Sierpinski Gasket

This object is self-similar, and its taking advantage of its self-similarity can aid in its construction. Let On be a set of cubes, with O0 containing a single unit cube. The following procedure will construct this object:

  1. Reduce all elements of On to 1/3rd their size.
  2. Let On+1 be the elements of On each copied to 20 positions as follows:

    BottomCenterTop
       
       
       
       
       
       
       
       
       

  3. Repeat until the desired level of refinement is reached.

The following images show six levels of refinement:

Interesting effects can be obtained by shifting some of the boxes during the copy operation. Shifting by only an extra tenth produces the following images - the left image at the second level of refinement and the right image at the fourth level: