Space-filling curves are related to fractals, in that they have self-similar patterns. Such space-filling curves were originally developed as conceptual mathematical "monsters", counter-examples to Weierstrassian and Reimannian treatments of calculus and continuity. These were curves that were everywhere-connected but nowhere-differentiable (or some similar paradoxical combination of conditions): that is, there were no breaks in the curves, but they were so extremely and discontinuously wiggly that ordinary differentiation did not apply to them. Moreover, they showed that a "line"--specifically a "curve", rather than a "straight line"--could fill two-dimensional space. As early as 1940, the great mathematics popularisers Kasner and Newman discussed the Koch snowflake, the anti-snowflake, and bizarre space-filling "curves" as examples of what Kasner and Newman called "pathological" shapes. Pathological, because the two-dimensional snowflake curve, for example, is contained within a finite area but is itself infinitely long, while the three-dimensional counterpart is a space-filling curve that is infinitely long and completely fills a finite volume. The author offers a few suggestions for materials to read about the great "popularisers" of mathematics.