It is well known that temporal first-derivative reaction-diffusion systems can produce various fascinating Turing patterns. However, it has been found that many physical, chemical and biological ...

We are providing an unedited version of this manuscript to give early access to its findings. Before final publication, the manuscript will undergo further editing. Please note there may be errors ...

Each filling factor ν = p/q is represented by a dot in a polar form with angle θ = 2πp/q. The radial axis represents the denominator q in an inverted manner: q = 1 is at the outermost edge, and larger ...