Tricorn

The Tricorn, also known as the Mandelbar set, is a variation of the Mandelbrot set that uses the complex conjugate of $z_n$ before squaring:

z_{n+1} = \bar{z}_n^2 + c
    

where $z_0 = 0$ and $c$ is the point on the complex plane being tested. The conjugate $\bar{z}$ of a complex number $z = a + bi$ is $a - bi$, which reflects it across the real axis.

This single change — negating the imaginary part before squaring — breaks the rotational symmetry of the Mandelbrot set and produces a shape with threefold symmetry, giving the fractal its name "Tricorn."

Multicorn

The Multicorn generalizes the Tricorn to arbitrary integer powers $d$:

z_{n+1} = \bar{z}_n^d + c
    

The power $d$ determines the rotational symmetry of the resulting fractal: a power of $d$ produces $(d+1)$-fold symmetry. The Tricorn is the special case $d = 2$, which has threefold symmetry.

Parameters

GUI Parameters

• Iterations: This integer value determines the maximum number of times the formula is applied for each point.
• Power (d): The exponent applied to the conjugate. $d = 2$ gives the classic Tricorn; higher values produce shapes with more symmetry axes.

Complex Parameters

This fractal does not have any complex parameters that can be controlled directly as control points. The complex number $c$ corresponds to each point on the plane being tested.