Inspired by some joint investigations I conducted with Steve Witham a year or so ago, I recently did a simulation of an even more natural "derandomized random walk" in the quadrant: see http://jamespropp.org/quincunx.gif This shows what happens when you use a non-random Galton board in which the pins are replaced by flip-flops, so that the 1st marble to arrive at a flip-flop gets routed to the right, the 2nd gets routed to the left, the 3rd gets routed to the right, and so on, in alternation. My quasi-Galton board has 1024 rows, but the picture is rotated by 45 degrees, so the "bottom row" actually appears as a diagonal. Hence: the 1st marble to arrive at a flip-flop gets routed rightward, the 2nd gets routed downward, the 3rd gets routed rightward, and so on. White pixels corresponds to flip-flops that lie outside the board, or have not yet been visited by a marble. Yellow pixels correspond to flip-flops that most recently sent a marble rightward. Blue pixels correspond to flip-flops that most recently sent a marble downward. The nth frame (where the initial frame is considered the 0th) shows the states of the flip-flops (or, as I tend to call them, rotors) after the first n particles have passed beyond the first 1024 rows. When viewed with a suitable browser, quincunx.gif shows the first 1000 steps of what happens to the rotors. (It's kind of small at first, but wait for it: the set of non-white pixels will eventually become big enough to look at without a magnifying glass.) As you can see, patterns spontaneously form and propagate in a kind of Jacob's ladder fashion, with some triangles moving outward and others moving inward, as in Steve's "rounder-router" simulations. I believe this is new, but I won't be surprised if it isn't, since it's such a natural thing to look at. Has anyone seen this before? I've also done some variants of this simulation (changing the initial conditions and in some cases incorporating randomness); I can post them if people are interested. Note that there's another way one might do the simulation that looks superficially different, in which you drop a new marble at (0,0) at each time step, so that after n marbles have been dropped in, there's a single marble in {(i,j): i+j=k} for all k between 0 and n-1. This resulting movie, viewed as a static three-dimensional object, is essentially the same as the movie I created; you're just slicing the space-time object differently using different notions of "time". Perhaps this way of looking at things is more informative; I haven't coded it yet. The object being studied can be written as a fairly simple three- dimensional celluluar automaton with specific simple boundary conditions, so perhaps Wolfram and his crowd know about it already. Any leads or observations will be appreciated! Jim Propp