Mandelbrot set

Definition

The Mandelbrot set is a complex fractal defined by the iteration of the complex quadratic polynomial $z_{n+1} = z_n^2 + c$. It is known for its intricate and infinitely detailed boundary.

5 Must Know Facts For Your Next Test

1. The Mandelbrot set is defined in the complex plane and consists of all points $c$ for which the orbit of 0 under iteration remains bounded.
2. A point $c$ belongs to the Mandelbrot set if, when iterating $z_{n+1} = z_n^2 + c$, starting with $z_0 = 0$, the sequence does not tend to infinity.
3. The boundary of the Mandelbrot set exhibits self-similarity, meaning smaller copies of the entire set can be found within it.
4. The escape time algorithm is commonly used to render images of the Mandelbrot set, coloring each point based on how quickly it escapes to infinity.
5. Newton’s Method can be applied in complex dynamics to find roots that help illustrate areas inside or outside the Mandelbrot set.

Review Questions

• What defines a point as belonging to the Mandelbrot set?
• Explain how Newton’s Method can relate to analyzing regions in or around the Mandelbrot set.
• Describe what is meant by self-similarity in the context of the Mandelbrot set.

Related terms

Complex Plane: A mathematical concept where each point represents a complex number; it has a real axis and an imaginary axis.

Fractal: A self-similar pattern that repeats at every scale and can be described by recursive algorithms.

Newton's Method: $$An iterative numerical technique used to find approximate solutions to real-valued functions. In calculus, it's often employed to find roots.$$