Blancmange Curve

In mathematics, the blancmange curve is a fractal curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1903, or as the Takagi–Landsberg curve, a generalization of the curve named after Takagi and Georg Landsberg. The name blancmange comes from its resemblance to a pudding of the same name. It is a special case of the more general de Rham curve.

The blancmange function is defined on the unit interval by

where is defined by, that is, is the distance from x to the nearest integer. The infinite sum defining converges absolutely for all x, but the resulting curve is a fractal. The blancmange function is continuous (indeed, uniformly continuous) but nowhere differentiable.

The Takagi–Landsberg curve is a slight generalization, given by

for a parameter w; thus the blancmange curve is the case . The value is known as the Hurst parameter. For, one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.

The function can be extended to all of the real line: applying the definition given above shows that the function repeats on each unit interval.