includes "Take a long walk on Pi: http://gigapan.com/gigapans/106803 " which looks sparser and stringier and more symmetrical than I expected. But it must not be qualitatively different from good random, else we'd be hearing more. At least it's not decimal. Correction: 3 EllipticE[1/9] - 8/3 EllipticK[1/9] + 8/3 EllipticPi[-1/3, 1/9] == (1/(54 (-I + 2 Sqrt[2])))(-243 I EllipticE[1/81 (17 - 56 I Sqrt[2])] + 27 Sqrt[-17 + 56 I Sqrt[2]] EllipticE[1/81 (17 + 56 I Sqrt[2])] + 2 (81 Sqrt[2] EllipticK[1/9] - 81 Sqrt[2] EllipticK[1/81 (17 + 56 I Sqrt[2])] + 3 (8 I - 7 Sqrt[2]) EllipticPi[1/27 (7 - 4 I Sqrt[2]), 1/81 (17 - 56 I Sqrt[2])] + (104 I + 71 Sqrt[2]) EllipticPi[ 1/3 (7 - 4 I Sqrt[2]), 1/81 (17 - 56 I Sqrt[2])])) ~3.99291 is ONE QUARTER of "the surface area of the convex hull of two edge-to-edge [perpendicular, with collinear diameters] unit disks. ( http://arxiv.org/abs/1211.4514)". But how the bleep do we get the above simplification? Or the even worse one in Finch's original paper? Are there more elliptic integral transformations like EllipticK[r] == E^(-I tan^-1(Sqrt[r/(1 - r)])) EllipticK[ 4 E^(I tan^-1(1/2 Sqrt[1/r - 4 + 1/(1 - r)])) Sqrt[(1 - r) r]] (which usually holds for two of the eight choices of √, and which I can't even prove)? Does this have anything to do with Complex Multiplication? --rwg Awful thought: What is the expected time at which a 3D random walk first becomes "knotted"? (W.r.t. pulling on the ends.)