If F(x) is a
(real) function, then the tangent of F at a point x0 is: y-F(x0)
= F'(x0) . (x-x0)Intersection of the tangent with the x-axis gives a point x1 = x0 - F(x0)/F'(x0) = f(x0)
The formula for this type of fractals is derived from Newton's algorithm to calculate the roots of a polynome:
If F(x) is a
(real) function, then the tangent of F at a point x0 is: y-F(x0)
= F'(x0) . (x-x0)
Intersection of the tangent with the x-axis gives a point x1
= x0 - F(x0)/F'(x0) = f(x0)
Newton's algorithm: the series x1 = f(x0), x2 = f(x1),... tends to a zero of F.
Initially, only for real functions, Cayley extended the result in 1879
for complex functions.
Let's take the complex function F(z)=z3 - 1, which has 3 roots: 1, j and j_ (the conjugate of j).
We will colour
a starting point x0 in red if the series converges to 1, in
green if it converges to j and in blue if it converges to j_. These three
regions are called the basins of attraction of the roots.
The image was drawn with fractint, type "newtbasin".
There is an anecdote about this image (The Beauty of Fractals, page 19):
A planet is divided between three powers. In order to avoid conflicts on the borderline,
wherever two regions are about to form a boundary, the third establishes a chain of
outposts...
The next figure shows the border of the three basins: ðA(1) = ðA(j) = ðA(j_).
In itself, this
figure is less spectacular than the Mandelbrot set: by zooming in
the Newton Fractal, there won't appear any spirals or little Mandelbrot
sets, but over and over the same crab-like structures (as they are called
in the book The Beauty of Fractals by Peitgen & Richter).
However, even if it is not obvious, there are mathematical relationships between
Mandelbrot and Newton fractals!
But this is enough theory for now; below are some pictures of Newton
Fractals. Depending on the choosen color palette, the image can take very
different aspects: it may look like a crab, a spider web, a plasma cell
or a flower. Watch out for the tunnel effect!
|
Degree 6: |
Degree 8: |
|
 |
|
Note the tunnel-effect by activating fractint's option color-cycling!
Details of degree 3 and degree 5:
