ÎçÒ¹¾ç³¡

ÎçÒ¹¾ç³¡
ÈÕ±¾Óï
ÈÕ±¾Óï

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

Publish: October 11, 2024

Writer Profile

  • Ichiro Tanioka

    Other : Chairman of the Board of Directors, Tanioka Gakuen Educational FoundationOther : President of Osaka University of Commerce

    ÎçÒ¹¾ç³¡ alumni

    Ichiro Tanioka

    Other : Chairman of the Board of Directors, Tanioka Gakuen Educational FoundationOther : President of Osaka University of Commerce

    ÎçÒ¹¾ç³¡ alumni

  • Yoshiaki Araki

    ÎçÒ¹¾ç³¡ alumni

    Yoshiaki Araki

    ÎçÒ¹¾ç³¡ alumni

In March 2023, a paper was published. It reported the discovery of a single type of tile that can "tile the plane indefinitely" and is "aperiodic overall." The discoverer, David Smith, was a private geometry enthusiast, but three other mathematicians collaborated to prove its aperiodicity, and the discovery report was made after utilizing computers and other tools. It is called the "Einstein tile," a term derived from the German words 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 capable of tiling only aperiodically. Breaking the common knowledge of the time, it was shown in 1964 that such a set of tiles existed, 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. Subsequently, mathematicians plotted the pursuit of an "aperiodic tile consisting of only one piece." Since it had not been found for half a century, it was thought that it "probably does not exist."

I was preparing a book with an editor from Blue Backs regarding "tessellation"¡ªthe knowledge of filling a plane¡ªand "symmetry." I learned of this new discovery during a period when my schedule was disrupted and delayed by the COVID-19 pandemic and other factors. We hurriedly 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 was able to gain a lot of new perspectives 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 acquiring knowledge about the world to pick it up. Incidentally, the title became what it is to capitalize on Penrose's name recognition.

(Written by Ichiro Tanioka)

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.

Writing Notes