The fractal curves are generated from an initial curve (often a regular polygon) and one or more replacement curves. Repeatedly, each line is replaced by a properly scaled copy of one of the replacement curves. The buttons show the initial curve and first iteration of the replacement curve used.
The code for describing the curves and computing the graphic output is from Roman E. Maeder, The Mathematica Programmer, New York: Academic Press, 1994.
Snapshot 1: the classic snowflake curve
Snapshot 2: W. Gosper's flowsnake, a space-filling curve
Snapshot 3: based on work by B. Mandelbrot