I’ve been fascinated by **aperiodic tilings** of the plane since **Martin Gardner** first wrote about them in **Scientific American**. In the 1960s, **Robert Berger** discovered a set of 20 426 **prototiles **or tile-types that can tile the plane but only with no translational periodicity — a wonderful mix of the expected and the surprising, a kind of visual music.

Over the years, the number of required prototiles has been greatly reduced. In the 1970s, **Roger Penrose** discovered a set of just** two concave** aperiodic prototiles. **Robert Ammann** then dissected these to discover a set of **three convex** aperiodic prototiles. Can a single prototile, one tile or stone, literally **ein stein** in German, force a nonperiodic tiling? Despite several near misses and potential applications to **quasicrystals**, the existence of an ein Stein remains a fascinating unsolved problem.

## 3 responses to “Ein Stein”

Excellent examples – thanks.

Still remains a fascinating but not unsolved problem anymore

David Smith’s ein stein is an awesome shape!