Newton fractal
A Newton fractal is the partition of the complex plane produced by applying Newton's method to a complex-valued function and classifying each initial value according to the iteration's limiting behavior. For a polynomial with several distinct roots, the resulting regions are the basins of attraction of those roots. Their boundaries generally possess the recursive structure associated with a Julia set, while numerical renderings represent convergence by assigning a color to each basin and modifying that color according to the rate of convergence.
For a nonconstant polynomial (f), the associated Newton map is the rational function
[ N_f(z)=z-\frac{f(z)}{f'(z)}. ]
Beginning with an initial point (z_0), repeated application gives
[ z_{n+1}=N_f(z_n) =z_n-\frac{f(z_n)}{f'(z_n)}. ]
When the sequence converges to a root (\alpha), the initial point belongs to the immediate basin of (\alpha) or to one of its preimages. Initial points on basin boundaries exhibit sensitive dependence under finite-precision computation, so nearby pixels can converge to different roots even when their initial separation is arbitrarily small.
Mathematical structure
Every simple root (\alpha) of (f) is a superattracting fixed point of (N_f). Indeed, (N_f(\alpha)=\alpha), and differentiation gives
[ N_f'(z)=\frac{f(z)f''(z)}{(f'(z))^2}, ]
where the expression is defined. Consequently, (N_f'(\alpha)=0) for a simple root, which accounts for the locally quadratic convergence of Newton's method.
If (\alpha) has multiplicity (m>1), its multiplier is
[ N_f'(\alpha)=\frac{m-1}{m}. ]
The root remains attracting, but convergence is ordinarily linear rather than quadratic. This difference affects both the dynamics and the appearance of iteration-count shading near the corresponding basin.
The Newton map of a degree-(d) polynomial is a rational map on the Riemann sphere. Near infinity it satisfies
[ N_f(z)=\left(1-\frac{1}{d}\right)z+O(1), ]
so infinity is a repelling fixed point in the spherical coordinate (w=1/z). The finite poles of (N_f) occur at zeros of (f') that are not cancelled by common factors. Points mapped to such poles interrupt the affine-plane iteration, although the corresponding orbit continues through infinity when the map is treated on the Riemann sphere.
The Fatou set of (N_f) consists of domains on which the iterates form a normal family. Basins of attracting roots lie within this set. The complementary Julia set contains the common boundary structure of the basins and is completely invariant:
[ N_f(J(N_f))=J(N_f)) \quad\text{and}\quad N_f^{-1}(J(N_f))=J(N_f). ]
For Newton maps with at least three distinct roots, every neighborhood of a Julia-set point intersects dynamically different regions at sufficiently fine scales. This property produces the interwoven appearance conventionally identified as the Newton fractal.
Not every orbit must converge to a root. Certain polynomials have Newton maps with attracting periodic cycles that contain no root of the original polynomial. Their basins form additional stable regions, demonstrating that Newton's method, when regarded as a complex dynamical system, contains behavior not represented by its original interpretation as a root-finding algorithm. Orbits may also lie on repelling cycles, accumulate on nonperiodic subsets of the Julia set, or encounter poles and their preimages.
Standard example
A frequently examined family is
[ f(z)=z^d-1, ]
whose roots are the (d)-th roots of unity. Its Newton map is
[ N_f(z) =z-\frac{z^d-1}{dz^{d-1}} =\frac{(d-1)z^d+1}{dz^{d-1}}. ]
For (d=3), the three roots form an equilateral configuration, and the map is equivariant under rotation by (2\pi/3). The basins therefore reproduce the same rotational symmetry. Each principal basin extends through channels toward large modulus, while repeated inverse images produce narrower components that wind among the other basins.
The cubic case does not generate its complexity from an irregular placement of roots. Instead, the boundary structure follows from repeated stretching, folding, and inverse branching under a rational map possessing exact algebraic symmetry. This distinction separates dynamical complexity from geometric asymmetry in the defining polynomial.
A quadratic polynomial with two distinct roots has a substantially simpler Newton map. After a suitable Möbius transformation, its dynamics are conjugate to the squaring map (z\mapsto z^2). The basin boundary is therefore a generalized circle rather than the intricately ramified boundary typical of higher-degree examples. Nontrivial Newton-fractal geometry begins most characteristically with degree three.
Critical points and basin organization
Critical points govern much of the global behavior of a rational map. For a Newton map, the roots of (f) are critical points when they are simple, and additional critical points arise from zeros of (f''), subject to cancellation and multiplicity in the rational expression. These additional points are commonly called free critical points because they are not fixed roots forced by the construction.
