The Mandelbrot is an image representing the behaviour of the series
z The above image is centered on (-0.75,0.0) and extends approximately 1.3 horizontally on either end, the left most point of the Mandelbrot is at (-2,0). Zoom 1 -tendrils (lightning)
The Mandelbrot set is considered to be a single connected set. The following zoom sequence concentrates on one of the tendrils that runs from the top portion of the set. The zoom factor increases by a factor of 5 going from left to right, the zoom is about the center of each image. The center of this image on the complex plane is (-0.170337,-1.06506).
The center of this sequence is (0.42884,-0.231345).
The center of this sequence is (-1.62917,-0.0203968)
The center of this sequence is (-0.761574,-0.0847596)
The following is a simple snippet of code that can be used to calculate the Mandelbrot by the somewhat inefficient brute force method. /* Iterate a single point x0 + i y0 of the Mandelbrot for "imax" iterations. Return the number of iterations at which the point escapes. Return "imax" if the point doesn't escape. */ int Iterate(double x0,double y0,long imax) { double x=0,y=0,xnew,ynew; int i; for (i=0;i<imax;i++) { xnew = x * x - y * y + x0; ynew = 2 * x * y + y0; if (xnew*xnew + ynew*ynew > 4) return(i); x = xnew; y = ynew; } return(imax); }
The Mandelbrot doesn't exist outside a circle of radius 2 on the
complex plane, this gives scope for turning it inside out by mapping
z
A relatively unexplored variation is to shade the plane based upon when the series decreases three times in succession. The Mandelbrot grow "wings" and discs at the peripheral points.
x = -1.04180483110546, y = 0.346342664848392
x = -0.751095959125087, y = -0.116817186889238
x = -0.812223315621338, y = -0.185453926110785 |