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.)
On 12/25/12, Bill Gosper <billgosper@gmail.com> wrote:
Awful thought: What is the expected time at which a 3D random walk first becomes "knotted"? (W.r.t. pulling on the ends.)
Bearing in mind that the ends may in general lie deep within the convex hull of the walk, perhaps this question requires rather more careful definition ... WFL
On 12/24/2012 4:47 PM, Fred lunnon wrote:
On 12/25/12, Bill Gosper<billgosper@gmail.com> wrote:
Awful thought: What is the expected time at which a 3D random walk first becomes "knotted"? (W.r.t. pulling on the ends.) Bearing in mind that the ends may in general lie deep within the convex hull of the walk, perhaps this question requires rather more careful definition ... WFL
You could constrain the question to those random walks that return to the starting point (although of measure zero) or you could try defining a different kind of 'random walk' that starts with a unit loop and randomly expands and rearranges it, with crossovers allowed. Brent
Or just draw a straight line from one end of the walk to the other. Maybe exclude cases where this line intersects the path-so-far. Rich --- Quoting meekerdb <meekerdb@verizon.net>:
On 12/24/2012 4:47 PM, Fred lunnon wrote:
On 12/25/12, Bill Gosper<billgosper@gmail.com> wrote:
Awful thought: What is the expected time at which a 3D random walk first becomes "knotted"? (W.r.t. pulling on the ends.) Bearing in mind that the ends may in general lie deep within the convex hull of the walk, perhaps this question requires rather more careful definition ... WFL
You could constrain the question to those random walks that return to the starting point (although of measure zero) or you could try defining a different kind of 'random walk' that starts with a unit loop and randomly expands and rearranges it, with crossovers allowed.
Brent
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