Complex dynamics
Tricorn (Mandelbar)
Crowe, Hasson, Rippon, and Strain-Clark introduced the Tricorn in 1989. Replace z² in the Mandelbrot iteration with the complex conjugate squared, z ← conj(z)² + c. The dynamics become anti-holomorphic and the resulting fractal has three horn-like extensions instead of two-fold symmetry.
At a glance
| Designer | Crowe, Hasson, Rippon, Strain-Clark, 1989 |
|---|---|
| Iteration | zn+1 = conj(zn)2 + c |
| Symmetry | 3-fold (D3) |
| Alias | Mandelbar |
Anti-holomorphic dynamics
Because z → conj(z) is anti-holomorphic, the Tricorn iteration is not a complex polynomial. Many Mandelbrot theorems (e.g. local connectivity of the set) become subtler. Nevertheless the Tricorn boundary is connected and exhibits characteristic horns where the Mandelbrot would show bulbs.
Higher-degree Mandelbars
More generally, iterating conj(z)n + c yields fractals with (n + 1)-fold symmetry. For n= 2 the symmetry is three-fold, hence “Tricorn”.
References
- Crowe, W. D., Hasson, R., Rippon, P. J., Strain-Clark, P. E. D., “On the structure of the Mandelbar set,” Nonlinearity, 1989.
- Nakane, S. & Schleicher, D., “On multicorns and unicorns,” Int. J. Bifurcation and Chaos, 2003.
- Mandelbrot set · Burning ship
Try it
Run an interactive playground at /tools/tricorn.