Newtonfraktálok

Newtonfraktálok, also known as Newton's fractals, are a type of fractal generated by applying Newton's method to find the roots of a polynomial. Newton's method is an iterative algorithm used to approximate the roots of a real-valued function. When applied to complex polynomials, the behavior of the iteration can be highly sensitive to the initial starting point.

The process involves choosing an initial complex number and repeatedly applying the Newton-Raphson formula: $z_{n+1} = z_n - f(z_n) / f'(z_n)$, where $f(z)$ is the polynomial and $f'(z)$ is its derivative. For each root of the polynomial, there is a region in the complex plane, called a basin of attraction, such that any initial point within this region will converge to that specific root. The boundaries between these basins of attraction form the fractal structure.

The visual representation of Newton's fractals typically involves coloring points in the complex plane based on which root they converge to and how quickly they converge. Different colors are assigned to different roots, and variations in shade or hue can indicate the number of iterations required for convergence. The resulting images often display intricate and self-similar patterns.

The complexity and beauty of Newton's fractals arise from the delicate balance between the stable convergence to roots and the chaotic behavior near the basin boundaries. They serve as a fascinating example of how simple mathematical rules can lead to complex emergent properties, and they have found applications in fields such as chaos theory and computer graphics.