Summary:
This video discusses the discovery of a shape known as the "Einstein tile" or the "Hat," which is a non-periodic tiling shape that can cover a surface without repeating predictably. The video explains the history of searching for aperiodic tiles, the discovery of the Hat, and its mathematical proof of non-periodicity. It also mentions the discovery of another aperiodic tile called the "Specter." The video discusses potential practical applications of aperiodic patterns in materials science and engineering. Overall, it highlights the exciting breakthrough in mathematics and its potential implications.
1. The episode is sponsored by Brilliant.
2. The discussion revolves around the concept of a shape that can completely cover a surface but never in a predictable repeating pattern.
3. Mathematicians have been searching for such a shape for over 50 years.
4. In March of the current year, a shape that fulfills this criteria was finally found.
5. The discovery has generated excitement within the mathematics community, even leading to a festival in its honor.
6. The shape discovered is referred to as aperiodic, meaning it cannot be tiled in a repeating pattern.
7. The shape, discovered by a retired printing technician named David Smith, is called a "polykite" as it's made up of eight kite shapes.
8. The shape was discovered through a process of trial and error, with Smith initially unable to tile it periodically.
9. The discovery was shared with computer scientist Craig Kaplan, who suggested it could be an answer to the Einstein problem.
10. The shape was named the "Hat" due to its appearance.
11. The existence of the shape was mathematically proven by a team of mathematicians and a software engineer.
12. The proof involved a technique called the unique hierarchy method, which relies on the hierarchical nature of a tiling.
13. The Hat tiling was shown to be periodic by demonstrating that it must be built in a unique hierarchy.
14. The researchers found another aperiodic monotile, called the "Turtle".
15. The Hat and Turtle were found to be related, with the lengths of their sides adjustable to morph one into the other.
16. This discovery gave an infinite continuum of periodic monotiles.
17. The result was published in March 2023, just five months after the discovery of the Hat.
18. The discovery was met with excitement in the math world, but some people criticized the shapes for using their reflections.
19. However, five days after the paper was published, Dave found another aperiodic monotile that doesn't need its mirror reflection.
20. This new shape, called the "Specter", tiles the plane periodically, but only when its sides are wiggled.
21. The practical applications of these periodic patterns are being explored, with potential uses in material science and engineering.
22. The video concludes with a suggestion for anyone wanting to tile their bathroom floor with these patterns.