Newton fractal

Each pixel is a complex starting point for Newton’s method, z → z − p(z)/p′(z). Its colour identifies the root it approaches; darker shades need more iterations. Slate pixels have not converged within the chosen budget or meet an undefined step.

• Render explicitly. Click the image to inspect a starting point. Use View coordinates to change the centre and scale; both axes have equal units.
• Custom coefficients are ordered a₀, a₁, …, aₙ: −1 0 1 means z²−1. Real decimal or scientific notation; commas separate coefficients, not decimal places. Degree 1–64.
• (z−1)³ has one distinct root with multiplicity three. z³−2z+2 sends 0 to 1 and back; more iterations do not make this cycle a root.
• Root locations use numerical arithmetic after exact decimal removal of repeated factors. Close or ill-conditioned roots can exceed the available precision; an incomplete or inseparable target set produces an error rather than a misleading plot.
• 1–512 iterations; centre and half-width up to 10⁶, minimum half-width 10⁻⁶. PNG saves the current computed pixels with polynomial, view and iteration parameters.
• After the first Render, changing inputs updates the image automatically. Ctrl/Meta+wheel zooms around the centre; drag with the mouse to move. Reset returns to the entered view coordinates. The temporary mouse view is not saved. Resizing recomputes the raster with equal axis units; large images are limited to 2,048 pixels on the longer side and about 1.6 million pixels.