Buzz
In this computer-generated visualization, the "einstein" tile is depicted as a novel geometric form, arranged in concentric circles around a central, darker "hat." Smith, Myers, Kaplan, and Goodman-Strauss (2023)Main Highlights
- Researchers have identified a groundbreaking 13-sided geometric shape known as the "einstein," which forms a tiling pattern that never repeats.
- Initially believed to be unattainable, this shape was uncovered by David Smith, a passionate "shape enthusiast," who named his first find "the Hat."
- Both "the Hat" and "the Turtle" belong to a category of einstein tiles, formed by altering side lengths, overturning prior beliefs about tiling patterns.
Picture a bathroom floor adorned with black and white tiles. Each tile shares the same shape, fitting together seamlessly without gaps or overlaps. While rectangles or hexagons might come to mind, envision a shape that generates a pattern which never repeats, regardless of the bathroom's size. This is the essence of an "einstein" tile.
What Is an Einstein?
Кажется невероятным; трудно представить одну фигуру, которая сочетается так многими способами, что создает узор, который никогда не повторяется. Но группа математиков утверждает, что нашла такую фигуру, известную как "апериодический монотиль" или эйнштейн. Эйнштейн может быть именем самого известного ученого 20-го века, но это также немецкое слово для "одного камня" или даже "одной фигуры".
Эйнштейн — забавная идея, потому что взаимосвязанные плитки кажутся по определению упорядоченными. Но эйнштейн также хаотичен — как бы вы их ни располагали, он не может создать периодический узор.
Потребовалось до 1960-х годов, чтобы геометры смогли представить даже набор фигур, "апериодический набор", который может покрыть бесконечную двумерную плоскость узором, который никогда не повторяется. Первый такой набор фигур был обнаружен в 1966 году — он состоял из 20 426 различных плиток.
Введите Дэвида Смита, "Шляпу" и "Черепаху"
Over the next decade, mathematicians refined the concept, reducing the number of shapes in the set from tens of thousands to just a few. Throughout this period, they sought The One — a solution to the "einstein problem." Some researchers even worked tirelessly to prove that such a single shape could not exist.
In November 2022, David Smith, a self-proclaimed "shape enthusiast" from East Yorkshire, England, announced that he had likely uncovered an einstein in the form of a 13-sided tile he named "the Hat," due to its resemblance to a fedora.
Smith, a retired printing technician, spent countless hours at home cutting shapes from paper and experimenting with them. He understood the mathematical theory required to find the einstein, and one day, he stumbled upon something highly promising. He reached out to computer scientists and a mathematician he knew, and together they worked to prove his discovery.
In March 2023, they published a preprint, sparking astonishment and excitement among mathematicians worldwide. Amid this breakthrough, Smith discovered another einstein: "the Turtle."
As it turns out, "the Hat" and "the Turtle" are just two members of a family of einstein tiles, formed by adjusting the lengths of the sides of the shapes relative to one another.
The Hat's most promising applications lie in developing more durable materials and in artistic endeavors.
