3D "einstein" Monotile: Chair44
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One 3D cube shape whose copies can fit together and fill all of space, without ever exactly repeating.
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Copies of this 3D cube shape can fit together to fill all of space without ever repeating, by forming exponentially larger and larger "chair" shapes. This sort of shape is known as an "einstein tile" (from "one stone", not Dr. Einstein!).
The tile was discovered this month by Ioannis Tsiokos and the OpenAI model Astra:
https://arxiv.org/abs/2609.19214
https://www.emergencecalculus.com/toys/chair44/
Discussion of the result:
https://www.quantamagazine.org/updates/transformation/#167384
The most readable writeup is by Dr. Goodman-Strauss:
https://arxiv.org/pdf/2609.24779
Another neat 3D printed model, built independently today:
https://www.printables.com/model/1853986-einstein3d-the-chair44-tile
Source code is included, requires OpenSCAD nightly (needs roof() support).
Recommended print settings: (should use under 10 grams of filament per 25mm cube)
2 perimeters
15% infill, cubic pattern
3 layers top and bottom
Dial in your support material gap, or use separate support material.
When aligned correctly, the tiles should gently pop together by hand. Extra support material may prevent full contact.
I wrote this OpenSCAD myself, by hand, by reading the papers. Please let me know if I introduced any errors!
(Note: the black cubes in the photos are an earlier version with one incorrect corner, causing problems at tile level 2 or higher. The gold cube tiles correctly, and matches both the papers and the models here.)