I hereby present the prize of some daring excursions into the very vast, terra incognita-extensions of the classic, cardioïd Mandelbrot set, consisting of all regular, ‘not quite-conformal’ bordering states, being a choice selection of twelve crisp & clear Julia-sets, harvested at exquisite positions in this giant parameter space.
Iterated conformal transformations generate the most fascinating and intricately woven objects in math, the Mandelbrot- and Julia-sets, their delicate structure due to the critical lines being reduced to points. But the transformation need not be strictly conformal to possess a degenerate critical line. A typical rule for updating an epicycloïd, ‘not necessarily-conformal’ system is:
r² ← x² + y²
φ ← arctan y/x, mod 2π
x ← r² · (cos 2φ – (cos ωφ) / c) + a
y ← r² · (sin 2φ – (sin ωφ) / c) + b
with ω ∈ N, ensuring closed curves.
The critical lines are found by solving
2r³ · (c1 – c2 · (cos 2φ
· cos ωφ + sin 2φ · sin ωφ)) =
with c1 = 2 + ω / c² and c2 = (2 + ω) / c.
Lim c→∞ is represented by the quadratic Mandelbrot-set, but now there exist extensions into c-space for each ω. To ensure that zero is the sole critical point, the lower c-bound should be chosen > 1. To produce the most filigrane Julia-sets, we require the curves to be strictly convex, suggesting c = ω² / 4 as starting point.
The rule is easily modified. Maintaining the general outline, harmonics are added by replacing ωφ with
k1π + ω1φ + sin(k2π + ω2φ) / c1.
The dynamics of a few such systems are mapped in the second row of Julia-sets. For the third row I concocted another variation:
x ← r² · cos 2φ ·
(c1 + cos(kxπ +
ωxφ) / ωx) / c2 +
y ← r² · sin 2φ · (c1 + cos(kyπ + ωyφ) / ωy) / c2 + b.
For the first row this concise rule was used (the claim for convexity no longer holds here):
x ← r² · cos 2φ + a
s ← ∑ sin(knπ + ωnφ) / ωn, n = 1, 2, ...
y ← r² · sin(2φ + s / c) + b.
To allow comparison, the coëfficiënts are chosen always so that the two period-one fixed points possess eigenvalues of nearly equal absolute magnitude. The constant vector was only slightly adjusted from case to case. Iteration time is much shortened using precalculated values, rescaled and stored in integer arrays:
x ← r² · f(pφ) + a
y ← r² · g(pφ) + b
a precision 1 / p = 0.0002 proves to be sufficient.
Finally, I devised a really circumlocutory way of extending Mandelbrot-sets, choosing at random the twelve coëfficiënts for the general quadratic equation
x’ = a1x² + a2xy +
a3y² + a4x + a5y +
y’ = a7y² + a8xy + a9x² + a10y + a11x + a12
and retaining those combinations for which the critical line is a (small) ellips. By tuning and trimming the coëfficiënts, the critical ellips is shrunk to a near-point. The resulting transformations obviously represent non-integer states bordering ω = 2 in the plain epicycloïd system. A sample Julia-set, located near the period-two bifurcation point, spreads like a panache, showing its concealed mocking bill in Mercator's projection: