I'd like to politely complain that I'm having trouble understanding exactly what is intended here. Regardless of how a math problem may be couched in real-world terms (like jaywalkers and infinitesimally wide roads), would it be okay to also please express the problem in unambiguous mathematical terms so that the meaning of the question doesn't have to be questioned? —Dan ----- A variant of taxicab geometry I came up with is jaywalker geometry. Envision the roads having, not zero width, but infinitesimal width. Distance is defined, not as how far a taxicab drives, but as how far a jaywalker walks. If a right turn is to be followed by a left turn or vice versa, to minimize his walking distance he crosses the street diagonally, over the whole length of it that he traverses, much to the annoyance of the taxicabs. In taxicab geometry there are in general lots of shortest paths between any two grid points, but in jaywalker geometry, of all those paths, the paths with the most turns are shortest. I think those paths are the ones that approximate Euclidean paths most closely. For instance instead of walking ten blocks north then ten blocks east, the jaywalker would alternate between walking one block north and one block east. -----