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Multibrot Set
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set. : z \mapsto z^d + c . \, where ''d'' ≥ 2. The exponent ''d'' may be further generalized to negative and fractional values. Examples The case of : d = 2\, is the classic Mandelbrot set from which the name is derived. The sets for other values of ''d'' also show fractal images when they are plotted on the complex plane. Each of the examples of various powers ''d'' shown below is plotted to the same scale. Values of ''c'' belonging to the set are black. Values of ''c'' that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show as contours, depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multibrot
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general Monic polynomial, monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set. : z \mapsto z^d + c . \, where ''d'' ≥ 2. The exponent ''d'' may be further generalized to negative and fractional values. Examples The case of : d = 2\, is the classic Mandelbrot set from which the name is derived. The sets for other values of ''d'' also show fractal images when they are plotted on the complex plane. Each of the examples of various powers ''d'' shown below is plotted to the same scale. Values of ''c'' belonging to the set are black. Values of ''c'' that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypocycloid
In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid created by rolling a circle on a line. Properties If the smaller circle has radius , and the larger circle has radius , then the parametric equations for the curve can be given by either: :\begin & x (\theta) = (R - r) \cos \theta + r \cos \left(\frac \theta \right) \\ & y (\theta) = (R - r) \sin \theta - r \sin \left( \frac \theta \right) \end or: :\begin & x (\theta) = r (k - 1) \cos \theta + r \cos \left( (k - 1) \theta \right) \\ & y (\theta) = r (k - 1) \sin \theta - r \sin \left( (k - 1) \theta \right) \end If is an integer, then the curve is closed, and has Cusp (singularity), cusps (i.e., sharp corners, where the curve is not Differentiable function, differentiable). Specially for the curve is a straight line and the circles are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractals
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 illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called Affine geometry, affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they Scaling (geometry), scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Dynamics
Complex dynamics is the study of dynamical systems defined by Iterated function, iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem **Poincaré metric **Schwarz lemma **Riemann mapping theorem **Carathéodory's theorem (conformal mapping) **Böttcher's equation *Combinatorics, Combinatorial ** Hubbard trees ** Spider algorithm ** Tuning **Lamination (topology), Laminations **Cantor function, Devil's Staircase algorithm (Cantor function) **Orbit portraits **Jean-Christophe Yoccoz, Yoccoz puzzles Parts * Holomorphic dynamics (dynamics of holomorphic functions) ** in one complex variable ** in several complex variables * Conformal dynamics unites holomorphic dynamics in one complex variable with differentiable dynamics in one real variable. See also *Arithmetic dynamics *Chaos theory *Complex analysis *Complex quadratic polynomial *Fatou set *Infinite co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multibrot Lupanov Power-2 Q1
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set. : z \mapsto z^d + c . \, where ''d'' ≥ 2. The exponent ''d'' may be further generalized to negative and fractional values. Examples The case of : d = 2\, is the classic Mandelbrot set from which the name is derived. The sets for other values of ''d'' also show fractal images when they are plotted on the complex plane. Each of the examples of various powers ''d'' shown below is plotted to the same scale. Values of ''c'' belonging to the set are black. Values of ''c'' that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show as contours, de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multibrot Rendered With Exponent On Veritical Axis In A 45-Degree Plane
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set. : z \mapsto z^d + c . \, where ''d'' ≥ 2. The exponent ''d'' may be further generalized to negative and fractional values. Examples The case of : d = 2\, is the classic Mandelbrot set from which the name is derived. The sets for other values of ''d'' also show fractal images when they are plotted on the complex plane. Each of the examples of various powers ''d'' shown below is plotted to the same scale. Values of ''c'' belonging to the set are black. Values of ''c'' that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show as contours, de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multibrot Rendered On Imaginary And Exponent
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set. : z \mapsto z^d + c . \, where ''d'' ≥ 2. The exponent ''d'' may be further generalized to negative and fractional values. Examples The case of : d = 2\, is the classic Mandelbrot set from which the name is derived. The sets for other values of ''d'' also show fractal images when they are plotted on the complex plane. Each of the examples of various powers ''d'' shown below is plotted to the same scale. Values of ''c'' belonging to the set are black. Values of ''c'' that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show as contours, de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multibrot Rendered On Real Axis And Exponent
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set. : z \mapsto z^d + c . \, where ''d'' ≥ 2. The exponent ''d'' may be further generalized to negative and fractional values. Examples The case of : d = 2\, is the classic Mandelbrot set from which the name is derived. The sets for other values of ''d'' also show fractal images when they are plotted on the complex plane. Each of the examples of various powers ''d'' shown below is plotted to the same scale. Values of ''c'' belonging to the set are black. Values of ''c'' that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show as contours, de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
