This paper describes a simple but flexible family of Haskell programs for drawing pictures of fractals such as Mandelbrot and Julia sets. Its main goal is to showcase the elegance of a compositional approach to program construction, and the benefits of a clean separation between different aspects of program behavior. Aimed at readers with relatively little experience of functional programming, the paper can be used as a tutorial on functional programming, as an overview of the Mandelbrot set, or as a motivating example for studies in computability.

The tedium of writing pickling and unpickling functions by hand is relieved using a combinator library similar in spirit to the well-known parser combinators. Picklers for primitive types are combined to support tupling, alternation, recursion, and structure sharing. Code is presented in Haskell; an alternative implementation in ML is discussed.

We derive a combinator library for non-deterministic parsers with a monadic interface, by means of successive refinements starting from a specification. The choice operator of the parser implements a breadth-first search rather than the more common depth-first search, and can be seen as a parallel composition between two parsing processes. The resulting library is simple and efficient for “almost deterministic” grammars, which are typical for programming languages and other computing science applications.

The Sieve of Eratosthenes is an efficient algorithm for computing the successive primes. Rendered informally, it is as follows:

1. Write down the successive “plurals”: 2, 3, 4, …

2. Repeat:

(a) Take the first number that is not circled or crossed out.

(b) Circle it.

(c) Cross out its proper multiples.

3. What is left (i.e. the circled numbers) are the successive prime numbers.