Tricorn (mathematics)

In mathematics, the tricorn, sometimes called the Mandelbar set, is a fractal defined in a similar way to the Mandelbrot set, but using the mapping z ↦ z ¯ 2 + c {\displaystyle z\mapsto {\bar {z}}^{2}+c} instead of z ↦ z 2 + c {\displaystyle z\mapsto z^{2}+c} used for the Mandelbrot set. It was introduced by W. D. Crowe, R. Hasson, P. J. Rippon, and P. E. D. Strain-Clark. John Milnor found tricorn-like sets as a prototypical configuration in the parameter space of real cubic polynomials, and in various other families of rational maps.
The characteristic three-cornered shape created by this fractal repeats with variations at different scales, showing the same sort of self-similarity as the Mandelbrot set. In addition to smaller tricorns, smaller versions of the Mandelbrot set are also contained within the tricorn fractal.
The tricorn T {\displaystyle T} is defined by a family of quadratic antiholomorphic polynomials
given by
where c {\displaystyle c} is a complex parameter. For each c {\displaystyle c} , one looks at the forward orbit
of the critical point 0 {\displaystyle 0} of the antiholomorphic polynomial p c {\displaystyle p_{c}} . In analogy with the Mandelbrot set, the tricorn is defined as the set of all parameters c {\displaystyle c} for which the forward orbit of the critical point is bounded. This is equivalent to saying that the tricorn is the connectedness locus of the family of quadratic antiholomorphic polynomials; i.e. the set of all parameters c {\displaystyle c} for which the Julia set J ( f c ) {\displaystyle J(f_{c})} is connected.
The higher degree analogues of the tricorn are known as the multicorns. These are the connectedness loci of the family of antiholomorphic polynomials f c : z ↦ z ¯ d + c {\displaystyle f_{c}:z\mapsto {\bar {z}}^{d}+c} .
Much like the Mandelbrot set, the tricorn has many complex and intricate designs. Due to their similarity, they share many features. However, in the tricorn such features appear to be squeezed and stretched along its boundary.
The following images are progressional zooms on a selected c {\displaystyle c} value where c = 0.48 + 0.58 i {\displaystyle c=0.48+0.58i} . The images are not stretched or altered, that is how they look on magnification.
The below pseudocode implementation hardcodes the complex operations for Z. Consider implementing complex number operations to allow for more dynamic and reusable code.
The tricorn is not path connected. Hubbard and Schleicher showed that there are hyperbolic components of odd period of the tricorn that cannot be connected to the hyperbolic component of period one by paths. A stronger statement to the effect that no two (non-real) odd period hyperbolic components of the tricorn can be connected by a path was proved by Inou and Mukherjee.
It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. On the other hand, the rational parameter rays at odd-periodic (except period one) angles of the tricorn accumulate on arcs of positive length consisting of parabolic parameters. Moreover, unlike the Mandelbrot set, the dynamically natural straightening map from a baby tricorn to the original tricorn is discontinuous at infinitely many parameters.