The Newton fractal is generated by applying Newton's method, a numerical algorithm for finding the roots (or zeroes) of a function, to a polynomial in the complex plane.
Newton's method is an iterative process. To find a root of a function $p(z)$, we start with an initial guess $z_0$ and repeatedly apply the formula:
z_{n+1} = z_n - \frac{p(z_n)}{p'(z_n)}
where $p'(z)$ is the derivative of $p(z)$. For a starting point $z_0$, this sequence will hopefully converge to one of the roots of the polynomial.
In the context of complex numbers, a polynomial can be defined by its roots $r_1, r_2, ..., r_k$. For such a polynomial, the Newton's method formula can be expressed without needing to compute the derivative directly:
z_{n+1} = z_n - \frac{1}{\sum_{i=1}^k \frac{1}{z_n - r_i}}
This is the formula used to generate this fractal.
The complex plane is colored based on which root each starting point $z_0$ converges to. The boundaries between these regions of convergence are fractal, creating intricate and beautiful patterns. Points that converge to the same root are typically given the same color, with the shade determined by how quickly they converge.
To know more, check the fantastic short video and its full version made by 3Blue1Brown.
Parameters
GUI Parameters
- Iterations: This integer value determines the maximum number of times the formula is applied for each point.
Complex Parameters (Control Points)
- Roots: These are the roots of the polynomial $p(z)$. You can add, remove, and drag these points on the complex plane. The number and position of the roots completely change the structure of the fractal. The fractal is generated for a polynomial with these specified roots.