, where $\| \cdot \|_A$ is the Manhattan norm and $\| \cdot \|_B$ the Euclidean norm.

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Hint: In the Euclidean plane, for a pair of points $x,y$ with distance $2$, how many points are there at distance $1$ from both $x$ and $y$? What about the Manhattan plane? (Look at $x = (0,0)$ and $y = (1,1)$.)
I guess not. Does this still hold for the euclidean $\mathbb{R}^3$ instead of $\mathbb{R}^2$? –  Chiel92 Apr 9 '13 at 11:15