Takagi–Landsberg curve - Revision history

A WASSERMANN at 07:56, 14 August 2025

2025-08-14T07:56:37Z

← Older revision		Revision as of 07:56, 14 August 2025
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	:<math> {\rm blanc}(x) = \sum_{n=0}^\infty {s(2^{n}x)\over 2^n},</math>		:<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.		where <math>s(x)</math> is defined by
			s(x)=\min_{n\in{\bold Z}}\|x-n\|,
			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		The Takagi–Landsberg curve is a slight generalization, given by

Michael at 07:26, 9 June 2011

2011-06-09T07:26:54Z

← Older revision		Revision as of 07:26, 9 June 2011
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	<jsxgraph width="500" height="500" box="box">		<jsxgraph width="500" height="500" box="box">
	var bd = JXG.JSXGraph.initBoard('box', {axis:true,~~originX~~: 25, ~~originY: 400~~, ~~unitX: 450~~, ~~unitY: 25~~});		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 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 N = bd.create('slider', [[0,7],[0.8,7],[0,5,40]], {name:'N'});
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	===The JavaScript code to produce this picture===		===The JavaScript code to produce this picture===

	<source lang="~~xml">~~		<source lang="javascript">
	~~<jsxgraph width="500" height="500" box="box~~">		var bd = JXG.JSXGraph.initBoard('box', {axis:true,boundingbox: [-0.05, 16, 1.27, -4]});
	var bd = JXG.JSXGraph.initBoard('box', {axis:true,~~originX~~: 25, ~~originY: 400~~, ~~unitX: 450~~, ~~unitY: 25~~});
	var w = bd.create('slider', [[0,8],[0.8,8],[0,0.25,1.5]], {name:'w'});		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 N = bd.create('slider', [[0,7],[0.8,7],[0,5,40]], {name:'N'});
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	return su;		return su;
	},0,1],{strokeColor:'red'});		},0,1],{strokeColor:'red'});
	~~</jsxgraph>~~
	</source>		</source>

A WASSERMANN at 10:14, 25 March 2011

2011-03-25T10:14:20Z

← Older revision		Revision as of 10:14, 25 March 2011
Line 14:		Line 14:
	<jsxgraph width="500" height="500" box="box">		<jsxgraph width="500" height="500" box="box">
	var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 400, unitX: 450, unitY: 25});		var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 400, unitX: 450, unitY: 25});
	var w = bd.~~createElement~~('slider', [[0,8],[0.8,8],[0,0.25,1.5]], {name:'w'});		var w = bd.create('slider', [[0,8],[0.8,8],[0,0.25,1.5]], {name:'w'});
	var N = bd.~~createElement~~('slider', [[0,7],[0.8,7],[0,5,40]], {name:'N'});		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 s = function(x){ return Math.abs(x-Math.round(x)); };
	var c = bd.~~createElement~~('functiongraph', [		var c = bd.create('functiongraph', [
	function(x){		function(x){
	var n, su, wval;		var n, su, wval;
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	<jsxgraph width="500" height="500" box="box">		<jsxgraph width="500" height="500" box="box">
	var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 400, unitX: 450, unitY: 25});		var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 400, unitX: 450, unitY: 25});
	var w = bd.~~createElement~~('slider', [[0,8],[0.8,8],[0,0.25,1.5]], {name:'w'});		var w = bd.create('slider', [[0,8],[0.8,8],[0,0.25,1.5]], {name:'w'});
	var N = bd.~~createElement~~('slider', [[0,7],[0.8,7],[0,5,40]], {name:'N'});		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 s = function(x){ return Math.abs(x-Math.round(x)); };
	var c = bd.~~createElement~~('functiongraph', [		var c = bd.create('functiongraph', [
	function(x){		function(x){
	var n, su, wval;		var n, su, wval;

A WASSERMANN at 17:40, 18 March 2009

2009-03-18T17:40:03Z

← Older revision		Revision as of 17:40, 18 March 2009
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	</source>		</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===		===External links===
	* [http://en.wikipedia.org/wiki/Blancmange_curve http://en.wikipedia.org/wiki/Blancmange_curve]		* [http://en.wikipedia.org/wiki/Blancmange_curve http://en.wikipedia.org/wiki/Blancmange_curve]

A WASSERMANN at 17:38, 18 March 2009

2009-03-18T17:38:47Z

@@ Line 13: / Line 13: @@
 <jsxgraph width="500" height="500" box="box">
-  var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 250, unitX: 450, unitY: 25});
+  var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 400, unitX: 450, unitY: 25});
-  var w = bd.createElement('slider', [[0,8],[0.8,8],[0,0.5,2]], {name:'w'});
+  var w = bd.createElement('slider', [[0,8],[0.8,8],[0,0.25,1.5]], {name:'w'});
   var N = bd.createElement('slider', [[0,7],[0.8,7],[0,5,40]], {name:'N'});
   var s = function(x){ return Math.abs(x-Math.round(x)); };
@@ Line 26: / Line 26: @@
+         }
          return su;
-      },0,1]);
+      },0,1],{strokeColor:'red'});
 </jsxgraph>

