Peano Curve

CORE DEFINITION

The Peano curve is a curve that can fill a square. In 1890, Italian mathematician Giuseppe Peano invented a curve that can fill a square, called the Peano curve. Its construction method is as follows: take a square and divide it into nine equal smaller squares, then starting from the lower-left square to the upper-right square, connect the centers of the small squares with line segments in sequence; next, divide each small square into nine equal squares, and connect their centers in the same way... If this operation is carried out infinitely, the resulting 'limiting curve' is called the Peano curve. Such a curve will fill the entire initially given square. In traditional concepts, the dimension of a curve is 1, and the dimension of a square is 2, and intuitively a 1-dimensional curve cannot fill a 2-dimensional square. But the Peano curve provides a counterexample. This shows that our understanding of dimension is flawed, and it is necessary to rethink the definition of dimension. This is the problem considered in fractal geometry. In fractal geometry, dimension can be fractional, called fractal dimension. In addition, the Peano curve is continuous but nowhere differentiable.