Yes, in a grid where the Manhattan distance is the minimum one, taking a single 90° turn is the fastest, since that path will have the length of the Manhattan distance.
However, it’s not the only path. The “ladder” one you said will be the same length.
While we are at it, if you wanna search for more. The same flawed assumption of “a ladder approximates a straight line” can also lead to π=2. Since you can enclose a circle in a square (Wich has perimeter 4R), then fold the corners recursively so there is a “stair” along the circle’s perimeter. That “stair” would have a length of 4R, but the circle’s perimeter is 2πR.
Ah. I originally missread your original comment.
Yes, in a grid where the Manhattan distance is the minimum one, taking a single 90° turn is the fastest, since that path will have the length of the Manhattan distance.
However, it’s not the only path. The “ladder” one you said will be the same length.
While we are at it, if you wanna search for more. The same flawed assumption of “a ladder approximates a straight line” can also lead to π=2. Since you can enclose a circle in a square (Wich has perimeter 4R), then fold the corners recursively so there is a “stair” along the circle’s perimeter. That “stair” would have a length of 4R, but the circle’s perimeter is 2πR.