Duh, I found my error. "The first image (δ = 43/100) is the first 10 million points of a 31 billion point orbit!" The period is apparently >> 31 billion = 9^11, which was a cutoff I hadn't realized I'd reached. We may not soon learn the true value of MinskyPeriod[34, 55, 43/100, (5/2 - Sqrt[5]/2)/(43/100)] because the Mathematica function is too slow, and Corey's C function MinskyPeriodC[34, 55, 43/100, Rationalize[(5/2 - Sqrt[5]/2)/(43/100), 2^-35]] // tim During evaluation of In[474]:= 1014.793072,23 Out[474]= {{34, 55, 43/100, 449335/139811, 1000000000}, {-52323, -37346}, {-82792, -57707}, {-57510, 77094}, {-66115, 69793}, {15026, 292613}, {43033, 302815}, {-7838, 298304}, {23672, 395987}, {21322, 499577}, {68999, 584307}, {14814, 462248}, {-23276, 488434}, {-71079, 353864}, {-51730, 276366}, {-77521, 250910}, {-113129, 176498}, {-82214, 109947}, {-77314, 141684}, {1269, 245606}, {-24070, 120762}, {-8254, 15225}, {21272418675}} (21G. The {x,y} are checkpoints every billion steps) only takes rational arguments. And each time I refine the rational approximation (i.e. raise the power of 1/2), the period increases. At some point, the terms of the rational approximation multiplied by the increasing {x,y} are likely to overflow, probably provoking an infinite loop via an irreversible step. So far, 2^-36 has run a couple of hundred billion, and diverged from 2^-48, which might run even longer, or overflow. So I just switched to 2^-49, which so far (87G) agrees with 2^-48. --rwg On Fri, Apr 15, 2016 at 12:50 AM, Bill Gosper <billgosper@gmail.com> wrote:
Regardless of whether the center is a true vertex, the odd number of spokes would require retracing somehow. But, apologies. My experimental procedure to produce starfish43.png must have been flawed. Running backwards (red) and forwards (green) 10^7 points and plotting every 101st: gosper.org/starfish43rg.png shows distinct inbound and outbound spokes, no problem. --rwg
I'm guessing that the apparent order-5 "vertex" is formed from two effects. First, that the scale of the picture is large -- a zoom on the center would reveal a void around the origin where the spokes seem to converge. Second, the points that are close in the plot are actually separated by a small multiple of 5, so adjacency in the plot is not the same as adjacency in the dynamic system.
On Thu, Apr 14, 2016 at 7:20 PM, rwg <rwg@sdf.org> wrote:
(Hopefully) I'd've said "puzzle" if I knew the answer. My only guess is that it retraces the spokes. Needs to be checked. -rwg
On 2016-04-14 12:06, Allan Wechsler wrote:
Was your last question a puzzle, or an honest question? I'm withholding my answer in case it was a puzzle.
On Thu, Apr 14, 2016 at 2:48 PM, Bill Gosper <billgosper@gmail.com> wrote:
For x₀ = 34, y₀ = 55, δ > 0, and ε = 4 (sin π/5]^2/δ, without the Floor
operation, the Minsky iteration simply repeats five points on an ellipse whose shape depends on δ. If we include the Floors, we get a variety of fivish orbits. δ = 1 produces those intriguing fractal pentagrams gosper.org/rastar.pdf discovered by Corey and Julian, and perhaps earlier by Steve Witham. A portion of the x-y rug plot is Figure 31 (p17) in http://www.blurb.com/books/2172660-minskys-trinskys-3rd-edition . A few months ago, Adam Goucher made the startling observation that this plot defines a legal kites-and-darts Penrose tiling. (See the last three images in gosper.org/g12a.pdf (102MB!), the stills from my (and Rohan Ridenour's) recent G4G12 talk. (Rohan did the animations on a separate screen.)) Varying δ gives a variety of orbits. δ = 33/100 simply loops over ten points in five close pairs. But δ = 17/50 makes five fuzzy patches of 100 points each. And δ = 17/50 + 1/googol makes a pretty doily with 30071 points! gosper.org/mnskplt.png Similarly, δ = 14/25 gives a 5794 point starfish, while δ = 14/25 + 1/googol gives a 72721 point obese starfish. gosper.org/mnskplt1.png . Various other starfish are in gosper.org/mnskplt2.png . Warning: The third one is actually five sea-cucumbers having a little fun at your expense. The last two (δ =.62 and .63) are just printouts of merely period 5 loops illustrating the experimental fact that every perturbation of δ by as little as .01 seems to produce a different orbit. The first image (δ = 43/100) is the first 10 million points of a 31 billion point orbit! Here is every 5001st point of it: gosper.org/starfish43.png . Note the radius grew from 55 to >
On 2016-04-14 17:53, Allan Wechsler wrote: 200000.
But something is (star)fishy. Finite Minsky orbits are unicursal. How can there be a graph vertex of degree 5 in the center?? --rwg _______________________________________________