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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:
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Sierpinski_all.stl is a carpet where the central square has been replaced by a triangle.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.
Sierpinski Triangles and Carpets by pmoews is licensed under the Creative Commons - Attribution license.
So what's this mean?
We're sure pmoews would love to see what you've printed - take a photo and share it on Thingiverse as a Make.
To post a Make simply visit this Thing again and click I Made One to start uploading your photo. You can also download the Thingiverse Mobile app (available via Google Play and Apple App Store) to take a photo and upload your Make right from the app!