A tricorn, created on a computer in Kalles Fraktaler.
Multicorns with the power going from 1 to 5

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 instead of used for the Mandelbrot set. It was introduced by W. D. Crowe, R. Hasson, P. J. Rippon, and P. E. D. Strain-Clark.[1] 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.[2]

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.

Formal definition

The tricorn is defined by a family of quadratic antiholomorphic polynomials

given by

where is a complex parameter. For each , one looks at the forward orbit

of the critical point of the antiholomorphic polynomial . In analogy with the Mandelbrot set, the tricorn is defined as the set of all parameters 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 for which the Julia set is connected.

The higher degree analogues of the tricorn are known as the multicorns.[3] These are the connectedness loci of the family of antiholomorphic polynomials .

Basic properties


The below pseudocode implementation hardcodes the complex operations for Z. Consider implementing complex number operations to allow for more dynamic and reusable code.

For each pixel (x, y) on the screen, do:
    x = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
    y = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
    zx = x; // zx represents the real part of z
    zy = y; // zy represents the imaginary part of z

    iteration = 0
    max_iteration = 1000
    while (zx*zx + zy*zy < 4  AND  iteration < max_iteration) 
        xtemp = zx*zx - zy*zy + x
        zy = -2*zx*zy + y
        zx = xtemp

        iteration = iteration + 1

    if (iteration == max_iteration) //Belongs to the set
        return insideColor;

    return iteration * color;

Further topological properties

The tricorn is not path connected.[5] 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.[7]

It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter.[8][9] 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.[10] Moreover, unlike the Mandelbrot set, the dynamically natural straightening map from a baby tricorn to the original tricorn is discontinuous at infinitely many parameters.[7]


  1. ^ Crowe, W. D.; Hasson, R.; Rippon, P. J.; Strain-Clark, P. E. D. (1 January 1989). "On the structure of the Mandelbar set". Nonlinearity. 2 (4): 541. Bibcode:1989Nonli...2..541C. doi:10.1088/0951-7715/2/4/003. S2CID 250790435.
  2. ^ Milnor, John (1 January 1992). "Remarks on iterated cubic maps". Experimental Mathematics. 1 (1): 5–24. Retrieved 6 May 2017 – via Project Euclid.
  3. ^ a b Nakane, Shizuo; Schleicher, Dierk (1 October 2003). "On multicorns and unicorns i: antiholomorphic dynamics, hyperbolic components and real cubic polynomials". International Journal of Bifurcation and Chaos. 13 (10): 2825–2844. Bibcode:2003IJBC...13.2825N. CiteSeerX doi:10.1142/S0218127403008259.
  4. ^ Nakane, Shizuo (1 June 1993). "Connectedness of the tricorn". Ergodic Theory and Dynamical Systems. 13 (2): 349–356. doi:10.1017/S0143385700007409. S2CID 123629440. Retrieved 6 May 2017.
  5. ^ a b "Multicorns are not path connected" (PDF). Math.cornell.edu. Retrieved 2017-05-06.
  6. ^ a b Mukherjee, Sabyasachi; Nakane, Shizuo; Schleicher, Dierk (1 May 2017). "On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials". Ergodic Theory and Dynamical Systems. 37 (3): 859–899. arXiv:1404.5031. doi:10.1017/etds.2015.65. S2CID 119524999.
  7. ^ a b Inou, Hiroyuki; Mukherjee, Sabyasachi (2021). "Discontinuity of Straightening in Anti-holomorphic Dynamics: I". Transactions of the American Mathematical Society. 374 (9): 6445–6481. arXiv:1605.08061v5. doi:10.1090/tran/8381. S2CID 53514019.
  8. ^ Goldberg, Lisa R.; Milnor, John (1993). "Fixed points of polynomial maps. Part II. Fixed point portraits". Annales Scientifiques de l'École Normale Supérieure. 26 (1): 51–98. doi:10.24033/asens.1667. Retrieved 6 May 2017.
  9. ^ Milnor, John W (1999). "Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account". arXiv:math/9905169.
  10. ^ Inou, Hiroyuki; Mukherjee, Sabyasachi (2015). "Non-landing parameter rays of the multicorns". Inventiones Mathematicae. 204 (3): 869–893. arXiv:1406.3428. Bibcode:2016InMat.204..869I. doi:10.1007/s00222-015-0627-3. S2CID 119633694.