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 \( z\mapsto {\bar {z}}^{2}+c \) instead of \( z\mapsto z^{2}+c \) used for the Mandelbrot set. It was introduced by W. D. Crowe, R. Hasson, P. J. Rippon, and P. E. D. Strain-Clark.[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.
A tricorn, created on a computer in C.
Formal definition
The tricorn T is defined by a family of quadratic antiholomorphic polynomials
\( f_c:\mathbb C\to\mathbb C \)
given by
\( f_c: z\mapsto \bar{z}^2 + c, \)
where c is a complex parameter. For each c, one looks at the forward orbit
\( (0, f_c(0), f_c(f_c(0)), f_c(f_c(f_c(0))), \ldots) \)
of the critical point \( {\displaystyle 0} \) of the antiholomorphic polynomial \( p_{c} \). In analogy with the Mandelbrot set, the tricorn is defined as the set of all parameters 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 for which the Julia set \( J(f_{c}) \) is connected.
Multicorns with the power going from 2 to 5
The higher degree analogues of the tricorn are known as the multicorns.[3] These are the connectedness loci of the family of antiholomorphic polynomials \( f_c: z\mapsto \bar{z}^d + c. \)
Basic properties
The tricorn is compact, and connected.[4] In fact, Nakane modified Douady and Hubbard's proof of the connectedness of the Mandelbrot set to construct a dynamically defined real-analytic diffeomorphism from the exterior of the tricorn onto the exterior of the closed unit disc in the complex plane. One can define external parameter rays of the tricorn as the inverse images of radial lines under this diffeomorphism.
Every hyperbolic component of the tricorn is simply connected.[3]
The boundary of every hyperbolic component of odd period of the tricorn contains real-analytic arcs consisting of quasi-conformally equivalent but conformally distinct parabolic parameters.[5][6] Such an arc is called a parabolic arc of the tricorn. This is in stark contrast with the corresponding situation for the Mandelbrot set, where parabolic parameters of a given period are known to be isolated.
The boundary of every odd period hyperbolic component consists only of parabolic parameters. More precisely, the boundary of every hyperbolic component of odd period of the tricorn is a simple closed curve consisting of exactly three parabolic cusp points as well as three parabolic arcs, each connecting two parabolic cusps.[6]
Every parabolic arc of period k has, at both ends, an interval of positive length across which bifurcation from a hyperbolic component of odd period k to a hyperbolic component of period 2k occurs.
Implementation
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.
It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter.[7][8] 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.[9]
References
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.
Milnor, John (1 January 1992). "Remarks on iterated cubic maps". Experimental Mathematics. 1 (1): 5–24. Retrieved 6 May 2017 – via Project Euclid.
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 10.1.1.32.4046. doi:10.1142/S0218127403008259.
Nakane, Shizuo (1 June 1993). "Connectedness of the tricorn". Ergodic Theory and Dynamical Systems. 13 (2): 349–356. doi:10.1017/S0143385700007409. Retrieved 6 May 2017.
"Multicorns are not path connected" (PDF). Math.cornell.edu. Retrieved 2017-05-06.
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.
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.
Milnor, John W (1999). "Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account". arXiv:math/9905169.
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.
Fractals
Characteristics
Fractal dimensions
Assouad Box-counting Correlation Hausdorff Packing Topological Recursion Self-similarity
Partial view of the Mandelbrot set.
An image of a fern which exhibits affine self-similarity.
6th iteration of the Sierpinski carpet
Iterated function
system
Barnsley fern Cantor set Koch snowflake Menger sponge Sierpinski carpet Sierpinski triangle Space-filling curve
Blancmange curve De Rham curve
Minkowski Dragon curve Hilbert curve Koch curve Lévy C curve Moore curve Peano curve Sierpiński curve T-square n-flake
Strange attractor
Multifractal system
L-system
Fractal canopy Space-filling curve
H tree
Escape-time
fractals
Burning Ship fractal Julia set
Filled Newton fractal Lyapunov fractal Mandelbrot set
Misiurewicz point Newton fractal Tricorn Mandelbox Mandelbulb
Rendering techniques
Buddhabrot Orbit trap Pickover stalk
Random fractals
Brownian motion
Brownian tree Brownian motor Fractal landscape Lévy flight Percolation theory Self-avoiding walk
People
Georg Cantor Felix Hausdorff Gaston Julia Helge von Koch Paul Lévy Aleksandr Lyapunov Benoit Mandelbrot Lewis Fry Richardson Wacław Sierpiński
Other
"How Long Is the Coast of Britain?"
Coastline paradox List of fractals by Hausdorff dimension The Beauty of Fractals (1986 book) Fractal art Chaos: Making a New Science (1987 book) The Fractal Geometry of Nature (1982 book) Kaleidoscope Chaos theory
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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