# Lessons from apparent paradoxes in geometric limits

1) Zeno's oxymoronic fleet, stationary arrow: One of the earliest infinity paradoxes, of course, is the flying arrow of Zeno which can't possibly be moving since it takes a finite amount of time to move half the distance AB from one point A to another C and again a finite time for half of that, the length AB/2, ad infinitum, and, since the sum of an infinite number of finite times must be infinite, the arrow could never arrive at C. The resolution comes with the introduction of the concept of the convergerce of an infinite sum, which the paradox points to. (Note Kalai's answer here.)

2) The two-equal-one paradox: Take a horizontal line segment L of length one, form an equilateral triangle from that, and then collapse the triangle by pushing its apex down to L, forming two triangles. Iterate on the apex of each of the two triangles, and then the next four triangles, ad infinitum, and we arrive at the apparent merging of a curve of length two at each iteration to a curve of length one. Resolution and lesson? You'd better impose smoothness, or differentialbility on the collapsing curve. (See this HSE question pointed out to me by David Renfro and "In Search of Infinity" by Vilenkin.) .

What other apparent paradoxes involving going to a geometric limit have been historically instructive and how, i.e., what can they teach us?

I was thinking of explicitly excluding Berkeley's ghosts, but that led to omega-epsilon gymnastics and eventually to Robinson's nonstandard analysis, I believe.

(I believe there are some very sophistcated topological paradoxes of a similar nature, but can't recall their names or where I read of them.)

Note that this excludes primarily set-theoretic or logical paradoxes such as Russell's Barbershop or the Hilbert/Cantor Hotel paradoxes although they are fascinating and important. Bertrand's probability paradox might suit the bill though since it relies on geometrical constructs.

• Copying my comment from the other thread since it's still relevant: You can equally well construct an example like in 2 using smooth curves (think plot of $\frac{1}{n}\sin(nx)$ from $0$ to $\pi$), unless you also assume suitable convergence of derivatives (in a suitable sense), but the actual lesson there in my opinion is "not everything is continuous" - this example shows length of a .rectifiable curve is not a continuous function under uniform metric on curves Jun 13 '20 at 21:02
• A couple of "geometric limit paradoxes" were instructive in the calculus of variations history. First is Weierstrass's objection to the "Dirichlet principle": while there are many smooth curves between two points passing through a third, not on the same line, the one of shortest length isn't smooth. Hence assuming that the isoperimetric problem has a solution isn't warranted. The other was popularized by Lebesgue (What is the history of staircase or 𝜋=4 paradox?): while sawtooths with decreasing teeth converge to a straight line their lengths do not Jun 15 '20 at 7:20
• @Wojowu: FYI, the example $\frac{1}{n}\sin(n^2 x)$ from $0$ to $\pi$ is given in: Nancy Edwards, An instance of intuition and lengths of limiting curves, The Pentagon 31 #1 (Fall 1971), 22-25 & 45. For these curves we even have their lengths approaching $\infty.$ I've cited this paper a few times in the past 15 or so years, but apparently only in places now no longer available (e.g. the ap-calculus discussion group at Math Forum). Jun 15 '20 at 15:28
• Closely related, but with a different flavor, is Mandelbrot's coastline paradox: length approximations for fractal curves do not have a limit at all, they diverge to infinity. Jun 17 '20 at 7:23
• Convergence of Fourier series to a function over a finite interval but divergence from it outside that interval was a point of contention among early developers and detractors, I believe. Jun 17 '20 at 13:54