Here is openSCAD code to make Sierpinski triangles and carpets. You can read about Sierpinski's work on Wikipedia or Wolfram Math. I like the article by Sylvie Champeaux at:

http://hmf.enseeiht.fr/travaux/CD9900/travaux/optmfn/hi/00pa/mfn03/champeau.htm

which explains how the Chaos Game leads to a Sierpinski triangle.

Sierpinski_triangle.stl is an ornament with a hole for hanging. The openSCAD code produced 81 separate triangles which were subtracted from a single triangle to make the ornament.

Sierpinski_carpet.stl is a carpet after three cycles of iteration.

Composite_carpet.stl is a composite of 3 carpets at different stages of iteration.

Sierpinski_all.stl is a carpet where the central square has been replaced by a triangle.

The triangles and carpets are limited to 3 or 4 iterations here; attempts to go further ran into openSCAD program limitations. Note that Polymaker, (thing:8711), has created an attractive three dimensional version of the triangle.

These objects are easily and quickly printed with or without a raft.

Sierpinski_carpet.stl has the smallest features and perhaps prints better with a single shell and 50% infill. The smallest squares had to be made larger than is required by the generating algorithm so as to be properly sized by ReplicatorG.