Adam Goucher has made a startling observation (quoted without permission): ---- I was reading the third edition of 'Minskys and Trinskys' by Gosper, Holloway and (Ziegler Hunts)^2: http://www.blurb.com/books/2172660-minskys-trinskys-3rd-edition The pattern of disjoint D10-symmetric snowflakes (or 'Rastermen', as they're referred to in the book) shown in Figure 31 appears to be affinely isomorphic to the pattern obtained by marking Penrose tilings: http://condellpark.com/kd/penrmark24bg.gif This suggests that it should be possible to create a very integer-esque method of drawing an affined Penrose tiling on a square grid: -- Run the Minsky algorithm on every square and mark any squares which participate in sufficiently long cycles. -- Exploit the fact that the Penrose tiling and its associated markings are mutually locally derivable. This could possibly be useful for running cellular automata on Penrose tilings in Golly -- just embed the tiling on a square grid, similar to Tim Hutton's 'slanted hexagon' approach. ------------ "P5d1" is the Minsky recurrence with d multiplier 1 and e chosen to make the unfloored period = 5. It makes those fractal pentagrams that fascinate me. It assigns to each gridpoint (x,y) the period of the minsky loop containing (x,y). Those periods are 5, 20, 30, 70, 130, 270, 530, 1070, ... Here <http://gosper.org/p5d1fun.pdf> is a raw sandbox of plotting the (x,y) with those periods. For some periods, the points form little figures (e.g., rhombi, hexagons, minipentagrams,...) whose centers are vertices of a rhombic Penrose tiling. Other periods produce patterns strongly resembling quasicrystal diffractograms. According to Wikipedia, Penrose began with gosper.org/6pents.bmp (minus the two Müller-Lyer segments). Hey kids: Animate this. Start with one chorded pentagon. Unfold it into two sharing the edge parallel to the chord. Then, with a Manipulate, slide the four blank pentagons in from the sides. Pause for effect. Slide them away. Finally, these same analyses led to gosper.org/pentfill.pdf and the earlier www.tweedledum.com/rwg/pentagonfill.pdf Sorry you have to click these yourselves. Even sorrier that every single gosper.org would-be attachment that I tried tonight was deleted, and needed to be restored. Just like 1970s quota enforcement. --rwg PS, I never got a copy of ---------- Forwarded message ---------- From: Bill Gosper <billgosper@gmail.com> Date: Sun, Jun 7, 2015 at 6:45 AM Subject: Fwd: What is the period of this integer sequence? To: math-fun@mailman.xmission.com I just noticed a typo in this item dated 2014-05-08 12:56: "Thus the only "restoring force" capable of pulling a[2n],a[2n+1] back to 0,1 ..." should say "back to (0,8)" Anyone who was confused by this is paying way too much attention. The item was primarily about the "Minsky Stock Index" restated toward the end. Last I heard, Tom Rokicki had chased it about 13 quadrillion steps without returning to (1,0). --rwg ----------------------------------------- [...] By way of couragement, here is a similar sequence we did track. ----- More xmission chupandose?