Mandelbrot and Julia Sets

Key idea

Fractals from iterating z² + c.

In depth

For c in C, iterate z ↦ z² + c from 0. The Mandelbrot set is the set of c for which the orbit stays bounded; Julia sets describe the boundary of bounded starting points for fixed c. They are fractals of infinite complexity from a simple rule.

Worked example

c = 0 gives bounded orbit (in the set); c = 1 gives 0, 1, 2, 5, 26, ... escaping, not in the set.

Formula

Real-world uses

• Creating fractal art and testing chaos in dynamical systems
• Computer graphics demos

Common mistake

Thinking the picture is made by a complex rule, when the rule is simple.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.