The paradox 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.[2] However, each staircase has length two, while the length of the diagonal is the square root of 2 (approximately 1.4142), so the sequence of staircase lengths does not converge to the length of the diagonal.[3][4]
Martin Gardner calls this "an ancient geometrical paradox".[5] The staircase paradox shows that, for curves under uniform convergence, the length of a curve is not a continuous function of the curve.[6]
Explanation
For any smooth curve, polygonal chains with segment lengths decreasing to zero, connecting consecutive vertices along the curve, always converge to the arc length. 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.[7]
Usage
A fallacious approach to calculating the circumference of a circle due to the staircase paradox
As well as highlighting the need for careful definitions of arc length in mathematics education,[8] the staircase 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.[9]
In higher dimensions, the Schwarz lantern provides an analogous example showing that polyhedral surfaces that converge pointwise to a curved surface do not necessarily converge to its area, even when the vertices all lie on the surface.[10]
See also
Coastline paradox, similar paradox where a straight segment approximation diverges
Aliasing, a more general phenomenon of inaccuracies caused by pixelation
Cantor staircase, a fractal curve along the diagonal of a unit square
Taxicab geometry, in which the lengths of the staircases and of the diagonal are equal
↑ Farrell, Margaret A. (February 1975), "An intuitive leap or an unscholarly lapse?", The Mathematics Teacher, 68 (2): 149–152, doi:10.5951/mt.68.2.0149, JSTOR27960047
↑ Ogilvy, C. Stanley (1962), "Note to page 7", Tomorrow's Math: Unsolved Problems for the Amateur, Oxford University Press, pp.155–161
Further reading
Gill, John (November 2022), A Short Note: Extending the Staircase Paradox, Abstract: There are sequences of continuously differentiable contours in the complex plane that converge uniformly to the line segment [0,1], even to z=0, but have lengths approaching any predetermined size, even infinite. Work (or potential energy) arising from these sequences is explored briefly.
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