
In
mathematics, the tricorn, sometimes called the Mandelbar set, is a
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as il ...
defined in a similar way to the
Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
, 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.
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Un ...
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
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically s ...
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
In mathematics, antiholomorphic functions (also called antianalytic functionsEncyclopedia of Mathematics, Springer and The European Mathematical Society, https://encyclopediaofmath.org/wiki/Anti-holomorphic_function, As of 11 September 2020, This ...
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s
:
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 Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
, 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
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values w ...
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
.
Basic properties
* The tricorn is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, and
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
. In fact, Nakane modified
Douady and
Hubbard Hubbard may refer to:
Places Canada
*Hubbard, Saskatchewan
*Hubbards, Nova Scotia
Canada/United States
* Mount Hubbard, a mountain on the Alaska/Yukon border
*Hubbard Glacier, a large freshwater glacier in Alaska and Yukon
Greenland
* Hubbard Gl ...
's proof of the connectedness of the
Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
to construct a dynamically defined
real-analytic
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given tw ...
from the exterior of the tricorn onto the exterior of the
closed unit disc
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose ...
in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
. One can define
external parameter rays of the tricorn as the inverse images of
radial line
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
s under this diffeomorphism.
* Every hyperbolic component of the tricorn is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
.
* 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.
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.
* 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.
Image gallery of various zooms
Much like the
Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
, 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
value where
. The images are not stretched or altered, that is how they look on magnification.
Tricornx1.png, Starting zoom. The Tricorn as a whole.
Tricornx6.png, 6 times the original zoom. Many "extrusions" or "seahorses" along the upper "head" are being converged onto the "neck". Aptly named "Seahorse convergence valley".
Tricornx20.png, 20 times the original zoom. Closer look at one side of the "valley". The "seahorses" begin to take shape, they look to be distorted from the original "seahorses" along the Mandelbrots boundary.
Tricornx40.png, 40 times the original zoom. The "seahorse" can be fully seen now.
Implementation
The below pseudocode implementation hardcodes the complex operations for Z. Consider implementing
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
operations to allow for more dynamic and reusable code.
For each pixel (x, y) on the screen, do:
Further topological properties
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.
References
{{DEFAULTSORT:Tricorn (Mathematics)
Fractals