Tool

Newton fractal playground

Pick a polynomial. Each pixel runs Newton’s iteration and is colored by which root it converges to, with brightness encoding the iteration count.

Max iterations: 60

polynomial: z³ − 1

roots: 3

render: 0 ms

Iteration rule

p(z) = z³ − 1
p'(z) = 3z²

Newton's iteration:
  z_{n+1} = z_n − p(z_n) / p'(z_n)

Roots of p (basin centres):
  root 1: 1.0000 + 0.0000 i
  root 2: -0.5000 + 0.8660 i
  root 3: -0.5000 + -0.8660 i

Each pixel is a starting z₀. Color = which root the orbit
converges to. Brightness = 1 − n/maxIter.

The polynomial z³ − 2z + 2is famous for an open disk near the origin where Newton’s method never converges to any root.

Background at /fractals/newton.

FAQ