Six pngs deleted in the name of "fun": gosper.org/3fold1.png, . . ., 3fold6.png plus gosper.org/minskytrispiral.png . ---------- Forwarded message ---------- From: Bill Gosper <billgosper@gmail.com> Date: Tue, Aug 8, 2017 at 6:53 AM To: mathfuneavesdroppers@googlegroups.com After finding this remarkable orbit, I'm sure Julian tried some obvious parameter tweaks, although he must have been using "circularized" Minsky rather than "tripled Trinsky". Again, the latter produces only integer coordinates. I just found these: ListPlot[{3, 1} # & /@ NestWhileList[{#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &@{#[[ 1]], #[[2]] + Floor[381 #[[1]]/128]} &@{#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &, {1, 0}, #2 != {1, 0} &, 2], AspectRatio -> 1, Axes -> False, Frame -> True] ListPlot[{3, 1} # & /@ NestWhileList[{#[[1]] - Floor[64 #[[2]]/(128 - 1/2)], #[[2]]} &@{#[[ 1]], #[[2]] + Floor[(384 - 3/2) #[[1]]/128]} &@{#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &, {1, 0}, #2 != {1, 0} &, 2], AspectRatio -> 1, Axes -> False, Frame -> True] ListPlot[{3, 1} # & /@ NestWhileList[{#[[1]] - Floor[64 #[[2]]/(128 - 1/2)], #[[2]]} &@{#[[ 1]], #[[2]] + Floor[(384 - 3/2) #[[1]]/128]} &@{#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &, {1, 2}, #2 != {1, 2} &, 2], AspectRatio -> 1, Axes -> False, Frame -> True] ListPlot[{3, 1} # & /@ NestWhileList[{#[[1]] - Floor[64 #[[2]]/(128 - 1/3)], #[[2]]} &@{#[[ 1]], #[[2]] + Floor[(384 - 1) #[[1]]/128]} &@{#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &, {6, 9}, #2 != {6, 9} &, 2], AspectRatio -> 1, Axes -> False, Frame -> True, PlotStyle -> PointSize[.001]] (Period 658047. Note increasing circularity with increasing radius, where the effects of the Floor functions diminish.) ListPlot[{3, 1} # & /@ NestWhileList[{#[[1]] - Floor[64 #[[2]]/(128 - 1/3)], #[[2]]} &@{#[[ 1]], #[[2]] + Floor[(384 - 1) #[[1]]/128]} &@{#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &, {5, 0}, #2 != {5, 0} &, 2], AspectRatio -> 1, Axes -> False, Frame -> True] (The vexing white vertical stripes are some artifact of SaveAs and are not on my screen.) ListPlot[{3, 1} # & /@ NestWhileList[{#[[1]] - Floor[64 #[[2]]/(128 - 1/4)], #[[2]]} &@{#[[ 1]], #[[2]] + Floor[(384 - 3/4) #[[1]]/128]} &@{#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &, {1, 2}, #2 != {1, 2} &, 2], AspectRatio -> 1, Axes -> False, Frame -> True] --rwg On Mon, Aug 7, 2017 at 3:56 AM, Bill Gosper <billgosper@gmail.com> wrote:

More generally, there is no threefold rotational symmetry. Julian's function can produce only gridpoints: ListPlot[{3, 1} # & /@ NestWhileList[{#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &@ {#[[1]], #[[2]] + Floor[381 #[[1]]/128]} &@ {#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &, {2, 1}, # != -{1, 4} &], AspectRatio -> 1, Axes -> False, Frame -> True]

And all the arguments to the Floors are merely rational. --Bill Gosper