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.
- 1 Definition
- 2 Properties
- 3 Graphical construction
- 4 Recursive definition
- 5 Other properties
- 6 See also
- 7 References
- 8 Further reading
- 9 External links
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 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.
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.
Convergence and continuity
The infinite sum defining converges absolutely for all x: since for all , we have:
- if .
Therefore, the Takagi curve of parameter w is defined on the unit interval (or ) if .
- for all x when .
This value can be made as small as we want by selecting a big enough value of n. Therefore, by the uniform limit theorem, is continuous if |w|<1.
The special case of the parabola
Fourier series expansion
The Takagi-Landsberg function admits an absolutely convergent Fourier series expansion:
with and, for
where is the maximum power of that divides . Indeed, the above triangle wave has an absolutely convergent Fourier series expansion
By absolute convergence, one can reorder the corresponding double series for :
putting yields the above Fourier series for .
The blancmange curve can be visually built up out of triangle wave functions if the infinite sum is approximated by finite sums of the first few terms. In the illustration below, progressively finer triangle functions (shown in red) are added to the curve at each stage.
|n = 0||n ≤ 1||n ≤ 2||n ≤ 3|
The periodic version of the Takagi curve can also be defined recursively by:
The version restricted to the unit interval can also be defined recursively by:
Integrating the Blancmange curve
Given that the integral of from 0 to 1 is 1/2, the identity allows the integral over any interval to be computed by the following relation. The computation is recursive with computing time on the order of log of the accuracy required.
Relation to simplicial complexes
Define the Kruskal–Katona function
The Kruskal–Katona theorem states that this is the minimum number of (t − 1)-simplexes that are faces of a set of N t-simplexes.
As t and N approach infinity, (suitably normalized) approaches the blancmange curve.
- Weisstein, Eric W., "Blancmange Function", MathWorld.
- Teiji Takagi, "A Simple Example of a Continuous Function without Derivative", Proc. Phys. Math. Japan, (1903) Vol. 1, pp. 176–177.
- Benoit Mandelbrot, "Fractal Landscapes without creases and with rivers", appearing in The Science of Fractal Images, ed. Heinz-Otto Peitgen, Dietmar Saupe; Springer-Verlag (1988) pp 243–260.
- Linas Vepstas, Symmetries of Period-Doubling Maps, (2004)
- Donald Knuth, The Art of Computer Programming, volume 4a. Combinatorial algorithms, part 1. ISBN 0-201-03804-8. See pages 372-375.
- Allaart, Pieter C.; Kawamura, Kiko (11 October 2011), The Takagi function: a survey, arXiv:1110.1691
- Lagarias, Jeffrey C. (17 December 2011), The Takagi Function and Its Properties, arXiv:1112.4205