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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
  • $\begingroup$ This sounds similar to the "disproof of the Pythagorean Theorem" I think that was the name I came across this. Perhaps I can find a link... Well, certainly math.stackexchange.com/q/1677958/36530 is quite relevant. I often give a version of this in Calculus II to show why definition and precision of concepts is essential to avoiding contradictory nonsense. $\endgroup$ – James S. Cook Dec 16 '19 at 2:36
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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
  • $\begingroup$ @DaveLRenfro Do you happen to have access to En marge du calcul des variations? I wonder where exactly this quote occurs. The first sentence sounds like it is from a preface, but if not he might have shared the story in one of those previously published papers. $\endgroup$ – Conifold Dec 16 '19 at 11:41
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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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