Newton's Fractal

Guide

Each pixel starts a calculation: repeatedly apply Newton’s method to find a root of a cubic polynomial. Its color shows which of the three roots it approaches. Move the third root to change the picture.

Try this: right-click in the parameter plane and move the mouse near the two fixed roots, or double-tap and drag on mobile. Increase the iterations to see the boundaries sharpen.

Controls

• Desktop: right-click a plot to switch between pan / zoom and point placement, independently for each plot. Drag to pan; in point placement mode the point follows the mouse. Scroll to zoom in either mode without moving the point.
• Mobile: drag with one finger to pan. To move a point, double-tap, keep the second tap held, and drag; lift your finger to finish. Use two fingers to pan and pinch to zoom. Adding a second finger cancels point placement, so pinching cannot leave the root or parameter accidentally moved.
• Orbit start: enable Show orbit, turn off its tracking/critical-point lock, and enable Move orbit start. Point placement in the dynamical plane then moves the orbit start instead of the parameter or root.
• Fullscreen: use Fullscreen to enter; use the browser’s Back gesture on mobile or Esc on desktop to leave. There is no exit button covering the plot.
• Keyboard: Tab to a plot, press M to switch modes, use the arrow keys to pan or move the point, + / − to zoom, and 0 to reset the view. Reset view also restores the initial pan and zoom.

Reading the picture

The dynamical plane colors starting points by the root they approach. You can choose the three root colors above the plots. Dark or blended regions indicate slower convergence or a calculation that has not settled within the chosen number of iterations.

The parameter plane tries a different third root at each pixel and follows the orbit of the roots’ centroid. Some choices lead to attracting cycles instead of convergence to a root.

• Black dots: the two fixed roots at −0.4 and +0.4.
• Gray dot: the third root, which you can move.
• Teal dot: the centroid of the three roots. For a cubic with distinct roots this is an additional critical point of its Newton map, except in degenerate cases.
• White path: the steps from one starting point when Show orbit is enabled. Choose Move orbit start to set that point, or Track centroid to follow the centroid as you move the third root.

The plotted calculation is N(z) = z − p(z)/p′(z), with p(z) = (z + 0.4)(z − 0.4)(z − r) and movable root r.