Peano[0] = DLine[{{0,0},{1,1}}];
Peano[n_] := Peano[n-1] /. DLine[{{x0_,y0_},{x1_,y1_}}] :> {
dx=x1-x0; dy=y1-y0;
DLine[{{x0,y0}, {x0+1/2 dx, y0+1/2 dy}}],
Line[{{x0+1/2 dx, y0+1/2 dy}, {x0+1/2 dx, y0}}],
DLine[{{x0+1/2 dx, y0}, {x0+dx, y0+1/2 dy}}],
Line[{{x0+dx, y0+1/2 dy}, {x0, y0+1/2 dy}}],
DLine[{{x0, y0+1/2 dy}, {x0+1/2 dx, y0+dy}}],
Line[{{x0+1/2 dx, y0+dy}, {x0+1/2 dx, y0+1/2 dy}}],
DLine[{ {x0+1/2 dx, y0+1/2 dy},  {x0+dx, y0+dy}}]
};
GraphicsGrid[Partition[
Table[Graphics[Peano[n] /. DLine -> Line], {n, 0, 3}],
2]]

AlmostPeano[0] = DLine[{{0,0},{1,1}}];
AlmostPeano[n_] := AlmostPeano[n-1] /. DLine[{{x0_,y0_},{x1_,y1_}}] :> {
dx=x1-x0; dy=y1-y0;
DLine[{{x0,y0}, {x0+0.45 dx, y0+0.45 dy}}],
Line[{{x0+0.45 dx, y0+0.45 dy}, {x0+0.55 dx, y0}}],
DLine[{{x0+0.55 dx, y0}, {x0+dx, y0+0.45 dy}}],
Line[{{x0+dx, y0+0.45 dy}, {x0, y0+0.55 dy}}],
DLine[{{x0, y0+0.55 dy}, {x0+0.45 dx, y0+dy}}],
Line[{{x0+0.45 dx, y0+dy}, {x0+0.55 dx, y0+0.55 dy}}],
DLine[{ {x0+0.55 dx, y0+0.55 dy},  {x0+dx, y0+dy}}]
};
GraphicsGrid[Partition[
Table[Graphics[AlmostPeano[n] /. DLine -> Line], {n, 0, 3}],
2]]