Hénon attractor as realised by Spirofractal, and a typical veil-like quadratic attractor.
Spirofractal generates this type of pattern by iterating the most general quadratic real polynomial in x and y with randomly chosen coefficients. The iteration works by evaluating the polynomial for the current x and y position, then assigning the value of the polynomial to x, and the previous value of x to y. The images produced by this operation are generally not symmetrical, and quite often look a little like a strangely draped veil. The Hénon Attractor is a famous case of this mode of operation. Complex attractors will be asymmetrical if both rotational symmetry and mirror symmetry have been switched off, and are occasionally asymmetrical anyway.
Hénon attractor as realised by Spirofractal, and a typical veil-like quadratic attractor.
This close up of the top left corner of the Hénon attractor reveals that what originally appears to be one thick line is in reality a set of much finer lines that are close together. This is typical of strange attractors.
A few polynomials produce symmetric images, though unfortunately I haven't found any interesting examples of this yet. However, there are many patterns which have have near symmetries in all or part of the image. These images are sometimes rather like the patterns produced by the Victorian "Harmonograph". I have no idea why this is.
Head on to Tiles (a.k.a. Symmetric Quilts)