smooth carpet

f[m_] := Map[
   ({{1 - #, 1 - #^2, 1 - #}, {1 - #^2, 1 - #, 1 - #^2}, {1 - #,
       1 - #^2, 1 - #}}) &
   , m,
   {2}];
next[m_] := ArrayFlatten[f[m]];
n = 5;
img[p_] := (
   ArrayPlot[
    Nest[next, {{p}}, n], ColorFunction -> "Rainbow",
    ColorFunctionScaling -> False,
    PixelConstrained -> True,
    ImageSize -> {3^n, 3^n}]);
Manipulate[
 img[p],
 {p, 0, 1}]