These are "space-filling curves" that twist and turn so much that they eventually pass through every single point in a 2D square, effectively turning a 1D line into a 2D area. Why They Were Called "Monsters"
Charles Hermite famously lamented his desire to "turn away in fear and horror from this lamentable plague of functions that have no derivatives." This paper argues, however, that the monster is not a plague, but a mirror—reflecting the true, jagged nature of the universe. monster curves
As you iterate, the "curve" gets longer and more tangled. After 1 step, it's a scribble. After 3 steps, it looks like a maze. After 10 steps, your computer screen can't tell the difference between the curve and the solid square. These are "space-filling curves" that twist and turn
The nickname "monster curves" was not a compliment. Prominent mathematicians of the era, such as , were initially repulsed by them, viewing them as a "gallery of monsters" that lacked the elegance of traditional Euclidean geometry. Because these curves were "nowhere differentiable," they couldn't be handled by the standard calculus of the time, making them feel like aberrations rather than natural parts of mathematics. From Pathology to Practicality After 1 step, it's a scribble