A One-Dimensional Line That Fills Space

A mathematical line can be drawn so intricately that it eventually covers every single point inside a square. This counterintuitive shape proves that a one-dimensional path can occupy two-dimensional area.

Discovered in 1890, this curve shattered the traditional belief that lines and surfaces occupy fundamentally different dimensions. By folding itself infinitely, the curve manages to pass through every point of a flat square without ever leaving the line. This bizarre concept challenges our basic geometric intuition of how lines and shapes interact. It serves as an early example of fractal geometry long before the field officially existed.