View source for Takagi–Landsberg curve

The blancmange function is defined on the unit interval by

:<math>    {\rm blanc}(x) = \sum_{n=0}^\infty {s(2^{n}x)\over 2^n},</math>

where <math>s(x)</math> is defined by <math>s(x)=\min_{n\in{\bold Z}}|x-n|</math>, that is, <math>s(x)</math> is the distance from x to the nearest integer. The infinite sum defining <math>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>    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>w = 1 / 2</math>. For <math>w = 1 / 4</math>, one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.


<jsxgraph width="500" height="500" box="box">
 var bd = JXG.JSXGraph.initBoard('box', {axis:true,boundingbox: [-0.05, 16, 1.27, -4]});
 var w = bd.create('slider', [[0,8],[0.8,8],[0,0.25,1.5]], {name:'w'});
 var N = bd.create('slider', [[0,7],[0.8,7],[0,5,40]], {name:'N'}); 
 var s = function(x){ return Math.abs(x-Math.round(x)); };
 var c = bd.create('functiongraph', [
     function(x){
        var n, su, wval;
        su = 0.0;
        wval = w.Value();
        for (n=0;n<N.Value();n++) {
           su += Math.pow(wval,n)*s(Math.pow(2,n)*x);
        }
        return su;
     },0,1],{strokeColor:'red'});      
</jsxgraph>

===The JavaScript code to produce this picture===

<source lang="javascript">
 var bd = JXG.JSXGraph.initBoard('box', {axis:true,boundingbox: [-0.05, 16, 1.27, -4]});
 var w = bd.create('slider', [[0,8],[0.8,8],[0,0.25,1.5]], {name:'w'});
 var N = bd.create('slider', [[0,7],[0.8,7],[0,5,40]], {name:'N'}); 
 var s = function(x){ return Math.abs(x-Math.round(x)); };
 var c = bd.create('functiongraph', [
     function(x){
        var n, su, wval;
        su = 0.0;
        wval = w.Value();
        for (n=0;n<N.Value();n++) {
           su += Math.pow(wval,n)*s(Math.pow(2,n)*x);
        }
        return su;
     },0,1],{strokeColor:'red'});      
</source>

===References===
* Teiji Takagi, "A Simple Example of a Continuous Function without Derivative", Proc. Phys. Math. Japan, (1903) Vol. 1, pp. 176-177.
===External links===
* [http://en.wikipedia.org/wiki/Blancmange_curve http://en.wikipedia.org/wiki/Blancmange_curve]

[[Category:Examples]]
[[Category:Curves]]