I randomly asked the kids for the limiting arclength of a period of a Fourier-approximated square-wave, and Neil surprised me by claiming it was infinite. Here's evidence in his favor. Just estimating the total variation of 2^n Gibbs wiggles (see http://www.tweedledum.com/rwg/gibbs.htm): In[45]:= Table[Block[{k = 0, p = N[π], p2}, p2 = p/2; Nest[(SinIntegral[k++*p] - p2)*(-1)^k + # &, 0, 2^n]], {n, 18}] The Nest is to avoid Mma's wonderful Sum and NSum, the latter of which reports substantial imaginary parts here. Out[45]= {1.85194, 2.10855, 2.3484, 2.57886, 2.80445, 3.02757, 3.24945, 3.4707, 3.69165, 3.91244, 4.13315, 4.35383, 4.57448, 4.79513, 5.01577, 5.23641, 5.45704, 5.67768} In[46]:= Differences[%] Out[46]= {0.25661, 0.239855, 0.230462, 0.225586, 0.223117, 0.221878, \ 0.221257, 0.220946, 0.220791, 0.220713, 0.220674, 0.220655, 0.220645, \ 0.22064, 0.220638, 0.220637, 0.220636} I.e., the nth Gibbs peak seems to die off like 1/n, too slowly to converge. So Sum[(SinIntegral[k*π] - π/2)*(-1)^k,{k,0,n}] ~ Log[b,n], where b ~ Log[2]/.220636 = 3.14159 . Hmm. I guess XKCD is OK with logs base π. --rwg