1890, the Italian mathematician piano invented a curve that can fill a square, called peano curve. Piano gave a detailed mathematical description of the corresponding relationship between the points on the interval [0, 1] and the points on the square. In fact, for t∈[0, 1], these points of the square can specify two continuous functions, x=f(t) and y=g(t), so that x and y take every value belonging to the unit square. Later Hilbert made this curve.
Generally speaking, it is impossible for one-dimensional things to fill two-dimensional squares. But peano curve only gave a counterexample.
This shows that our understanding of dimensions is flawed, and it is necessary to re-examine the definition of dimensions. This is the problem of fractal geometry. In fractal geometry, the dimension can be a fraction called fractal dimension.
In addition, peano curve is a continuous but non-derivative curve. Therefore, if we want to study the curve in the traditional sense, we must add the condition of derivative, so as to exclude special cases like peano curve.