Penrose Geometry: From Symmetry to the Golden Ratio and the Einstein Tile

2024/10/11

In March 2023, a certain paper was published. It reported the discovery of a single type of tile of the same size that can "tile the plane indefinitely" and "never repeat periodically." The discoverer, David Smith, was a private geometry enthusiast, but three other mathematicians collaborated to prove its aperiodicity, and the discovery report was the result of extensive use of computers. It is called the "Einstein Tile," a term derived from the German for "one stone," though it certainly also alludes to the famous physicist.

Shapes that can only tile a plane aperiodically are a relatively new topic.

To begin with, until the first half of the 20th century, it was not known whether it was even possible for a set of tiles to be arranged only aperiodically. It was in 1964 that the existence of such a set of tiles was shown, breaking the common knowledge of the time, but at that time the set consisted of over 20,000 tiles. The number of tiles was gradually reduced, progressing smoothly down to the two tiles of Penrose tiling. After that, mathematicians plotted the pursuit of an "aperiodic tile consisting of only one piece." Because it had not been found for half a century, it was thought that it "probably does not exist."

I was preparing a book on "tessellation" (the knowledge of filling a plane) and "symmetry" with an editor from Blue Backs. I learned of this new discovery during a period when the schedule had been disrupted and delayed by the COVID-19 pandemic and other factors. We hastily added Yoshiaki Araki, an expert in this field, as a co-author and rewrote most of the chapters to complete this book.

I have been a fan of Blue Backs since I was a junior high and high school student, and as a result, I have gained a great deal of new perspective and knowledge. I feel that I owe much of who I am today to Blue Backs. This is my second book from Blue Backs, and if possible, I would love for junior high and high school students who will be gaining knowledge about the world to pick it up. Incidentally, we ended up with the title above to capitalize on Penrose's name recognition.

(Written by Ichiro Tanioka)

Penrose Geometry: From Symmetry to the Golden Ratio and the Einstein Tile

Ichiro Tanioka / Yoshiaki Araki

Kodansha Blue Backs

208 pages, 1,100 yen (tax included)

*Affiliations and titles are as of the time this magazine was published.