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.
Read more about Blancmange Curve: Graphical Construction, Integrating The Blancmange Curve, Relation To Simplicial Complexes
Famous quotes containing the word curve:
“I have been photographing our toilet, that glossy enameled receptacle of extraordinary beauty.... Here was every sensuous curve of the “human figure divine” but minus the imperfections. Never did the Greeks reach a more significant consummation to their culture, and it somehow reminded me, in the glory of its chaste convulsions and in its swelling, sweeping, forward movement of finely progressing contours, of the Victory of Samothrace.”
—Edward Weston (1886–1958)