The Penrose tiling is self-similar as many other quasi-periodic tilings. It matches a copy of itself inflated by the golden ratio τ=(1+√5)/2≅1.618, see “Penrose tiling tied up in ribbons“. Noting that our quasi-periodic designs of 5-fold symmetry are closely related to the Penrose tiling, we think that the self-similarity might show up somehow in these designs.
To show you a simple example, I made a design using wave packages based on cos(x) and sin(x) functions. It should match wave packages of a scale inflated by τ. Thus I used an additional wave package based on cos(x/τ). Wherever it has a negative value, the color of the design gets inverted. This matches the basic design and nicely accentuates its structure:
The center of perfect 5-fold symmetry lies in the lower left part.
The matching between the two different wavelengths is not surprising. The wave vectors of the mapping between output image and input image are of the form k_m=(cos(2πm/5),sin(2πm/5)). Noting that cos(2π/5)=1/(2τ) we get immediately that k_1+k_4=(1/τ,0), which is one of the waves used for controlling the color change. Thus sums of basic wave vectors give the smaller wave vectors of the inflated tiling.