mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, the staircase paradox is a pathological example showing that

limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

of curves do not necessarily preserve their

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...

. It consists of a sequence of "staircase"

polygonal chain In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...

s in a

unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordina ...

, formed from horizontal and vertical

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

s of decreasing length, so that these staircases converge uniformly to the diagonal of the square. However, each staircase has length two, while the length of the diagonal is the

square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...

, so the sequence of staircase lengths does not converge to the length of the diagonal.

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of L ...

calls this "an ancient geometrical paradox". It shows that, for curves under uniform convergence, the length of a curve is not a continuous function of the curve. For any

smooth curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

, polygonal chains with segment lengths decreasing to zero, connecting consecutive vertices along the curve, always converge to the

arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services ...

. 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. In higher dimensions, the

Schwarz lantern In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra. It is formed by stack ...

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. As well as highlighting the need for careful definitions of arc length in mathematics education, the paradox has applications in

digital geometry Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean space. Simply put, digitizing is replacing an object by a discrete set of its points. The ...

, where it motivates methods of estimating the perimeter of pixelated shapes that do not merely sum the lengths of boundaries between pixels.

References

{{reflist, refs= {{citation , last = Bennett , first = Albert A. , date = February 10, 1920 , issue = 2 , journal = The Texas Mathematics Teachers' Bulletin , pages = 12-22 , title = Limit proofs in geometry , url = https://books.google.com/books?id=1svxAAAAMAAJ&pg=RA12-PA12 , volume = 5; see especially p. 16 {{citation , last = Farrell , first = Margaret A. , date = February 1975 , doi = 10.5951/mt.68.2.0149 , issue = 2 , journal = The Mathematics Teacher , jstor = 27960047 , pages = 149–152 , title = An intuitive leap or an unscholarly lapse? , volume = 68 {{citation , last1 = Thompson , first1 = Silvanus P. , author1-link = Silvanus P. Thompson , last2 = Gardner , first2 = Martin , author2-link = Martin Gardner , contribution = Appendix: Some recreational problems related to calculus , pages = 296–325 , publisher = Palgrave , title = Calculus Made Easy , year = 1998. Se
pp. 305–306
{{citation , last1 = Klette , first1 = Reinhard , last2 = Yip , first2 = Ben , issue = 3 , journal = Machine Graphics and Vision , pages = 673–703 , title = The length of digital curves , url = https://www.researchgate.net/profile/Reinhard-Klette/publication/37986806_The_Length_of_Digital_Curves/links/0fcfd50d82684d1de7000000/The-Length-of-Digital-Curves.pdf , volume = 9 , year = 2000 {{citation , last = Krantz , first = Steven G. , contribution = 15.1: How to measure the length of a curve , isbn = 978-0-88385-766-3 , mr = 2604456 , pages = https://books.google.com/books?id=ulmAH-6IzNoC&pg=PA249 , publisher = Mathematical Association of America , location = Washington, DC , series = MAA Textbooks , title = An Episodic History of Mathematics: Mathematical Culture Through Problem Solving , year = 2010 {{citation , last = Moscovich , first = Ivan , author-link = Ivan Moscovich , location = New York , page = 23 , publisher = Sterling Publishing , title = Loopy Logic Problems and Other Puzzles , url = https://books.google.com/books?id=enkLi5z-WbAC&pg=PA23 , year = 2006 {{citation , last = Ogilvy , first = C. Stanley , author-link = C. Stanley Ogilvy , contribution = Note to page 7 , pages = 155–161 , publisher = Oxford University Press , title = Tomorrow's Math: Unsolved Problems for the Amateur , year = 1962 {{citation , last = Sedaghat , first = H. , isbn = 9780192895622 , page = 9 , publisher = Oxford University Press , title = Real Analysis and Infinity , url = https://books.google.com/books?id=sWNjEAAAQBAJ&pg=PA9 , year = 2022 {{citation , last1 = Sinitsky , first1 = Ilya , last2 = Ilany , first2 = Bat-Sheva , doi = 10.1007/978-94-6300-699-6 , pages = 375–376 , publisher = Sense Publishers , title = Change and Invariance: A Textbook on Algebraic Insight into Numbers and Shapes , url = https://books.google.com/books?id=RimcDQAAQBAJ&pg=PA37 , year = 2016 {{citation , last = Stewart , first = Ian , author-link = Ian Stewart (mathematician) , contribution = Diagonal of a square , isbn = 9780191071515 , pages = 43 & 54 , publisher = Oxford University Press , title = Infinity: A Very Short Introduction , year = 2017

External links

A Short Note: Extending the Staircase Paradox
Length Mathematical paradoxes Limits (mathematics)

See also

References

External links