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 > 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun