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The staircase 'paradox' has been discussed here and elsewhere a few times (search for staircase + paradox).

My question is whether this puzzle has been discussed in the academic literature or historically in mathematics. I can't find any reference to it except in fora, but I want to reference it in a paper.

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    $\begingroup$ I recall reading somewhere (an actual publication, since I think I read it in the 1980s) that Lebesgue was fascinated by this "paradox" and it was apparently making the rounds among students and faculty when Lebesgue was a student (mid 1890s), but it's surely much older than this. I wrote about this once in the ap-calculus discussion group at Math Forum, but it seems none of the Math Forum posts are currently available. However, much of that particular post can also be found here. $\endgroup$ – Dave L Renfro Dec 15 '19 at 18:33
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    $\begingroup$ Incidentally, this non-continuous aspect of the length of curves actually comes up in the study of general relativity and black holes --- see the comments to Lower semicontinuity of length of graph: $L(g)\le\liminf_{n\to\infty}L(f_n)$. See also The staircase paradox, or why $\pi\ne4$ and the many questions linked to it. $\endgroup$ – Dave L Renfro Dec 15 '19 at 18:41
  • $\begingroup$ Thanks all I will check these out. $\endgroup$ – buckner Dec 15 '19 at 18:48
  • $\begingroup$ Dave, a further question. If we lay the staircase on the ground, i.e. the x axis, so that in the discrete case it zigs up from y=0 then zags back again to the x axis, then at the limit is every point on the staircase at y=0? That seems truly bizarre. $\endgroup$ – buckner Dec 15 '19 at 20:52
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    $\begingroup$ That seems truly bizarre --- This is simply a consequence of the fact that "limiting behavior properties" can sometimes be different from the properties of the approaching objects. For example, you can easily define a sequence of finite sets of points in a plane that (in most any reasonable sense) limits to the the plane itself (not enough space here to describe such an example, however). In this case each of the objects is a finite and discrete set of points, but the limit is a continuous and infinite plane. $\endgroup$ – Dave L Renfro Dec 16 '19 at 12:00
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The "staircase paradox" (or "Pythagoras paradox") name appears to be recent, so it is hard to search for it. Wolfram calls it "diagonal paradox", but that may be conflating it with a different paradox due to Leibniz, which he used to argue against the actual existence of indivisibles, see The Philosophical Assumptions Underlying Leibniz's Use of the Diagonal Paradox in 1672. The horizontal zigzag/sawtooth variant appears to be older, Lebesgue reports it from as early as 1890-s. Mathpages calls it limit paradox and Lukowski (p.13) paradox of approximation. Koch's snowflake construction (1904) uses a related idea, as does Mandelbrot's "coastline paradox", which he attributes to Richardson (1961). In 1890 Schwarz gave a construction, which can be seen as a clever 2D generalization of the sawtooth paradox, of polyhedral surfaces inscribed into a cylinder and converging to it (Schwarz lanterns), with the surface areas growing to infinity, see Surface Area and the Cylinder Area Paradox.

Friend in Numbers: fun & facts (1954), p.72 gives the sawtooth paradox under the title "the field of barley". Lakoff and Núñez in Where mathematics comes from (2000) discuss a variation with semicircles as a "classic paradox of infinity". It "shows" that $π=2$. Their version, along with the sawtooth version "showing" that $2=1$, appears in Paradoxes and Sophisms in Calculus, pp. 30-31. In 1997 there was a lively pedagogical discussion involving it in MAA journals, see On Arc Length by Barry and references therein. Barbeau in the Fallacies, Flaws, and Flimflam section of CMJ mentions the Lebesgue story that Dave Renfro is probably recalling:

"A request to the newsgroup math-history-list@maa.org elicited responses from John Conway, Roger Cooke, Mark McKinzie, and Rick Otten, who provided the following references. L. C. Young [7, 8] quotes an anecdote from Lebesgue's book, In the Margin of the Calculus of Variations, in which the paradox was presented as a "joke" at the College de Beauvais."

The Young's book is Lectures on the Calculus of Variations and Optimal Control theory (1981), p.152, and he quotes Lebesgue directly (see image below):

"All my papers [on this subject] are connected with a schoolboy's 'joke.' At the College de Beauvais, we used to show that, in a triangle, one side is equal to the sum of the other two. Let $ABC$ be a triangle. If $A_1, B_1, C_1$ are the mid-points of its sides, we have $$ BA+AC = BC_1 + C_1A_1 + A_1B_1 + B_1C. $$ On each of the triangles $BC_1A_1, A_1B_1C$, proceed as on $ABC$. We obtain a broken line, formed of eight segments, and equal to $BA + AC$. By continuing in this way, we obtain a sequence of broken lines, which stray less and less from the side $BC$, and which still have as length the sum of the two other sides of our original triangle. The pupils at Beauvais concluded from this, that the segment BC, the geometrical limit of our broken lines, had as length the sum of the two other sides $BA + AC$. My schoolfellows saw there no more than a good joke. To me, the argument appeared most disturbing, since I could see no difference between it and proofs relating to the areas and surfaces of cylinders, cones, spheres, and to the length of a circumference."