A WASSERMANN at 17:36, 18 March 2009

2009-03-18T17:36:27Z

← Older revision		Revision as of 17:36, 18 March 2009
Line 14:		Line 14:
	<jsxgraph width="500" height="500" box="box">		<jsxgraph width="500" height="500" box="box">
	var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 250, unitX: 450, unitY: 25});		var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 250, unitX: 450, unitY: 25});
	var w = bd.createElement('slider', [[0,8],[1,8],[0,0.5,1]], {name:'w'});		var w = bd.createElement('slider', [[0,8],[0.8,8],[0,0.5,2]], {name:'w'});
	var N = bd.createElement('slider', [[0,7],[1,7],[0,5,40]], {name:'N'});		var N = bd.createElement('slider', [[0,7],[0.8,7],[0,5,40]], {name:'N'});
	var s = function(x){ return Math.abs(x-Math.round(x)); };		var s = function(x){ return Math.abs(x-Math.round(x)); };
	var c = bd.createElement('functiongraph', [		var c = bd.createElement('functiongraph', [

A WASSERMANN at 17:35, 18 March 2009

2009-03-18T17:35:56Z

← Older revision		Revision as of 17:35, 18 March 2009
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	<jsxgraph width="500" height="500" box="box">		<jsxgraph width="500" height="500" box="box">
	var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 250, unitX: ~~500~~, unitY: 25});		var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 250, unitX: 450, unitY: 25});
	var w = bd.createElement('slider', [[0,8],[1,8],[0,0.5,1]], {name:'w'});		var w = bd.createElement('slider', [[0,8],[1,8],[0,0.5,1]], {name:'w'});
	var N = bd.createElement('slider', [[0,7],[1,7],[0,5,40]], {name:'N'});		var N = bd.createElement('slider', [[0,7],[1,7],[0,5,40]], {name:'N'});

A WASSERMANN at 17:35, 18 March 2009

2009-03-18T17:35:32Z

← Older revision		Revision as of 17:35, 18 March 2009
Line 13:		Line 13:

	<jsxgraph width="500" height="500" box="box">		<jsxgraph width="500" height="500" box="box">
	var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 250, unitX: ~~200~~, unitY: 25});		var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 250, unitX: 500, unitY: 25});
	var w = bd.createElement('slider', [[0,8],[1,8],[0,4,6]], {name:'w'});		var w = bd.createElement('slider', [[0,8],[1,8],[0,0.5,1]], {name:'w'});
	var N = bd.createElement('slider', [[0,7],[1,7],[0,5,20]], {name:'N'});		var N = bd.createElement('slider', [[0,7],[1,7],[0,5,40]], {name:'N'});
	var s = function(x){ return Math.abs(x-Math.round(x)); };		var s = function(x){ return Math.abs(x-Math.round(x)); };
	var c = bd.createElement('functiongraph', [		var c = bd.createElement('functiongraph', [

A WASSERMANN at 17:34, 18 March 2009

2009-03-18T17:34:23Z

← Older revision		Revision as of 17:34, 18 March 2009
Line 14:		Line 14:
	<jsxgraph width="500" height="500" box="box">		<jsxgraph width="500" height="500" box="box">
	var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 250, unitX: 200, unitY: 25});		var bd = JXG.JSXGraph.initBoard('box', {axis:true,originX: 25, originY: 250, unitX: 200, unitY: 25});
	var w = bd.createElement('slider', [[1,8],[5,8],[0,4,6]], {name:'w'});		var w = bd.createElement('slider', [[0,8],[1,8],[0,4,6]], {name:'w'});
	var N = bd.createElement('slider', [[1,7],[5,7],[0,5,20]], {name:'N'});		var N = bd.createElement('slider', [[0,7],[1,7],[0,5,20]], {name:'N'});
	var s = function(x){ return Math.abs(x-Math.round(x)); };		var s = function(x){ return Math.abs(x-Math.round(x)); };
	var c = bd.createElement('functiongraph', [		var c = bd.createElement('functiongraph', [
Line 22:		Line 22:
	su = 0.0;		su = 0.0;
	wval = w.Value();		wval = w.Value();
	for (n=0;n<N.~~value~~();n++) {		for (n=0;n<N.Value();n++) {
	su += Math.pow(wval,n)s(Math.pow(2,n)x);		su += Math.pow(wval,n)s(Math.pow(2,n)x);
	}		}

A WASSERMANN at 17:33, 18 March 2009

2009-03-18T17:33:51Z

← Older revision		Revision as of 17:33, 18 March 2009
Line 26:		Line 26:
	}		}
	return su;		return su;
	},0,1}]);		},0,1]);
	</jsxgraph>		</jsxgraph>