Incredibly, this shape can cover an infinite area with a pattern that never repeats. A small sample of that pattern is shown above.

The 13-sided tile also called “the hat” can fully cover an infinite plane without gaps. The tile’s pattern never repeats, making it an example of a long-sought shape known as an “Einstein.”  This hat was the first true example of an “einstein.” That’s the name for a special type of shape that can tile a plane. Much like common ceramic wall or floor tile, it can cover an entire surface with no gaps or overlaps. It can even tile a plane that’s infinitely big. But the magic is that an Einstein tile does this with a pattern that never repeats.

Scientifically, they are described as being aperiodic shapes. That means the Einstein or hat shape cannot form a repeating pattern.