Staircase paradox

In mathematical analysis, the staircase paradox is a pathological example showing that limits of curves do not necessarily preserve their length.

[1] It consists of a sequence of "staircase" polygonal chains in a unit square, formed from horizontal and vertical line segments of decreasing length, so that these staircases converge uniformly to the diagonal of the square.

[3][4] Martin Gardner calls this "an ancient geometrical paradox".

The failure of the staircase curves to converge to the correct length can be explained by the fact that some of their vertices do not lie on the diagonal.

[8] As well as highlighting the need for careful definitions of arc length in mathematics education,[9] the paradox has applications in digital geometry, where it motivates methods of estimating the perimeter of pixelated shapes that do not merely sum the lengths of boundaries between pixels.