Click on the image name see the animation.
These pictures are visualisations of valleys, inlets, clouds, lakes and precipices. The border points are incredibly hard to compute with the iteration z<-z^2+c taken hundreds of times. Border points are found by successive bisection of a line which has one end in the Mandelbrot set and one point outside. Bisection occurs 52 times, because this is bit-size of the mantissa of a double precision number. The viewport decreases in size as a geometric series. |
The Mandelbrot set is sometimes called the 'bug'. Computer rotating algorithms show this. When the image is rotated the bug appears to be sniffing or eating something. As it happens the proboscis is longest when it points along a line with many rational numbers on it. See www.linas.org for a more detailed discussion on the distribution of rationals on the unit interval / circle.
These .gif animations are made from 'affine' transformations of the Mandelbrot Set. An affine transformation consists of a translation and a rotation. Written in complex number notation this is essentially T(z) = a + bz, where a and b are complex numbers. b itself can be written as b = R (cos (t)+ i sin(t)) where t is a rotation. Each frame of the animation uses the same displacement (a), while R falls off as a geometric series R=k^n with |k|<1 while the phase angle t changes at a constant rate. The thumbnail images show a portion of the Mandelbrot set and its boundary at 10^-14 of its original size.