![]() ![]() He reasoned that the length of a coastline depends on the length of the measurement tool. Lewis Fry Richardson was an English mathematician in the early 20th century studying the length of the English coastline. Local dynamics near rationally indifferent fixed point/cycle ( parabolic ).One of the earliest applications of fractals came about well before the term was even used.Local dynamics near indifferent fixed point/cycle.Basin of attraction of attracting periodic/fixed point - Koenigs coordinate.Basilica Julia set ( c = -1 is a center of period 2 component).Circle Julia set ( c = 0 is a center of period 1 component).Basin of attraction of superattracting periodic/fixed point - Boettchers coordinate, c is a center of period n component of Mandelbrot set.Orbit portraits and lamination of dynamical plane.exterior of all Julia sets = basin of attraction of superattracting fixed point (infinity).Feigenbaum Julia sets ( c is Generalized Feigenbaum point: the limit of the period-q cascade of bifurcations and landing points of parameter ray or rays with irrational angles ).Misiurewicz Julia sets (c is a Misiurewicz point ).dendrits ( Julia set is connected and locally connected ).Cremer Julia sets -a non-linearizable irrationaly indifferent fixed point.connected ( c is inside Mandelbrot set ).disconnected ( c is outside of Mandelbrot set ).Elliptic Julia set: Siegel disc - a linearizable irrationaly indifferent fixed point.Examples : Airplane Julia set, Douady's Rabbit, Basillica. superattracting : filled Julia set have superattracting cycle( c is in the center of hyperbolic component ).attracting : filled Julia set have attracting cycle ( c is inside hyperbolic component ).with an non-empty interior ( connected ).Polynomials Chebyshev polynomials Complex quadratic polynomials non-complex-analytic formulas (like Mandelbar and Burning Ship).complex-analytic formulas (like Mandelbrot set and Julia set).Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection.Period doubling cascade and the Myrberg-Feigenbaum point in the 1/2 family.roots, islands and Douady tuning - How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is not accesible from main cardioid ( M0) by a finite number of boundary crossing ?.principle Misiurewicz points of the wake k/r, tuning: How to find angles of external rays landing on the principle Misurewicz point of the wake ?.wake : How to find angles of external rays that land on the root point of the wake ( = angles of the wake)?.Parameter plane: combinatorial algorithms.Discrete Velocity of non-attracting Basins and Petals by Chris King.Triangle Inequality Average Coloring = TIA and curvature average algorithm ( CAA).Average Colorings "are a family of coloring functions that use the decimal part of the smooth iteration count to interpolate between average sums." Jussi Harkonen.Boettcher map, complex potential and external ray.Maping component to the unit disk ( Riemann map ):.external ray ( parameter and dynamic) trace.unroll a closed curve and then stretch out into an infinite strip.Inverse iteration method ( IIM) for drawing:.Decomposition of the target set: Binary Decomposition Method ( BDM) which in parabolic case gives: zeros of Qn or parabolic checkerboard ( chessboard).escape and attracting time for (level sets method (LSM), level curves method (LCM).How to construct map with desired properities ?Īlgorithms: Algorithms, methods of drawing/computing or representation finctions ( for space transformations see here).periodic points of complex quadratic map.Iterations : forward and backward ( inverse ) and critical orbit.Programming computer graphic: files, plane, transformations, curves.Mathematics for computer graphic: numbers, sequences, functions, numerical methods, fields.2.2.1.1.2 Complex quadratic polynomials. ![]()
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