Why Penrose Tiles Never Repeat  Summary and Q&A
TL;DR
Penrose tilings are quasiperiodic patterns that never repeat themselves, and the key to understanding them lies in a hidden pattern called a pentagrid.
Key Insights
 ❓ Penrose tilings are quasiperiodic patterns that have fascinated mathematicians for many years.
 The pentagrid is a hidden pattern within Penrose tilings and consists of five sets of parallel lines.
 By using the pentagrid, one can create Penrose tilings by drawing tiles at the intersections of the lines.
 🥳 The ratio of thin tiles to wide tiles in Penrose tilings is determined by the golden ratio, which is an irrational number.
 🤔 The pentagrid allows for the calculation of the ratio of thin tiles to wide tiles along any ribbon in a Penrose tiling.
 ❓ Penrose tilings can also be created using different grids, such as the heptagrid or Deca Grid.
 💁 The patterns formed by Penrose tilings are not random but follow specific geometric rules.
Transcript
these incredibly pretty geometric patterns are Penrose tilings and if you've heard anything about them it's probably that they never repeat themselves I mean they look pretty similar all over and there are patches that are perfect matches but if you slide the whole thing over and around it will never completely line up with itself again patterns li... Read More
Questions & Answers
Q: What are Penrose tilings?
Penrose tilings are geometric patterns that never repeat themselves, making them quasiperiodic.
Q: What is the pentagrid?
The pentagrid is a hidden pattern within Penrose tilings, consisting of five sets of parallel lines. It helps in understanding how the tilings are created.
Q: How can the pentagrid be used to create Penrose tilings?
By drawing tiles at the intersections of the lines in the pentagrid, one can create Penrose tilings. The orientation of the tiles is determined by the perpendicularity to the two intersecting lines.
Q: Why do Penrose tilings never repeat?
Penrose tilings never repeat because the ratio of thin tiles to wide tiles in the ribbons of tiles follows the irrational number known as the golden ratio. If the pattern were to repeat, the ratio would have to be rational.
Summary & Key Takeaways

Penrose tilings are geometric patterns that do not repeat and are known as quasiperiodic patterns.

The key to understanding Penrose tilings is the pentagrid, which consists of five sets of parallel lines.

By using the pentagrid, one can create Penrose tilings by drawing tiles at the intersections of the lines.