Coalesce

Continuing with the theme of selectively breaking and re-forming a tiling pattern (1, 2, 3), though this time without any rotation.

Here's the (very slow) code:

DynamicModule[{BasicTileCoords, f, cols, centerpoint, ψ, layering}, 
 BasicTileCoords = {{0, -1}, {0, 0}, {Sqrt[3]/2, 1/2}, {Sqrt[3]/2, 0}, {Sqrt[3]/4, -1/4}, {Sqrt[3]/4, -3/4}};
 cols = ColorData["BlueGreenYellow"] /@ Range[1, 0, -1/4];
 f = 3 Piecewise[{{0, # < 0}, {1/2 - 1/2 Cos[#], 0 <= # < π}, {1, # >= π}}] &;
 layering = Append[Riffle[Reverse[Range[5]], -Reverse[Range[5]]], 0];
 Manipulate[
  Graphics[{EdgeForm[Directive[JoinForm["Round"], cols[[4]], Thickness[.006]]], 
    Table[
     {centerpoint = {3 Sqrt[3]/4 i, 3/2 j + (-1)^i*3/8};
      ψ = f[θ - π/(3 Sqrt[7]) Norm[centerpoint - {0, 3/8}]];
      FaceForm[Directive[cols[[t]]]], 
      Polygon[centerpoint + # & /@ (RotationMatrix[2 π t/3].(ψ {Sqrt[3]/4, -1/4} + #) & /@ BasicTileCoords)]}, 
      {j, layering}, {i, layering}, {t, 1, 3}]}, 
   PlotRange -> {{-5, 5}, {-5 + 3/8, 5 + 3/8}}, Background -> cols[[5]], ImageSize -> 540],
{θ, 0., 2 π}]
 ]