Fractal Guide
What is the Tricorn (Mandelbar Set)?
The Tricorn is a fractal born by taking the complex conjugate in the Mandelbrot recurrence. Also known as the Mandelbar set, it features distinctive three-fold symmetry.
Contents
1. Definition of the Tricorn
The Tricorn is defined by z(n+1) = conj(z(n))ยฒ + c, where conj(z) is the complex conjugate of z (the imaginary part's sign is flipped).
The only difference from the Mandelbrot set is taking the complex conjugate at each iteration. This small change transforms the Mandelbrot's two-fold symmetry into three-fold symmetry, producing an entirely different fractal.
2. The Secret of Three-Fold Symmetry
The Mandelbrot set is symmetric about the real axis (two-fold symmetry), but the Tricorn has 120ยฐ rotational symmetry (three-fold). This arises from the combination of complex conjugation and squaring producing third-order symmetry.
This three-fold symmetry appears not only in the overall set but also in zoomed details. The miniature copies familiar from the Mandelbrot set appear in three directions in the Tricorn.
3. Comparison with the Mandelbrot Set
The Tricorn's boundary appears more angular, with pointed protrusions extending in three directions. The main 'buds' are roughly triangular rather than cardioid-shaped.
Interestingly, zooming into the Mandelbrot set's boundary can reveal hidden tiny Tricorns. This geometric interplay reflects deep structures in complex dynamical systems.
4. Rendering and Exploration
The rendering algorithm is nearly identical to the Mandelbrot set โ the only difference is taking the complex conjugate of z at each iteration. Computational cost is similar, enabling real-time interactive exploration.
Try zooming into Tool Palette's Fractal Gallery while observing the three-fold symmetry. Near the boundaries, you'll find unique fractal structures different from the Mandelbrot set.
Explore the Tricorn
Explore the Tricorn's three-fold symmetry in the Fractal Gallery.
Go to Fractal Gallery