Problem #DES-tilings0

Problem

We’ll explore how to tile the plane using rectangles, more general quadrilaterals, pentagons and more unconventional shapes. In this exercise sheet, we define a plane tiling as a covering of the entire plane, without any gaps or overlaps, using identical geometric shapes that can be rotated or reflected to one another. Usually, it is sufficient to cover a small portion of the plane with a particular pattern that we see can be extended to cover the entire plane.

In the majority of today’s problems, one only needs to create a pattern for a small section of the plane, which can then be extended to cover the entire plane through repetition. Such tilings are referred to as periodic. Finding a tiling of the plane using identical shapes that is not periodic, meaning the pattern never repeats itself, has been a long-standing open problem. However, this problem was solved in an article published in March 2023 by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. The idea of their solution is the following diagram from the article:

The hard part is to prove that this tiling is aperiodic.