Penrose Tiles — Definition, Formula & Examples
Penrose tiles are a small set of shapes (usually two) that can cover an entire flat surface with no gaps or overlaps, yet the pattern never repeats. They were discovered by mathematician Roger Penrose in the 1970s.
A Penrose tiling is an aperiodic tiling of the Euclidean plane constructed from a finite set of prototiles such that the tiling admits no translational symmetry. The most common versions use two quadrilateral prototiles — either the 'kite' and 'dart' or two rhombi with interior angles related to multiples of (36° and 72°) — subject to specific matching rules that enforce aperiodicity.
How It Works
You start with two tile shapes and a set of matching rules (often shown as colored edges or arrows) that dictate which edges can be placed next to each other. Following these rules, you can cover any region of the plane, but no matter how far you extend the tiling, you will never find a section that you can shift to perfectly overlay another section. Despite having no translational symmetry, Penrose tilings do exhibit local five-fold rotational symmetry, which is impossible in periodic tilings. The ratio of kites to darts (or thick to thin rhombi) in an infinite Penrose tiling approaches the golden ratio .
Worked Example
Problem: In a Penrose rhombus tiling, the two tiles are a thick rhombus (interior angles 72° and 108°) and a thin rhombus (interior angles 36° and 144°). If a large patch contains 100 thick rhombi, approximately how many thin rhombi does it contain?
Step 1: In an infinite Penrose tiling, the ratio of thick to thin rhombi equals the golden ratio.
Step 2: Solve for the number of thin rhombi given 100 thick rhombi.
Step 3: Since we are estimating a large patch (not an infinite tiling), the count rounds to approximately 62 thin rhombi.
Answer: A patch with 100 thick rhombi contains approximately 62 thin rhombi, reflecting the golden ratio relationship between the two tile types.
Why It Matters
Penrose tilings provided the theoretical framework for understanding quasicrystals, a discovery that earned Dan Shechtman the 2011 Nobel Prize in Chemistry. They also appear in architecture, art, and computer graphics wherever non-repeating but orderly patterns are desired.
Common Mistakes
Mistake: Assuming Penrose tiles are random because they are non-repeating.
Correction: Aperiodic does not mean random. Penrose tilings follow strict matching rules and exhibit long-range order, including five-fold symmetry. They are highly structured — just never periodic.