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Home/Math Visualization/Newton Fractal

Newton Fractal

This interactive simulator explores Newton Fractal in Math Visualization. Basins of attraction for Newton iteration on zⁿ−1 with adjustable relaxation ω. Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.

Who it's for: For learners comfortable with heavier math or second-level detail. Typical context: Math Visualization.

Key terms

  • newton
  • fractal
  • newton fractal
  • math
  • visualization

Parameters

Polynomial

40
1

Shortcuts

  • •Drag to pan, wheel to zoom

Measured values

roots n3
max iter40
ω (1=Newton)1.00
view span3.20e+0

How it works

Newton’s method for p(z) = zⁿ − 1 colours each starting point by which root the iteration zₙ₊₁ = zₙ − ω·p(zₙ)/p′(zₙ) converges to. The boundary between the basins of attraction is a fractal with the property that near any point of three different colours all three colours meet — Newton himself never imagined this. Try n = 4 or 5 for the most photogenic basins; ω ≠ 1 turns it into a relaxed scheme with very different boundary structure.

Key equations

zₙ₊₁ = zₙ − ω · p(zₙ) / p′(zₙ)
p(z) = zⁿ − 1, p′(z) = n z^{n−1}
roots: ζₖ = exp(2πi k / n), k = 0…n−1