Length of the straight path connecting two points in the taxicab geometry? Are such straight paths even permissible in this geometry? Let's equip $\mathbb R^2$ with the taxicab metric induced by the norm $||\cdot||_1$.

Now let's look at this image on the Wikipedia page. The caption says "Taxicab geometry versus Euclidean distance: In taxicab geometry, the red, yellow, and blue paths all have the same shortest path length of 12. In Euclidean geometry, the green line has length $6\sqrt 2 \approx 8.49$ and is the unique shortest path." Let's name the black point at the bottom left as $A$ and the black point at the top right as $B$.
Indeed, I understand that the red path, blue path and the yellow path all have the same length of $12$ units in this taxicab geometry. However, is there a notion of path length of the straight green line connecting $A$ and $B$ in this taxicab geometry? What's the length of the green line connecting $A$ and $B$ in this geometry? Or can such a path not even exist in this geometry? Are other "curved" paths between $A$ and $B$ permissible? Why or why not?
I'm a bit confused about this issue. I keep seeing only these grid-like paths in all texts explaining taxicab geometry, but none of them explain the reason. Are other kinds of paths not permissible somehow? Some clarification would help.
 A: It's a reasonable thing to be confused about.  Presentations of taxicab geometry don't do any favors when they model it too closely after grid-based cities.  The distance  from $(0,0.5)$ to $(1,0.5)$ is one, even though a cab couldn't get you from the middle of a block of 14th Street to the middle of the same block of 15th street in a straight line.
In a taxicab geometry, you can certainly talk about the green line segment, just like you could think about any other set of points in the plane.  But the length of the line segment is $d_T(A,B)=12$, as all of the other paths demonstrate.
There are other interesting phenomena that distinguish horizontal and vertical line segments from oblique line segments like $\overline{AB}$.  For instance, if two points $X,Y$ form a horizontal or vertical line segment, then that line segment is $\{Z\mid d_T(X,Z)+d_T(Y,Z)=d_T(X,Y)\}$, like we're used to for the Euclidean metric.    But, in your diagram, $\{C\mid d_T(A,C)+d_T(B,C)=d_T(A,B)\}$ is the entire rectangle whose opposite corners are $A$ and $B$.
