einstein hat - world's first aperiodic monotile

This is a 3D model of the world's first "einstein", an aperiodic monotile.

Update: There's a chiral (handed) aperiodic monotile now, see: https://www.printables.com/model/494205-spectre-worlds-first-chrial-aperiodic-monotile

You will likely have seen various examples of https://en.wikipedia.org/wiki/Tessellation , endless patterns made up of only a few or even just one single tile. You may also have seen https://en.wikipedia.org/wiki/Penrose_tiling , an endless pattern which can never repeat, based on only two tiles.

In 2023, the https://en.wikipedia.org/wiki/Einstein_problem (from German “one stone”, but also a play of words on a certain scientist) was solved for the first time: Is there a single geometry which allows for perfect tessellation in a pattern that will never repeat? The answer is (likely, the paper still needs peer review as of 2023-03-22) yes: https://arxiv.org/pdf/2303.10798.pdf

Feel invited to print a few billions pieces to try it for yourself.

And please: Upload as many makes with as many pieces in as many colors as you can.

The measurements are precise to micrometers; 1.73 =~ sqrt(3), an irrational number. The physical model is not perfect in the mathematical and it can never be. In practice, this does not matter; warping of your prints and other imperfections will become a problem way before you manage to print an infinite amount of pieces.

If your printer makes imperfect corners, consider printing the fillets version; all corners have a 0.5mm radius fillet in those models.

Extra / fun files:

• DXF for to CNC/laser cuting ALL your scraps
• Coaster for your favorite beverage
• Holder for dice, pens, or other things - in both orientations so you can endlessly tile your holders
• Dogtag for use with PrusaSlicer's text embossing tool (still fits with the rest of the normal-sized tiles)

If you want to learn more about penrose tiling, there's a few nice videos on YouTube:

• Minute Physics: 

• Numberphile: 

• Veritasium: