Math
A curve that fills a square
A line has no area, and yet a curve can pass through every point of a square - Peano found one in 1890, Hilbert drew his in 1891. At each order the curve visits every cell exactly once, moving one cell at a time, and the limit is continuous. Drag the marker: a tenth of the curve stays inside about a third of the side, which is why databases and image formats index space along this curve. The reverse is false, and the panel measures that too - two touching cells can be five sixths of the curve apart.
What happens
A line has as many points as a square - Cantor proved it in 1878 and nobody wanted to believe it, and his matching tore the segment to dust. Peano and Hilbert found a continuous one: a curve that visits every cell of a grid once, refined by a rule that keeps every step one cell long, and in the limit passes through every point of the square. Its length doubles with every order while the area it fills stays one. The same rule lifts to the cube. Raise the order, follow the curve with the marker, and watch nearby points on the line stay nearby in the square - mostly.
Equations used in the module
Equations Hilbert: the cell at distance d from the bits of d, two at a time Peano: base 3, three cells per digit every curve is a permutation of the cells, each step one cell length at order n, unit square: (4^n − 1)/2^n → ∞ while the area stays 1
