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

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 is "aperiodic overall." The discoverer, Mr. David Smith, was a private geometry enthusiast, but three other mathematicians collaborated to prove its aperiodicity, and the discovery report was made using computers and other tools. It is called the "Einstein tile," a term derived from the German for "one stone," though it certainly also brings to mind the famous physicist.

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

Until the first half of the 20th century, it was not even known whether it was possible for a set of tiles to be arranged only aperiodically. Breaking the common sense of the time, it was shown in 1964 that such a set of tiles existed, but at that time, the set consisted of more than 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."

The author was preparing a book with an editor from Blue Backs regarding "tessellation"—the knowledge of filling a plane—and "symmetry." I learned of this new discovery during a period when the schedule was disrupted and delayed due to the COVID-19 pandemic and other factors. We hurriedly added Mr. 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 was able to gain a lot of new worlds 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 like junior high and high school students who will be gaining knowledge of the world from now on to definitely pick it up. Incidentally, taking advantage of Penrose's name recognition, the title of the work became as stated above.

(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 of publication.