介提出者
謝爾賓斯基地毯是由瓦茨瓦夫·謝爾賓斯基於1916年提出的一種分形，是自相似集的一種。它的豪斯多夫維是 log 8/log 3 ≈ 1.8928。門格海綿是它在三維空間中的推廣。
構造
謝爾賓斯基地毯的構造與謝爾賓斯基三角形相似，區別僅在於謝爾賓斯基地毯是以正方形而非等邊三角形為基礎的。將一個實心正方形劃分為的9個小正方形，去掉中間的小正方形，再對餘下的小正方形重複這一操作便能得到謝爾賓斯基地毯。如下圖
謝爾賓斯基地毯謝爾賓斯基地毯可以由以下電腦程式構造：
/**
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);
}
