# How a Hobbyist Solved a 50-Year-Old Math Problem (Einstein Tile)

TLDRA retired printing technician, David Smith, discovered a shape that can tile a surface without repeating patterns, solving a 50-year-old math problem known as the Einstein Tile. The polykite shape, dubbed 'the hat,' along with another he found called 'the turtle,' introduced the possibility of an infinite continuum of aperiodic monotiles. Despite initial skepticism, Smith's findings were confirmed with mathematical proofs and have potential applications in material science, offering a new direction in the study of aperiodic patterns.

### Takeaways

- 🧩 The discovery of a shape that can cover a surface without repeating patterns is a 50-year-old unsolved math problem now solved by a hobbyist.
- 🎉 A retired printing technician named David Smith, who calls himself a 'shape hobbyist,' found the first ever aperiodic monotile, now known as the 'hat'.
- 🛠 The 'hat' is a polykite, a shape made up of eight kite shapes, and was found to tile non-periodically by hand and later mathematically proven.
- 🔍 The search for a single aperiodic tile was dubbed the 'Einstein Tile' quest, symbolizing a significant leap in understanding tiling patterns.
- 📈 The discovery was so impactful that a festival was held in its honor, celebrating the breakthrough in mathematics.
- 🔬 The proof of the 'hat' being an aperiodic monotile involved the unique hierarchy method, demonstrating its non-repeating tiling nature.
- 📚 The process of proving the 'hat' involved collaboration between mathematicians and computer scientists, showcasing interdisciplinary efforts.
- 🐢 Following the 'hat,' Smith discovered another aperiodic monotile called the 'turtle,' and later, a shape that does not require mirror reflection, named 'spectre'.
- 🔗 The 'turtle' and 'hat' are part of an infinite continuum of aperiodic monotiles, indicating a vast potential for new discoveries.
- 🏡 While the immediate application might be seen in tiling designs, aperiodic patterns are being explored for their properties in material science, potentially leading to new materials.
- 🌐 The story of David Smith highlights that anyone, regardless of formal training, can make significant contributions to the field of mathematics.

### Q & A

### What is the significance of the discovery of the Einstein Tile?

-The Einstein Tile is significant because it represents the first known shape that can completely cover a surface without ever forming a predictable repeating pattern, solving a 50-year-old mathematical problem and sparking excitement in the mathematics community.

### What is the difference between periodic and nonperiodic tiling?

-Periodic tiling refers to a pattern that can be copied and translated across itself to form a repeating pattern, exhibiting translational symmetry. Nonperiodic tiling, on the other hand, cannot form such a repeating pattern no matter how the tiles are arranged.

### What is the unique hierarchy method used in the proof of the hat's aperiodicity?

-The unique hierarchy method relies on the hierarchical or fractal nature of a tiling, where each smaller tile belongs to one and only one unique larger tile, ensuring non-periodicity by preventing the tiling from aligning perfectly upon translation.

### Who discovered the first aperiodic monotile, and what was his background?

-David Smith, a retired printing technician who describes himself as a shape hobbyist, discovered the first aperiodic monotile. His discovery was made by playing with shapes on his computer.

### How did David Smith come to collaborate with Craig Kaplan?

-David Smith emailed Craig Kaplan, a computer scientist, on November 20th, 2022, to inquire whether his discovered shape could be an answer to the Einstein problem, leading to their collaboration.

### What is the name of the first aperiodic monotile discovered by David Smith, and why was it named so?

-The first aperiodic monotile discovered by David Smith is called the 'hat' because both Smith and Craig Kaplan's family members thought it resembled a shirt, and when turned upside down, a hat.

### What is the importance of the mathematical proof in confirming the hat as the Einstein Tile?

-The mathematical proof is crucial as it confirms that the hat tiling is aperiodic, ensuring that the pattern never repeats, even on an infinite plane, which is necessary to validate it as the sought-after Einstein Tile.

### How did the discovery of the hat lead to the discovery of the turtle and the realization of an infinite continuum of aperiodic monotiles?

-The discovery of the turtle came shortly after the hat when Joseph Samuel Meyers had the insight that the hat and the turtle were related shapes. By adjusting the lengths of the sides, they could morph one into the other, revealing an infinite continuum of aperiodic monotiles.

### What is the difference between the hat and the turtle in terms of their aperiodicity?

-The hat and the turtle are both aperiodic monotiles, but the turtle is considered a stronger example because it does not require its mirror reflection to tile the plane aperiodically, unlike the hat, which is considered weakly aperiodic.

### What is the 'spectre' tile, and why was it named as such?

-The 'spectre' is an undisputable Einstein Tile that does not require its mirror reflection to tile the plane aperiodically. It was named 'spectre' by David Smith due to the feeling of a presence or spirit during the time of its discovery, which coincided with the death of his oldest brother.

