Newton Fractal Generator
Explore Newton's method visually. Pick a polynomial, adjust iterations and tolerance, then render a real, pixel-computed Newton fractal you can zoom, pan, and download.
Drag to pan Β· Scroll or pinch to zoom Β· Settings update live results, no simulated data.
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Why Use This Newton Fractal Maker
Built for students, educators, and generative-art enthusiasts who want real, adjustable output.
Live Computation
Every pixel runs Newton's iteration in real time in your browser β no pre-rendered images or mock previews.
Adjustable Parameters
Change the polynomial, iteration depth, tolerance, and zoom to see how convergence basins shift.
Custom Color Palettes
Switch between vivid, sunset, and monochrome schemes to highlight different root basins.
Pan & Zoom
Drag to explore the complex plane and scroll to zoom into fractal boundaries in fine detail.
One-Click Export
Copy the rendered image to your clipboard or download it as a high-resolution PNG.
Mobile Friendly
Fully responsive layout that works cleanly on phones, tablets, and desktops alike.
How the Newton Fractal Generator Works
Four steps from polynomial to rendered image.
Pick a complex polynomial f(z), such as zΒ³ β 1, whose roots define the basins of attraction.
Each pixel maps to a starting complex number zβ on the plane you're viewing.
Newton's formula z = z β f(z)/fβ²(z) repeats until the value converges within your tolerance.
The pixel is colored by the root it reached, shaded by how many iterations that took.
What Is a Newton Fractal?
A Newton fractal is one of the more approachable doors into complex dynamics because it starts from something familiar: Newton's method for finding roots. Most people first meet Newton's method in a calculus class as a way to approximate where a real function crosses zero. Extend that same idea into the complex plane, apply it to every point on a grid instead of a single starting guess, and color each point by which root it eventually settles on. That's the entire recipe behind a Newton fractal β and the reason the result looks nothing like the tidy convergence you'd expect from a first-year math exercise.
The polynomial zΒ³ β 1 = 0 is the classic starting example, since it has exactly three roots evenly spaced around the unit circle. Color each root a different hue and you'd expect three clean pie-slice regions. Instead, the boundaries between those regions twist into infinitely repeating, self-similar detail. Zoom into any edge and you'll find smaller versions of the same three-color pattern repeating forever β a direct consequence of how sensitive Newton's iteration is near a boundary, where a tiny nudge in the starting point can send the sequence toward a completely different root.
This generator lets you experiment with that sensitivity directly. Switch the polynomial degree to see how adding more roots changes the basin geometry, tighten the convergence tolerance to sharpen edge detail, or increase the iteration cap to resolve slower-converging regions near the boundaries. Because every frame is computed live from the polynomial you select rather than loaded from a static image, changing any single setting β palette, zoom, tolerance β produces a genuinely new render, not a filtered copy of the same picture.
Newton fractals show up well beyond recreational math. Numerical analysts use basin-of-attraction visualizations like this one to reason about where root-finding algorithms are reliable and where they're chaotic, which matters for anything from engineering simulations to control systems. For most visitors, though, the appeal is simpler: it's a rare case where a genuinely useful piece of numerical mathematics also happens to be beautiful to look at. Use the controls above to generate your own view, then download or share whatever you find.
Frequently Asked Questions
What is a Newton fractal and how is it different from the Mandelbrot set?
A Newton fractal is generated by applying Newton's root-finding method to a complex polynomial from every point on the plane and coloring each point by which root it converges to. Unlike the Mandelbrot set, which tracks escape to infinity, a Newton fractal tracks convergence toward one of several fixed roots, producing basin-of-attraction patterns with intricate boundaries.
How do I generate a Newton fractal for the equation zΒ³ β 1 = 0?
Choose the "zΒ³ β 1" option in the tool above. The generator computes the three complex cube roots of unity, then iterates z = z β f(z)/fβ²(z) for every pixel until convergence, coloring each point by the root it lands on and shading it by how many iterations were required.
Why does the Newton fractal boundary look so chaotic even though the roots are simple?
Even with only a few simple roots, the boundaries between their basins of attraction are fractal because Newton's method is extremely sensitive to starting position near those boundaries. Tiny changes in a starting point can send the iteration to a different root entirely, which produces the self-similar, infinitely detailed edges Newton fractals are known for.
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