Below is a short summary and detailed review of this video written by FutureFactual:
Understanding Penrose Tilings and the Pentagrid: How Quasi-Periodic Patterns Never Repeat
Penrose tilings are non-repeating, quasi-periodic patterns formed by five-direction ribbons derived from a pentagrid. This video reveals how a hidden pentagrid underpins Penrose tilings, how ribbons align along grid intersections to build the tiling, and why the ratio of wide to thin tiles is governed by the irrational golden ratio, guaranteeing no full repetition. It also nods to variations with other grids and points to an interactive tool for exploring these patterns.
Introduction: What Penrose Tilings Are and Why They Don’t Repeat
Penrose tilings are famous for their beauty and their stubborn non-repetition. They look almost periodic, but slide or shift will never recreate the exact pattern. The video explains that these are quasi-periodic patterns, and it introduces a hidden structure called the pentagrid that helps make sense of them.
"Penrose tilings never repeat themselves," - MinutePhysics
The Pentagrid Concept: Five Sets of Intersecting Lines
A pentagrid is a grid made of five sets of equally spaced parallel lines, each set rotated relative to the others by 36 or 72 degrees. The ribbons seen in Penrose tilings correspond to these line intersections, forming five infinite, parallel ribbon directions that together encode the tiling.
"The ribbons form a pentagrid, and Penrose tilings are made from these five directions of ribbons," - A presenter (MinutePhysics)
From Grids to Tilings: Building the Tiles
To construct a Penrose tiling, you place a tile at every intersection of the pentagrid, oriented so its sides are perpendicular to the two crossing lines. As you move along a line, the tiles align in sequence, letting you slide them into ribbons. Doing this for all lines in all directions yields the tiling pattern. You can also simply place a tile at each intersection and slide along the grid lines to obtain the same Penrose tiling.
"You draw a tile at every intersection and slide them along the grid lines to form a Penrose tiling," - MinutePhysics
Why the Pattern Doesn’t Repeat: The Golden Ratio
By analyzing the spacing of the lines, the video shows that the ratio of wide to thin tiles is linked to the spacing of lines at 72 and 36 degrees. Basic trigonometry reveals that this ratio equals the golden ratio, which is irrational. Therefore, the tiling cannot repeat in any finite direction; if it did, the ratio would have to be rational, which it is not. The nested structure of intersections ensures the non-repetition persists as you go along any ribbon, with the long-run proportion of tile types converging to the golden ratio.
"The ratio of wide tiles to thin tiles is the golden ratio, which is irrational," - MinutePhysics
Extensions and Interactive Exploration
Beyond the classic Penrose tiling, one can create pentagrid-like patterns with other grid families such as hepta, octa, deca, and more. The ribbons and grid lines reveal quasi-periodic patterns across these grids as well. An interactive website by Aish lets you highlight grid lines, map them to tiles, and experiment with different colorings, enabling exploration of Penrose-like tilings and related patterns.
"Other grids can produce Penrose-like tilings," - MinutePhysics
Takeaways
Quasi-periodic tilings like Penrose tilings are not random; they are structured by pentagrids and governed by irrational ratios. The golden ratio explains why repetition does not occur, while the pentagrid framework makes the construction and non-repetition intuitive. The video also points to hands-on tools to experiment with these fascinating geometric patterns and to see how different grids influence the resulting tilings.
"Sometimes there are slightly more wide tiles for every 10 thin ones, in a way that is perfectly predicted by the golden ratio, but never repeats," - MinutePhysics
