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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 1901, 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.
Contents
Definition
The blancmange function is defined on the unit interval by
where
The Takagi–Landsberg curve is a slight generalization, given by
for a parameter w; thus the blancmange curve is the case
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.
Graphical construction
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.
Convergence and continuity
The infinite sum defining
Therefore, the Takagi curve of parameter w is defined on the unit interval (or
The Takagi function of parameter w is continuous. Indeed, the functions
This value can be made as small as we want by selecting a big enough value of n. Therefore, by the uniform limit theorem,
The special case of the parabola
For
Differentiability
The Takagi curve is a fractal for parameters
one may immediately write the formal sum
This sum is manifestly convergent for
Equivalently, one may apply the discrete wavelet transform to obtain a representation in terms of Haar wavelets
Fourier series expansion
The Takagi-Landsberg function admits an absolutely convergent Fourier series expansion:
with
where
By absolute convergence, one can reorder the corresponding double series for
putting
Recursive definition
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:
Proof:
Observe that this definition has the form of a discrete wavelet transform; thus the coefficients of the wavelet transform take a particularly simple form.
Self similarity
The above recursive definition allows the monoid of self-symmetries of the curve to be given. This monoid is given by two generators, g and r, which act on the curve (restricted to the unit interval) as
and
A general element of the monoid then has the form
In this representation, the action of g and r are given by
and
That is, the action of a general element
Note that
The monoid generated by g and r is sometimes called the dyadic monoid; it is a sub-monoid of the modular group. When discussing the modular group, the more common notation for g and r is T and S, but that notation conflicts with the symbols used here.
The above three-dimensional representation is just one of many representations it can have; it shows that the blancmange curve is one possible realization of the action. That is, there are representations for any dimension, not just 3; some of these give the de Rham curves.
Integrating the Blancmange curve
Given that the integral of
one has that
The definite integral is given by:
A more general expression can be obtained by defining
which, combined with the series representation, gives
Note that
This integral is also self-similar on the unit interval, under an action of the same dyadic monoid as described above. Here, the representation is 4-dimensional, having the basis
From this, one can then immediately read off the generators of the four-dimensional representation:
and
Repeated integrals transform under a 5,6,... dimensional representation.
Relation to simplicial complexes
Let
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,