Blancmange curve

In mathematics, the blancmange curve is a self-affine fractal curve constructible by midpoint subdivision.

It is a special case of the more general de Rham curve.

The blancmange function is defined on the unit interval by where

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

The periodic version of the Takagi curve can also be defined as the unique bounded solution

is certainly bounded, and solves the functional equation, since Conversely, if

is a bounded solution of the functional equation, iterating the equality one has for any N whence

Incidentally, the above functional equations possesses infinitely many continuous, non-bounded solutions, e.g.

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 illustrations below, progressively finer triangle functions (shown in red) are added to the curve at each stage.

defined by the partial sums are continuous and converge uniformly toward

By derivation under the sign of series, for any non dyadic rational

is the sequence of binary digits in the base 2 expansion of

: Equivalently, the bits in the binary expansion can be understood as a sequence of square waves, the Haar wavelets, scaled to width

it is of bounded variation on no non-empty open set; it is not even locally Lipschitz, but it is quasi-Lipschitz, indeed, it admits the function

The Takagi–Landsberg function admits an absolutely convergent Fourier series expansion: with

has an absolutely convergent Fourier series expansion By absolute convergence, one can reorder the corresponding double series for

The 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

This acts on the curve as a linear function:

for some constants a, b and c. Because the action is linear, it can be described in terms of a vector space, with the vector space basis: In this representation, the action of g and r are given by and That is, the action of a general element

maps the blancmange curve on the unit interval [0,1] to a sub-interval

where the values of a, b and c can be obtained directly by multiplying out the above matrices.

That is, there are representations for any dimension, not just 3; some of these give the de Rham curves.

allows the integral over any interval to be computed by the following relation.

Defining 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 dyadic monoid described in the section Self similarity.

The action of g on the unit interval is the commuting diagram 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.

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.

(suitably normalized) approaches the blancmange curve.