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.[2] However, each staircase has length two, while the length of the diagonal is the square root of 2, 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] It shows that, for curves under uniform convergence, the length of a curve is not a continuous function of the curve.[6]
Staircases converging pointwise to the diagonal of a unit square, but not converging to its length
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] 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.[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.[10]
See alsoedit
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, JSTOR 27960047
^Sedaghat, H. (2022), Real Analysis and Infinity, Oxford University Press, p. 9, ISBN9780192895622
^Stewart, Ian (2017), "Diagonal of a square", Infinity: A Very Short Introduction, Oxford University Press, pp. 43 & 54, ISBN9780191071515
^Thompson, Silvanus P.; Gardner, Martin (1998), "Appendix: Some recreational problems related to calculus", Calculus Made Easy, Palgrave, pp. 296–325. See pp. 305–306.
^Sinitsky, Ilya; Ilany, Bat-Sheva (2016), Change and Invariance: A Textbook on Algebraic Insight into Numbers and Shapes, Sense Publishers, pp. 375–376, doi:10.1007/978-94-6300-699-6, ISBN978-94-6300-699-6
^Krantz, Steven G. (2010), "15.1: How to measure the length of a curve", An Episodic History of Mathematics: Mathematical Culture Through Problem Solving, MAA Textbooks, Washington, DC: Mathematical Association of America, pp. https://books.google.com/books?id=ulmAH-6IzNoC&pg=PA249, ISBN978-0-88385-766-3, MR 2604456
^Ogilvy, C. Stanley (1962), "Note to page 7", Tomorrow's Math: Unsolved Problems for the Amateur, Oxford University Press, pp. 155–161
^Bennett, Albert A. (February 10, 1920), "Limit proofs in geometry", The Texas Mathematics Teachers' Bulletin, 5 (2): 12–22; see especially p. 16
^Klette, Reinhard; Yip, Ben (2000), "The length of digital curves", Machine Graphics and Vision, 9 (3): 673–703
External linksedit
A Short Note: Extending the Staircase Paradox
April 12, 2024
staircase, paradox, mathematical, analysis, staircase, paradox, pathological, example, showing, that, limits, curves, necessarily, preserve, their, length, consists, sequence, staircase, polygonal, chains, unit, square, formed, from, horizontal, vertical, line. 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 2 However each staircase has length two while the length of the diagonal is the square root of 2 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 It shows that for curves under uniform convergence the length of a curve is not a continuous function of the curve 6 Staircases converging pointwise to the diagonal of a unit square but not converging to its lengthFor 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 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 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 10 See also editCoastline 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 equalReferences edit Moscovich Ivan 2006 Loopy Logic Problems and Other Puzzles New York Sterling Publishing p 23 ISBN 9780486490694 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 JSTOR 27960047 Sedaghat H 2022 Real Analysis and Infinity Oxford University Press p 9 ISBN 9780192895622 Stewart Ian 2017 Diagonal of a square Infinity A Very Short Introduction Oxford University Press pp 43 amp 54 ISBN 9780191071515 Thompson Silvanus P Gardner Martin 1998 Appendix Some recreational problems related to calculus Calculus Made Easy Palgrave pp 296 325 See pp 305 306 Sinitsky Ilya Ilany Bat Sheva 2016 Change and Invariance A Textbook on Algebraic Insight into Numbers and Shapes Sense Publishers pp 375 376 doi 10 1007 978 94 6300 699 6 ISBN 978 94 6300 699 6 Krantz Steven G 2010 15 1 How to measure the length of a curve An Episodic History of Mathematics Mathematical Culture Through Problem Solving MAA Textbooks Washington DC Mathematical Association of America pp https books google com books id ulmAH 6IzNoC amp pg PA249 ISBN 978 0 88385 766 3 MR 2604456 Ogilvy C Stanley 1962 Note to page 7 Tomorrow s Math Unsolved Problems for the Amateur Oxford University Press pp 155 161 Bennett Albert A February 10 1920 Limit proofs in geometry The Texas Mathematics Teachers Bulletin 5 2 12 22 see especially p 16 Klette Reinhard Yip Ben 2000 The length of digital curves Machine Graphics and Vision 9 3 673 703External links editA Short Note Extending the Staircase Paradox Retrieved from https en wikipedia org w index php title Staircase paradox amp oldid 1192817459, wikipedia, wiki, book, books, library,
