Question

What is the mandelbrot set?

Answer 1

See explanation...

Explanation:

For any complex number `c`, define the function:

`f_c(x) = x^2+c`

Consider the sequence formed by repeatedly applying `f_c`, starting from `0`...

`0`, `f_c(0)`, `f_c(f_c(0))`, `f_c(f_c(f_c(0)))`,...

If the sequence is bounded, then `c` is in the Mandelbrot set, otherwise it isn't.

When generating an image of the Mandelbrot set on a computer, the test for boundedness is usually crossing a circle of some radius like `2` in the complex plane, and the colour of a pixel is based on how many iterations it takes for the sequence for the corresponding complex number to cross the circle.

[Image generated using fraqtive]

In this image, the black portion represents the actual Mandelbrot set.