Unconventional Space-filling Curves

A 2-dimensional space-filling curve, here, is a surjective continuous map from an interval to a 2-dimensional space. To construct a new space-filling curve distinct and much different from Hilbert’s, Peano’s, and the Z-order, we first observe that all three of these curves are based on self-similar fractals–in particular, the recursive definition of the sequences of functions that converge to these curves systematically replaces smaller units of the whole with the entire map. For instance, the space-filling curve given in Munkres, , which is there referred to as the ”Peano space-filling curve” (271) but which follows more closely Hilbert’s design , begins with H0, a parametrized curve from t = 0 to t = 1. The function increases linearly from (0, 0) to ( 2 , 1 2 ) at t = 1 2 and then back to (1, 0) at t = 0. H1, which replaces H0, is a slightly more complicated function.