Explore some mathematical ideas without doing the maths.
Interesting patterns can be made by choosing the colour for each point by some rule.
The simplest ones are based on distances.
The distance from a point - circles
Adding the distances from two points - ellipses
Multiplying the distances, and weighting one of the points.
This one might look similar, but it is calculated
using the rule that generates the Mandelbrot set. See below.
Inside the Mandelbrot Set
Most pictures of the Mandelbrot Set show the set in black.
The interesting colours are in the area outside the set.
First steps, inspired by an animation shared by jerekt.
Orbit periods.
Limit set periods.
A Filled-In Julia set, associated with the Mandelbrot set..
See below.
Explore Julia sets - and their connection with the Mandelbrot set.
The Filled-In Julia set with C = (0.25,0.55)
This set has C = (-0.54, 0.6) which is in part of the Mandelbrot that has a
limit set period of 10.
For C = (0, -1) the Julia set has no interior. It is thread-like.
Here C = (-0.752, -0.04) which is not inside the Mandelbrot set.
The Julia set is a collection of disconnected points.
C = (-0.752, -0.04) as in the previous example.
We have zoomed in on part of the set.
Cubic Mandelbrot and Julia sets
Julia sets can be created for any polynomial. So, taking it one step at a time, we move from
quadratic polynomials (where the highest power of Z is Z2)
to cubic polynomials (where the highest power of Z is Z3)
The cubic Mandelbrot set for Z3+Z
The Mandelbrot set for Z3+1.6875Z showing the regions
generated by the two critical points
The Mandelbrot set for Z3 coloured according
to the limit set periods.
Here we have zoomed in on the middle of the cubic Mandelbrot set
for Z3+Z showing the limit set periods.
What next?
Here are some other patterns. These are generated by more complicated rules.
