ACW> I would like to see a version of this plot where period is mapped to grayscale, so that I could see which areas had larger periods. Corey's new plotter uses lightness ~ 1/period, e.g., this cave<http://gosper.org/x=1,y=2,13o20%E2%89%A4%CE%B4%E2%89%A461o60,119o60%E2%89%A4%E2%84%87%E2%89%A4151o60,%CE%B4g=%CE%B5g=1o13860.png>, which, although it starts out with a simple-looking recursion, must grow arbitrary dark by Julian's theorem. But usually these δ-ε plots are too confusing, even with color. E.g., the ("randomly" colored) http://nbickford.files.wordpress.com/2011/03/neighborhood.png . Note the isolated point (period 634) at (13/6,10/7). I believe the 18.7 trillion region is a very short line segment. --rwg I believe that Corey and Julian have shown that the function Period(x0,y0,δ,ε) partitions R^4 into (countably many?) "congruence regions" (producing Minsky orbits pointwise equivalent under rigid translation), where each region is the Cartesian product of two "n-gons" (3 ≤ n ≤ 6) with hyperbolically curved sides. If this seems simpler than the Mandelbrot set, look at some of the x-y "rug-plots", each of which is unbounded, and consider there's a different one for each δ-ε region.