Nice. I don't have the prerequisites to understand what \( A(\cdot) \), Antimatter etc. is. If you could elaborate I'd be interested. Do you remember which dusty old library book it was?

A(

*x*) means the attracting basin of

*x*. Antimatter is the fractal formula given for "supernova" higher up in this thread, without the "+

*b*" (so, with

*b* fixed at zero). The library book was some old highly technical mathematical research volume about iterated rational functions of the complex numbers. I went into the stacks specifically to find any formula that might produce Herman rings, after hearing in multiple places that it had been proven (the Sullivan classification) that such functions divide the complex plane into only a few types of regions: attracting basins for attractive fixed points and cycles (corresponds to parameters inside Mandelbrot bulbs and cardioids); attracting basins for parabolic fixed points and cycles (corresponds to parameters at cardioid cusps and where bulbs are tangent to cardioids or other bulbs); attracting basins for Siegel disks (points on the boundaries of bulbs and cardioids where no bulb attaches and that are not cusps) or cycles of same; attracting basins for Herman rings or cycles of same; and the Julia set itself, which is the boundary of any one of the above (thus, any point belonging to the Julia set has points belonging to all of the attracting basins of that function arbitrarily close by). If a function produces no attractors at all the Julia set is the full Riemann sphere. Furthermore, every attracting basin must capture at least one critical point or value.

The ordinary Mandelbrot fractal maps the possibilities for quadratic polynomials. Infinity always attracts, and counts as a critical point. The other critical point is zero (derivative of

*z*^{2} +

*c* is 2

*z*, only zero at zero) and may thus either also go to infinity, or else may go to one finite attractor. The Mandelbrot Set itself contains the points where the latter happens, and is the closure of that set. But it cannot produce Herman rings, only the other possibilities named above. I got curious to see one of these things and dug around in the university library and card catalogue (the internet was barely a thing yet back then!) and eventually found the formula I named "Antimatter" presented as an example that could generate Herman rings, if

*a* was an irrational unit (i.e. cos

*x* +

*i* sin

*x* for

*x*/2pi irrational) and

*c* was either pure real or pure imaginary and of magnitude exceeding about 3 or so.

I took the formula and coded it as a Fractint formula, and later as UF formulae (and derived the Supernova variation the same way Nova derives from the Newton's method Julia set). It turned out that formulas capable of producing Herman rings produce some unique and beautiful patterns, both in parameter and dynamic space images, such as seahorses with lakes in the center and smoke-swirl turbulence patterns that alternate spirals on each side along an arc or around a ring. Spacefilling Julia sets from parameters close to ones that produce Herman rings can be especially spectacular ...