15625 dots

vis[l_, \[Alpha]_] := vis[l, {0, 0}, 0, \[Alpha]];
vis[{}, p_, n_, \[Alpha]_] := {p};
vis[l_, p_, j_, \[Alpha]_] :=
 Module[{rest = l[[2 ;;]], n = First[l]},
  Table[
   vis[rest,
    p + \[Alpha]^j {Sin[2 Pi i /n + j Pi/2], Cos[2 Pi i /n + j Pi/2]},
     j - 1, \[Alpha]], {i, n}]
  ];
frame[\[Alpha]_] := Module[{l},
  (*l=Flatten@Map[Table[First@#,{i,Last@#}]&,FactorInteger[n]];*)

  l = Table[5, {i, 6}];
  Graphics[{Opacity[0.5],
    Map[Point, Partition[Flatten[vis[l, \[Alpha]]], 2]]},
   PlotRange -> 2]
  ];
frame[1.2]