Blancmange curve

  (Redirected from Midpoint displacement)
Graph of the blancmange function

In mathematics, the blancmange curve is a self-affine 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.

DefinitionEdit

The blancmange function is defined on the unit interval by

 

where   is the triangle wave, 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  ; 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.

The function could also be defined by the series in the section Fourier series expansion.

Functional equation definitionEdit

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

 .


Indeed, the blancmange function   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

 , for  

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

Graphical constructionEdit

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

PropertiesEdit

Convergence and continuityEdit

The infinite sum defining   converges absolutely for all  : since   for all  , we have:

  if  .

Therefore, the Takagi curve of parameter   is defined on the unit interval (or  ) if  .

The Takagi function of parameter   is continuous. Indeed, the functions   defined by the partial sums   are continuous and converge uniformly toward  , since:

  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.


SubadditivityEdit

Since the absolute value is a subadditive function so is the function  , and its dilations  ; since positive linear combinations and point-wise limits of subadditive functions are subadditive, the Takagi function is subadditive for any value of the parameter  .

The special case of the parabolaEdit

For  , one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.

DifferentiabilityEdit

For values of the parameter   the Takagi function   is differentiable in classical sense at any   which is not a dyadic rational. Precisely, by derivation under the sign of series, for any non dyadic rational   one finds

 

where   is the sequence of binary digits in the base 2 expansion of  , that is,  . Moreover, for these values of   the function   is Lipschitz of constant  . In particular for the special value   one finds, for any non dyadic rational    , according with the mentioned  

For   the blancmange function   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   as a modulus of continuity .

Fourier series expansionEdit

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  

Self similarityEdit

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   for some integers   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   for some integers m, n, p. The mapping is given exactly by   where the values of a, b and c can be obtained directly by multiplying out the above matrices. That is:

 

Note that   is immediate.

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 curveEdit

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. 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. Here, the representation is 4-dimensional, having the basis  . Re-writing the above to make the action of g more clear: on the unit interval, one has

 .

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 complexesEdit

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,   (suitably normalized) approaches the blancmange curve.

See alsoEdit

ReferencesEdit

  • Weisstein, Eric W. "Blancmange Function". MathWorld.
  • Takagi, Teiji (1901), "A Simple Example of the Continuous Function without Derivative", Proc. Phys.-Math. Soc. Jpn., 1: 176–177, doi:10.11429/subutsuhokoku1901.1.F176
  • 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.

Further readingEdit

External linksEdit