For years, mathematicians have been fascinated by the concept of "aperiodic tiling" — creating infinite, non-repetitive patterns by cleverly arranging geometric shapes. However, until recently, it was unclear whether this could be achieved with a single shape alone. Now, after years of dedicated research, mathematicians have finally discovered a unique monomeric shape that can completely cover an entire surface without ever repeating a pattern. This seemingly simple breakthrough holds vast potential applications in fields such as materials science and decorative arts.

Recently, a major discovery by researchers has unveiled the mystery of a fascinating geometric shape called the "hat" - a shape that had previously only existed in theoretical concepts. The "hat" is a polygon composed of 13 sides, which can complete a plane tiling without repeating itself. More amazingly, the "hat" belongs to a non-periodic monotile tiling, which means it does not have any translational symmetry, and its tiling pattern will never appear in a periodic repetition.

As is well known, the Penrose tiling is a typical non-periodic tiling form, which uses two different geometric shapes to construct a non-periodic pattern. In contrast, the "hat" tiling only requires the use of a single shape called "Einstein" - in German, "Einstein" means "a stone". For this reason, the "hat" pattern is a true non-periodic monomeric tiling.

The "hat" shape is a porous structure composed of eight kite-shaped polygons connected along their edges. Until recently, the existence of aperiodic monomeric tiling was just a theoretical conjecture. However, a research team led by mathematician David Smith and his team have successfully proven the existence of this shape in a preprint paper published online this month