Let that sink in.
Take a 1x1 square. It contains an infinite number of points. Peano built a single, continuous line that touches every single one of them . monster curves
A line can, in fact, behave like a square. The distinction between one and two dimensions depends on how you define "distance" and "covering." Let that sink in
As mathematician Hans Hahn once put it: "The concept of a curve is far richer and more terrifying than anyone had imagined." You don't need infinite iterations to see the beauty. Open a simple Python environment (or even a spreadsheet) and generate the first 4 iterations of the Hilbert curve. Plot the points. You'll see a beautiful, orderly maze that slowly begins to eat the empty space. Peano built a single, continuous line that touches
This was mathematical heresy. How can a one-dimensional object cover a two-dimensional area without crossing itself (infinitely many times) or turning into a blob? You don't need a PhD to understand the construction. It's built on a simple "copy and replace" rule, much like a fractal.
Meet the . The Problem With "Simple" For most of mathematical history, "curve" meant something tidy: a circle, a sine wave, a parabola. But in 1890, Italian mathematician Giuseppe Peano dropped a bomb. He constructed a curve that passes through every point of a unit square.