These are simple tiles to demonstrate the so-called “EinStein Problem” in geometry: https://en.wikipedia.org/wiki/Einstein_problem

The problem was whether there exists a simple geometric tile that can cover an endless plane with no repeating patterns using only this one type of tile.

In 2022 David Smith found a pattern, nicknamed “the Hat”, that did this. Together with mathematicians Craig S. Kaplan, Joseph Samuel Myers, and Chaim Goodman-Strauss the team discovered a whole class of related tiles with similar properties.

Here I provide a FreeCAD model, STLs and GCode for the Hat and the most basic member of that class (Tile(1,1)). As well as a variation of that tile with curved sides, called “Spectre”.

The top side of those tiles is marked, so that you don't accidentally flip them while puzzling (flipping would make it a second kind of tile and violate the premise of the problem).

Happy puzzling!