TIL about the inner magic of Penrose tiling
Today I learned and obsessed over Penrose tiling.
A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and aperiodic means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.
Reading about types of symmetries I run into an explanation of the Penrose patterns of tiles and a demonstration of why it never repeats.
- Why Penrose Tiles Never Repeat, a good short explanation with clear visuals.
- In other words, a lack of repetition mathematically means a lack of transactional symmetry.
- The ratio at which the two tiles of the Penrose pattern appear in any particular direction is the golden ratio, which is an irrational number and therefore lacks repetition (by definition).
- The Penrose tiling is a pentagrid, and other variations of the tiling could be generated by shifting the lines of the grid.
- Additionally similar grids that generate Penrose-like tiling can be created by increasing the number of lines: heptagrid, octagrid, nanogrid, decagrid, etc
- Here’s an interactive playground site: Pattern Collider