Lebesgue's En marge du calcul des variations does not seem to be translated into English. It was only found after his death and published in 1963. Four of the six chapters were previously published as papers.

At the end, Lebesgue probably refers to the classical approximations of arc lengths and surface areas by the "method of exhaustion". In the calculus of variations there was active work at the time on the isoperimetric problem that raised related analytic issues. It turned out that the classical proofs, from Zenodorus to Steiner, had gaps concerning existence of limit figures. Weierstrass and Edler gave first rigorous proofs for curves in 1879 and 1882, and Schwarz for surfaces in 1890.

enter image description here

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    $\begingroup$ since I could see no difference between it and proofs relating to the areas and surfaces of cylinders, cones, spheres, and to the length of a circumference --- I definitely recall this part in what I read many years ago, because I remember thinking that in this part Lebesgue went on to explain how the same kind of reasoning (clearly incorrect here) was used elsewhere. I might have seen it in Kenneth O. May's biographical essay in the 1966 translation of Lebesgue's Measure and Integral, but I don't own a copy of this book, so I can't check this now. $\endgroup$ – Dave L Renfro Dec 16 '19 at 11:21
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    $\begingroup$ Regarding the Schwarz area problem, see also this note by V. Frederick Rickey for a very nice and detailed historical discussion. And for an indication of later developments arising out of this, see my answer to How is area defined? $\endgroup$ – Dave L Renfro Dec 16 '19 at 11:28
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    $\begingroup$ No, I don't have a copy, and I assumed you've googled for it. A google scholar search turned up this review in Mathematical Gazette (the first page of which is freely available when the search is done in google scholar), in case you're interested. $\endgroup$ – Dave L Renfro Dec 16 '19 at 12:06
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    $\begingroup$ @Spencer I did not aim to discuss the resolution of the paradox, this is already done at length on Math SE.The issue is that uniform convergence does not imply convergence of derivatives, which enter the length formula. That the derivatives are then integrated is a side issue, and the Riemann, or even Cauchy, integral suffices there. Lebesgue did go on to do some work on isoperimetry that might have been related to this (I could not ascertain the connection yet), but his measure theory is tangential to it. $\endgroup$ – Conifold Dec 17 '19 at 0:23
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    $\begingroup$ @Spencer: Nothing specifically relevant to measure theory is needed to resolve the "paradox". It is simply a fact that the standard method of assigning lengths to curves has the property that two curves can be arbitrary close together without their lengths being close together. Incidentally, there is more than one way to define what it means for curves to be "close together", and for some of these ways (e.g. 1 and 2) it is true that curves close together have lengths that are close together. $\endgroup$ – Dave L Renfro Dec 17 '19 at 14:51
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This is a follow-up to some of my comments to the OP and to the answer that @Conifold gave. A few days ago I purchased a copy of the book that I had mentioned, Measure and the Integral by Lebesgue (1966). I was correct that this book is where, many years ago (1980s, possibly even late 1970s), I had read about Lebesgue's fascination with this “paradox”. However, contrary to what I thought, Lebesgue's fascination is not discussed in Kenneth O. May's biographical essay, but instead it appears in Section 66 on pp. 97-98. Those wishing to read more about these issues involving arc length and surface area will find Chapter V: Lengths of Curves. Areas of Surfaces (Sections 62-83; pp. 92-124) especially instructive. Below is all of Section 66, followed by some bibliographic information about this 1966 book.

66 A similar paradox for lengths. $\;\;\;\;$ If mathematicians had not been hypnotized by the word “inscribed,” if they had not forgotten that inscribing had been chosen only as one way of approximating, they would have seen that the difficulty encountered for areas existed equally for curves. Now it was just this difference between curves and surfaces that was most shocking. Allow me to refer to my own recollections.

When I was a schoolboy, it was agreed in France, as I have said [previously], that one could evaluate lengths, areas, and volumes by passing to the limit. Soon doubts began to appear in the textbooks. The students who had heard Schwarz’ objections in Hermite’s analysis course had now in their turn become teachers. Besides, everything then predisposed us to a critical analysis of concepts: researches on functions of a real variable and on sets, which people were beginning to consider, Tannery’s teaching, which had aroused in many of his students the desire for complete comprehension or at least verbal precision. People began to doubt, sometimes without knowing what they doubted. For example, the determination of the area of a circle by means of the areas of the polygons that it contained or that contained it (see section 42) was confused with an argument about limits.

