Takagi–Landsberg curve: Difference between revisions
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Revision as of 17:36, 18 March 2009
The blancmange function is defined on the unit interval by
- [math]\displaystyle{ {\rm blanc}(x) = \sum_{n=0}^\infty {s(2^{n}x)\over 2^n}, }[/math]
where [math]\displaystyle{ s(x) }[/math] is defined by [math]\displaystyle{ s(x)=\min_{n\in{\bold Z}}|x-n| }[/math], that is, [math]\displaystyle{ s(x) }[/math] is the distance from x to the nearest integer. The infinite sum defining [math]\displaystyle{ blanc(x) }[/math] converges absolutely for all x, but the resulting curve is a fractal. The blancmange function is continuous but nowhere differentiable.
The Takagi–Landsberg curve is a slight generalization, given by
- [math]\displaystyle{ T_w(x) = \sum_{n=0}^\infty w^n s(2^{n}x) }[/math]
for a parameter w; thus the blancmange curve is the case [math]\displaystyle{ w = 1 / 2 }[/math]. For [math]\displaystyle{ w = 1 / 4 }[/math], one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.
