The black region shown is a single polygon (completed by the black region below the figure). Its boundary in the figure is an open, simply connected polygonal curve, the fifth step in the creation of a Peano curve that fills the Koch snowflake. It's based on the family of curves described by Mandelbrot in The Fractal Geometry of Nature (see Pl. 68). As he there says: "This advanced teragon, shown as boundary between two fantastically intertwined domains serves better than any number of words to explain what plane-filling means."
This applet is not interactive, given the amount of memory used in the creation of the figure; but the generation of the curve may be explored in my Peano curve applet.