http://gosper.org/x=1, y=1, dmin=emin=3o2, dmax=emax=8o3, dg=eg=1o2574, de<=49o12.png , corrects and refines the "δ-ℇ plot" on p23 of http://www.blurb.com/books/2172660 (also Neil's blog, http://nbickford.files.wordpress.com/2011/03/x1-y1-dminemin3o2-dmaxemax8o3-d... ). It's a map of the periods of the Minsky circle algorithm, starting with x0=1, y0=1, with multipliers 3/2 ≤ δ,ℇ ≤ 8/3. With a zoomable, nonblurring viewer, I find it fascinating and surprising. The (δ,ℇ) which blow up exponentially (because δ ℇ>4) are shown in white. But note that some colored rectangles transgress δ ℇ = 4 ! The floor function in the Minsky iteration can actually tame an infinite sequence. Contrariwise, imposing the floor operation on x0=1, y0=1/2, δ 3^(n+1), ℇ = 3^-n changes a mere period 3 into infinite linear growth! Any such growers in the graphic appear black, along with any orbits longer than 10000, of which there are >57000 (in this plot). Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the hyperbola. Julian completed my humiliation by finding the period 159 region {x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four. Zooming at a crack between large rectangles often shows a single layer of variegated pixels, due to the sampler landing exactly on the crack, which, remarkably, is often composed of numerous truly 1-dimensional line segment subregions. Now zoom at the top of the upper largish pink rectangle. Euclid's orchard? Other high-frequency patches resemble peering into factory windows or out of skyscrapers. Also note the apparent, possibly provable accumulation point at (2,2). See book, p 23, for Julian's proof why this would guarantee unbounded periods, despite the lack of thin black rectangles at this sampling density (1/2574)). Also, note the optically illusory reverse curvature when you zoom at (2,2). The plot was computed by Corey's old program, and took 14 hrs, and produced a blank screen due to a bug in Mathematica. Exporting it took several more minutes, returning the filename, but writing no file! I spent a day fighting back creeping insanity, until Julian found an ingenious workaround. --rwg I have proofread this message, and disavow it. --Julian