谢尔宾斯基地毯

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Menger 5.PNG

谢尔宾斯基地毯是由瓦茨瓦夫·谢尔宾斯基于1916年提出的一种分形,是自相似集的一种。它的豪斯多夫维是 log 8/log 3 ≈ 1.8928。门格海绵是它在三维空间中的推广。

构造[编辑]

谢尔宾斯基地毯的构造与谢尔宾斯基三角形相似,区别仅在于谢尔宾斯基地毯是以正方形而非等边三角形为基础的。将一个实心正方形划分为3 \times 3的9个小正方形,去掉中间的小正方形,再对余下的小正方形重复这一操作便能得到谢尔宾斯基地毯。[1]

Menger 0.PNG Menger 1.PNG Menger 2.PNG Menger 3.PNG Menger 4.PNG Menger 5.PNG

谢尔宾斯基地毯可以由以下计算机程序构造:

 /**
 Decides if a point at a specific location is filled or not.
 
 @param x is the x coordinate of the point being checked
 @param y is the y coordinate of the point being checked
 @param width is the width of the Sierpinski Carpet being checked
 @param height is the height of the Sierpinski Carpet being checked
 @return 1 if it is to be filled or 0 if it is not
 */  int isSierpinskiCarpetPixelFilled(int x,int y,int width,int height)
 {
     // base case
  if (x<1)
  {
       return 0;
  }
     
   // general case
   {
    /*
      If the grid was split in 9 parts, what part(x2,y2) would x,y fit into?
    */
    int x2 = x*3/width; // an integer from 0..2 inclusive
    int y2 = y*3/height; // an integer from 0..2 inclusive
    
      if (x2==1 && y2==1) // if in the centre squaure, it should be filled.
         return 1;
     
      /* offset x and y so it becomes bounded by 0..width/3 and 0..height/3 
        and prepares for recursive call
      */ 
      x-=x2*width/3;
      y-=y2*height/3;
   
  }
  
  return isSierpinskiCarpetPixelFilled(x,y,width/3,height/3);
 }

参考[编辑]