### What are some potential practical applications of aperiodic patterns like the Einstein Tile?

-Aperiodic patterns, including the Einstein Tile, are being explored in material science and engineering for their unique properties. They may offer more elasticity and less prone to failure compared to periodic lattices, and their applications are still being discovered and developed.

### Outlines

### 🧩 Discovery of the Aperiodic Monotile

The script introduces the groundbreaking discovery of a shape that can tile a surface without repeating patterns, known as an aperiodic monotile. Mathematicians had been searching for such a shape for over 50 years. The script discusses the concept of nonperiodic tiling, explaining how periodic tilings have translational symmetry, while aperiodic ones do not. The first set of aperiodic tiles was found in 1964 with 20,426 different tiles, which was later reduced to just two by Roger Penrose. The script then introduces the 'Ein Stein Tile' and the recent discovery by a retired printing technician named David Smith, who found a simple shape, the polykite, that could not tile periodically.

### 📉 Mathematical Proof of Aperiodicity

This section delves into the process of proving that the newly discovered shape, named 'the hat', is indeed an aperiodic monotile. Dave and Craig enlisted the help of Chaim Goodman-Strauss and Joseph Samuel Meyers to provide a mathematical proof. The proof relies on the unique hierarchy method, which is based on the hierarchical or fractal nature of a tiling. The team identified four recurring clusters of hats, which they replaced with 'metatiles' to simplify the proof. They demonstrated that these metatiles formed a unique hierarchy, ensuring non-periodicity. The proof was completed in just eight days, confirming the hat as the sought-after Einstein Tile.

### 🐢 The Continuum of Aperiodic Monotiles

Following the proof of the hat's aperiodicity, Dave discovered another aperiodic monotile, the 'turtle'. This led to Joseph's realization that the hat and the turtle were related, and by adjusting side lengths, one could morph into the other, creating an infinite continuum of aperiodic monotiles. This was a significant development, as it provided not just one, but infinitely many aperiodic monotiles. However, some controversy arose as some argued that the hat's use of reflection meant it was not a true single tile. Dave then found a tile that did not require reflection for aperiodicity, which he called the 'spectre'.

### 🛠 Practical Applications and Future Prospects

The script concludes by discussing the potential practical applications of aperiodic patterns, such as the development of new materials with aperiodic lattices that may be more elastic and less prone to failure. It also highlights the importance of the discovery for the field of mathematics and encourages viewers to explore the subject further through a geometry course on Brilliant. The course promises to teach about various tilings and their real-world applications, offering an interactive learning experience. The script ends with a promotional offer for Brilliant, encouraging viewers to start their own journey into mathematical discovery.

### Mindmap

### Keywords

### 💡Nonperiodic Tiling

### 💡Aperiodic Tiles

### 💡Einstein Tile

### 💡Polykite

### 💡Unique Hierarchy Method

### 💡Metatiles

### 💡Aperiodic Monotile

### 💡Turtle

### 💡Spectre

### 💡Weakly Aperiodic

### 💡Practical Applications

### Highlights

A hobbyist has solved a 50-year-old math problem, discovering a shape that can cover a surface without repeating patterns, known as the Einstein Tile.

The discovery of the Einstein Tile has caused a sensation in the mathematics community, leading to a celebratory festival.

The concept of nonperiodic tiling, where shapes cover a plane without translational symmetry, is central to the Einstein Tile.

The first set of aperiodic tiles was found in 1964, but it wasn't until 2022 that a single shape was discovered to have this property.

David Smith, a retired printing technician, found the polykite shape that couldn't tile periodically, leading to the Einstein Tile discovery.

Craig Kaplan, a computer scientist, confirmed the potential of the polykite shape to be an aperiodic monotile using computer software.

The naming of the shape as 'the hat' was inspired by its appearance and the collaborative effort to understand its properties.

A mathematical proof using the unique hierarchy method was developed to confirm the hat's aperiodicity.

The team expanded to include Chaim Goodman-Strauss and Joseph Samuel Meyers, who helped derive the proof for the hat's aperiodicity.

The proof's technique relies on the hierarchical nature of tiling, ensuring that smaller tiles belong to one unique larger tile.

Dave found another aperiodic monotile, the turtle, during the proof development process.

Joseph Meyers discovered a morphing continuum between the hat and the turtle, suggesting an infinite number of aperiodic monotiles.

The publication of the discovery in March 2023 was celebrated, but also sparked debate about the definition of a single tile.

Dave found a new aperiodic monotile, the spectre, which does not require its mirror reflection to tile aperiodically.

The spectre was adjusted to be weakly aperiodic, solving the issue of tiling with its mirror reflection.

Aperiodic patterns have practical applications in material science, with potential for new elastic and durable materials.

The story of the Einstein Tile emphasizes the accessibility of mathematics and the potential for anyone to contribute to its discoveries.

Brilliant's geometry course is recommended for those interested in exploring the world of tilings and mathematical concepts.