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.
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.