The »Newton fractal« (flower) for a given complex function is constructed by coloring each complex number (or rather each point on a finite grid). We iteratively apply Newton’s method to construct the sequence . If we detect that the sequence converges after terms (that is, if the difference between -th and previous term is smaller than a fixed convergence threshold constant), we will paint the pixel corresponding to according to , and . If the sequence does not converge after a fixed number of iterations, we deem it divergent and paint the pixel black. Note that if the sequence does converge, it is to some root of function .

The renderer implements two generalizations of »Newton fractals«.

Nova fractals

The Newton method step in the above is substituted by a generalized

for some complex constants .


A new layer of complexity can be implanted into the fractal in the following way. Repeated a fixed amount of times, when we detect convergence at the term , we simply restart the sequence by setting and continue with (restarting simultaneously the counter of the maximal number of iterations). The effects can be seen below, for repetitions set to 1 (usual Newton fractal), 2 and 5 respectively.

Function and its derivative

Must be a JavaScript expression. Available complex functions: .add(), .sub(), .mul(), .div(), .pow(), .sin(), .cos(), .abs(), .tan(); Examples:

Complex number must be entered as “new Complex(real, imag)”.
f(z) =
f'(z) =

Parameters for the Newton’s method

Sets parameters a, c for the modified Newton’s method. Increase repetitions for added complexity.
a =
c =
Maximum iterations
Convergence threshold =

Color scheme

Sets the color for a point as a function of n, the number of iterations until convergence. Needs to be a JavaScrip expression.
Red(n) =
Green(n) =
Blue(n) =

Display window and resolution

Sets the part of complex plane to be displayed and the number of pixels. More pixels will cause slower rendering.
Controls: left-click on canvas to zoom, right-click to unzoom. Click on the below arrows for shift.

Click on image to render the fractal.