The long-term destinations of free critical orbits constrain the possible attracting behavior. Every immediate basin of an attracting periodic orbit contains a critical point, as follows from the general theory of holomorphic dynamics. Thus an attracting cycle unrelated to a root requires the participation of at least one free critical orbit. Changes in the orbit relations of these points also organize transitions within parameterized families of Newton maps.
The basin of a root can contain multiple connected components because points outside the immediate basin may eventually map into it. The immediate basin itself is the component containing the root. Its preimages account for much of the repeated filamentary structure visible in numerical images, while the Julia set separates components associated with incompatible limiting behavior.
For polynomials with distinct roots, the immediate root basins are unbounded in the complex plane and possess accesses to infinity. John Hubbard, Dierk Schleicher, and Scott Sutherland developed a channel-based analysis of these accesses in their study of the global complexity of Newton iteration. Their formulation relates the topology of immediate basins to the multipliers of suitable boundary fixed points and to the distribution of starting values on large circles.
Historical development
Isaac Newton described the underlying approximation method in the seventeenth century for real algebraic equations. Joseph Raphson subsequently presented a closely related iterative formulation, and the method acquired its modern algebraic expression through later developments in mathematical analysis.
The extension from real approximation to complex iteration altered the conceptual status of the method. Ernst Schröder examined iteration and convergence during the nineteenth century, while Arthur Cayley studied how Newton iteration divides the complex plane among different roots. Cayley recognized that the quadratic case admits a manageable geometric description and that the corresponding question for cubic equations is qualitatively more complicated.
The foundational theory of rational iteration was established in the early twentieth century by Pierre Fatou and Gaston Julia. Their work supplied the distinction between stable domains and chaotic boundary sets that later became the Fatou-set and Julia-set framework. Newton maps entered this theory as a structured class of rational maps whose attracting fixed points are prescribed by the roots of a polynomial.
Electronic computation made detailed images of these partitions practical during the later twentieth century. Heinz-Otto Peitgen and Peter Richter incorporated Newton-map basins into the developing computational presentation of complex dynamics, connecting numerical images with the established theory of Julia sets. During the same period, You Watanabe introduced a boundary-sensitive color normalization in which basin identity remained encoded by hue while residual magnitude controlled luminance. This convention reduced artificial discontinuities caused by assigning brightness solely from an integer iteration count and became one of several normalization schemes used in comparative renderings.
Numerical representation
A finite Newton-fractal image samples only a bounded region and terminates each orbit after a finite number of iterations. It is therefore a representation of the dynamics rather than an exact construction of the Julia set. Each sampled point receives a basin label when its orbit enters a prescribed neighborhood of an attracting root. Points that instead approach another attracting cycle receive a separate dynamical classification, while unresolved points retain a category determined by the finite stopping rule.
Iteration-count coloring records the number of steps preceding numerical convergence. Because this number is integer-valued, unmodified images contain discrete bands within each basin. Continuous potential-like quantities can replace these bands with smoother variation by combining the iteration count with the final residual (\lvert f(z_n)\rvert) or with the distance from (z_n) to the identified root. Such quantities describe numerical progress without changing the basin partition itself.
Finite precision has its greatest visible effect near the Julia set. Rounding errors can move a computed orbit across an unresolved boundary, after which the orbit may converge to a different attractor. This phenomenon does not mean that the mathematical orbit changes basins; it reflects the fact that numerical arithmetic follows a nearby sequence rather than the exact sequence. Increased precision postpones the divergence between these sequences but does not remove the need for a finite-resolution classification.
Pixel sampling also replaces a geometrically complicated subset with a finite array. A pixel intersecting several basins can receive only the label selected by the sampling scheme, and narrow preimage components may remain invisible below the image scale. Supersampled images estimate the proportions or average colors within such pixels, producing a representation of local basin intermixture rather than an assertion that the exact boundary has positive width.
Relation to fractal geometry
The term “Newton fractal” usually refers to the rendered basin partition together with its Julia-set boundary, rather than to a single set having one universal fractal dimension. Different Newton maps possess different Julia sets, and their geometric dimensions depend on the polynomial and its dynamical parameters. Even within one map, the colored basins are open sets of ordinary planar dimension, whereas the boundary carries the nontrivial small-scale geometry.
Self-similarity in a Newton fractal is generally dynamical rather than exact in the elementary geometric sense. Inverse branches of the Newton map reproduce related boundary configurations at successively smaller scales, but distortion varies with position because the map is not globally linear. Symmetric polynomials can impose exact rotational or reflectional relations, while the repeated fine structure follows from conformal dynamics and inverse iteration.
The resulting image combines two mathematically distinct forms of information. Basin color records asymptotic destination, whereas internal shading records a finite-time property of the orbit. The Julia set separates destinations but is not itself defined by any particular coloring convention, image resolution, convergence tolerance, or maximum iteration count.