To keep things simple I am creating rosette with six-fold rotational symmetry. The mapping functions are, using polar coordinates: X = r³ cos(6*φ) and Y = r³ sin(6*φ). An input image of a single butterfly results in 6 distorted butterflies:
For a two-color symmetry I use a control function of three-fold rotational symmetry
W = r cos(3*φ).
Using simple color inversion I get:
Clearly, this image has only three-fold rotational symmetry. The black wings become white in the parts with the inverted color and this destroys the propeller shape. The image now has two rather disparate parts.
With the improved inversion of my last post, that only changes the hue and not the grey part, I get a subtler effect:
The propeller shape is not destroyed and the image seems to be more coherent. We see that the rather bright yellow of the centers of the flowers becomes a much darker blue. Thus, even with this method we get changes in brightness. The false colors are interesting or disturbing – that depends on your taste.