Playing with infinities is dangerous business, and it's very easy to yield paradoxes like the OP. But note that before the 20th century, mathematicians like Gau

Playing with infinities is dangerous business, and it's very easy to yield paradoxes like the OP. But note that before the 20th century, mathematicians like Gauss would avoid playing with such ill-defined objects, for precisely this reason[1]. Their rule was that you should first derive an expression in the finite case, then take the limit to infinity at the end. In this case, it's mathematically simple that a Manhattan line should have the same length, no matter what N (the number of steps) is. [1] http://en.wikipedia.org/wiki/Russell%27s_paradox http://en.wikipedia.org/wiki/Russell%27s_paradox for instance. The Banach-Tarski paradox is also worthwhile: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox I think nowadays most mathematicians are "comfortable" with such aberrant results, but that was not always the case. And there's still a chance that future generations of mathematicians will dismiss this work.