Multibrot sets and multicorns
The quadratic Mandelbrot set is one member of a larger family. Replacing z² with zᵈ produces Multibrot sets; conjugating z before the power creates antiholomorphic relatives called multicorns.
Power determines symmetry
For integer d greater than one, zₙ₊₁ = zₙᵈ + c produces a parameter set with d − 1 rotational symmetries around the origin, together with reflection symmetry. Cubic and higher powers reorganize bulbs and filaments while preserving the bounded-orbit question.
For z↦zᵈ+c, a Multibrot set has (d−1)-fold rotational symmetry. Degree also changes the number and arrangement of major components. A fair visual comparison holds crop, iteration logic and color normalization constant; otherwise a palette change can masquerade as geometry. Non-integer powers additionally require a choice of complex branch and do not simply belong to the same family.
Conjugation changes the theory
Antiholomorphic iteration uses the complex conjugate of z. The tricorn, the quadratic case, has threefold appearance and bifurcation behavior that differs significantly from the Mandelbrot set; familiar holomorphic theorems cannot simply be copied over.
Multicorns use z̄ᵈ+c and are antiholomorphic. Their hyperbolic components have different boundary phenomena, including arcs of parabolic parameters rather than only isolated cusps. Familiar holomorphic vocabulary therefore does not transfer mechanically. Visible multiple symmetry is an entry point, but conjugation genuinely changes parameter geometry and the nature of bifurcations.
A controlled way to compare formulas
Keeping color, viewport and iteration policy consistent while changing only the formula reveals which structures come from exponent, symmetry or conjugation. This is more informative than treating every silhouette as an unrelated decorative preset.
A controlled investigation chooses a degree, sets an appropriate escape condition and compares identical pixel coordinates in parameter and Julia planes. Smoothing and color source come afterward. This preserves which changes arise from exponent or conjugation. Style presets should not silently couple those mathematical parameters to effects when the purpose is formula comparison.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Multicorns and Unicorns II: Bifurcations in Antiholomorphic DynamicsErgodic Theory and Dynamical Systems
- Mandelbrot SetWolfram MathWorld
Continue the idea
The Mandelbrot set
What the Mandelbrot set contains, how z² + c creates its boundary and why deep zoom reveals related rather than identical copies.
The Burning Ship fractal
Why taking absolute values before squaring breaks rotational symmetry and creates the Burning Ship’s flame-like landscape.
Julia sets
How fixing c in z² + c creates a dynamical plane, from connected dendrites to disconnected Cantor dust.