Formerly, when I was a schoolboy, the teachers and pupils had been satisfied with this reasoning by passage to the limit. However, it ceased to satisfy me when some of my schoolmates showed me, along about my fifteenth year, that one side of a triangle is equal to the sum of the other two and that $\pi = 2.$ Suppose that $ABC$ is an equilateral triangle and that $D,$ $E,$ and $F$ are the midpoints of $BA,$ $BC,$ and $CA.$ The length of the broken line [= polygonal path] $BDEFC$ is $AB + AC.$ If we repeat this procedure with the triangles $DBE$ and $FEC,$ we get a broken line of the same length made up of eight segments, etc. Now these broken lines have $BC$ as their limit, and hence the limit of their lengths, that is, their common length $AB + AC,$ is equal to $BC.$ The reasoning with regard to $\pi$ is analogous.

Nothing, absolutely nothing, distinguishes this reasoning from what we used to evaluate the circumference and area of a circle, the surface and volume of a cylinder, a cone, and a sphere. This result has been extremely instructive to me.

Besides, every paradox is highly instructive. In my opinion, the critical examination of paradoxes and the correction of erroneous reasoning should be standard exercises, frequently repeated at the secondary level.

The preceding example shows that passing to the limit for lengths, areas, or volumes requires justification, and, like Schwarz’ example, it is enough to arouse all one’s suspicions.

Henri Léon Lebesgue (1875-1941), Measure and the Integral, edited with a biographical essay by Kenneth Ownsworth May (1915-1977), The Mathesis Series, Holden-Day, 1966, xii + 194 pages.

This book is an English translation of two works by Lebesgue. The first work is on pp. 12-175 and the second work is on pp. 178-194.

The first work was originally published in L’Enseignement Mathématique with the title Sur la mesure des grandeurs and consists of an Introduction and 8 sections published in 6 parts: (i) L’E. M. (1) 31 #2 (1932), pp. 173-206 [Introduction (pp. 173-174); I. Comparaison des Collections; Nombres Entiers (pp. 175-181); II. Longueurs; Nombres (pp. 182-206)]. (ii) L’E. M. (1) 32 #1 (1933), pp. 23-51 [III. Aires (pp. 23-51)]. (iii) L’E. M. (1) 33 #1 (1934), pp. 22-48 [IV. Volumes (pp. 22-48)]. (iv) L’E. M. (1) 33 #2 (1934), pp. 177-213 [V. Longueurs des Courbes. Aires des Surfaces (pp. 177-213)]. (v) L’E. M. (1) 33 #3 (1934), pp. 270-284 [VI. Grandeurs Mesurables (pp. 270-284)]. (vi) L’E. M. (1) 34 #2 (1935), pp. 176-219 [VII. Intégration et Dérivation (pp. 176-212); VIII. Conclusions (pp. 212-219)]. [[ Note: “(1) 33 #3” means “series 1, volume 33, issue 3”. I do not know the precise dates of the issues, or even whether such more precise dates exist, so the years are for the volumes. ]]

The first work was published as a book with the title Sur la Mesure des Grandeurs by Gauthier-Villars (Paris) and L’Enseignement Mathématique (Geneva) in 1956 (iv + 184 pages), which was reprinted with the title La Mesure des Grandeurs by Albert Blanchard (Paris) in 1975 (iv + 184 pages).

The second work is the published version of a conference talk that Lebesgue gave in Copenhagen on 8 May 1926 and it was originally published with the title Sur le développement de la notion d’intégrale in Matematisk Tidsskrift B [after 1952: Mathematica Scandinavica] 1926 (1926), pp. 54-74, and reprinted with the same title in Revue de Métaphysique et de Morale 34 #2 (April-June 1927), pp. 149-167, and translated into Spanish and published in Revista Matemática Hispano-Americana with the title Evolución de la noción de integral and published in 2 parts: (i) R. M. H.-M. (2) 2 #3 (March 1927), pp. 65-74. (ii) R. M. H.-M. (2) 2 #4 (April 1927), pp. 97-106.

Book Reviews I know of: Truman Arthur Botts, Science (N.S.) 155 #3765 (24 February 1967), p. 992; A. S. G., Current Science 36 #7 (5 April 1967), p. 194; Thomas William Hawkins, American Mathematical Monthly 75 #6 (June-July 1968), pp. 696-697; Roger Philip Rigelhof, Canadian Mathematical Bulletin 11 #5 (December 1968), pp. 753-754; Mark Edward Noble, Mathematical Gazette 52 #382 (December 1968), 412-413; André Reix, Revue Philosophique de la France et de l'Étranger 166 #4 (October-December 1976), 437-438 (in French).

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  • $\begingroup$ "every paradox is highly instructive". Recall Russell's remarks on the Russian barber paradox. Some saw it as playground logic, Russell saw it as undermining Frege's programme of logicism. $\endgroup$ – buckner Jan 19 at 17:45

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