Tricorn/Multicorn — Fantastic Fractals

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} = \overset{z}{ˉ}_{n}^{2} + c

where $z_{0} = 0$ and $c$ is the point on the complex plane being tested. The conjugate $\overset{z}{ˉ}$ 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} = \overset{z}{ˉ}_{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.