Penrose Tilings: A Journey into Aperiodic Patterns

Penrose tilings are a fascinating class of aperiodic tilings that cover the plane in a non-repeating yet orderly fashion. Discovered by the mathematician and physicist Sir Roger Penrose in the 1970s, these tilings are an important part of geometry, mathematical physics, and crystallography.

Unlike regular periodic tilings (like those made of squares or triangles), Penrose tilings exhibit quasi-periodicity, meaning they never repeat exactly but display local patterns that recur with no global repetition. They provide insight into the mathematical foundations of symmetry and have even been observed in the physical world through quasicrystals.

1. What Is a Penrose Tiling?

A Penrose tiling is a non-periodic covering of the plane using a finite set of prototiles (basic shapes). Penrose constructed several versions of these tilings, but the two most famous are:

• P2 Tiling (Kites and Darts): This tiling uses two shapes:
  • The kite, which resembles a wide diamond.
  • The dart, which is a thin, arrow-like shape.
• P3 Tiling (Rhombuses): This tiling uses two rhombic (diamond-shaped) tiles with angles in the ratio of the golden ratio.

These tilings enforce aperiodicity through matching rules that constrain how the tiles can fit together.

2. Mathematical Structure of Penrose Tilings

To understand the beauty of Penrose tilings, we delve into their mathematical structure:

• Aperiodicity: Penrose tilings are non-periodic, meaning no translational symmetry exists. If you shift the tiling in any direction, it will never match up with itself exactly.
• Local Isomorphism Theorem: Any two Penrose tilings share the same local patterns, despite being globally different. This means you will always find similar patches in different Penrose tilings.
• Inflation and Deflation: Penrose tilings can be generated by a recursive process called substitution. This involves breaking tiles into smaller versions (deflation) or combining them into larger ones (inflation).

The ratio governing the size of these substitutions is the golden ratio:


This constant is deeply connected to the geometry of Penrose tilings.

3. Construction Methods

There are several ways to construct Penrose tilings:

• Substitution Method: Start with a few tiles and recursively subdivide them according to specific rules.
• Cut-and-Project Method: This involves projecting a higher-dimensional lattice (from four-dimensional space) onto the plane to generate a Penrose tiling.
• Matching Rules: Each tile has marked edges that must match in specific ways to enforce aperiodicity.

For example, in the kite-and-dart tiling, the tiles must meet edge-to-edge in particular ways to prevent periodic patterns from forming.

4. Golden Ratio and Penrose Tilings

The golden ratio plays a central role in Penrose tilings. Many properties of the tiling relate to , including:

• The ratio of areas of the kite and dart is .
• The angles in the rhombus tiling are related to : , , and .
• The frequency of occurrence of tiles in a large Penrose tiling converges to the golden ratio.

The golden ratio appears naturally because of its unique algebraic properties:


5. Symmetry in Penrose Tilings

Penrose tilings exhibit a form of symmetry called five-fold rotational symmetry. Although no periodic tiling of the plane can have five-fold symmetry (by the crystallographic restriction theorem), Penrose tilings approximate this symmetry locally.

For the P3 rhombus tiling, rotating the pattern by leaves the structure largely unchanged, even though the pattern never repeats exactly.

6. Applications of Penrose Tilings

Penrose tilings are not just mathematical curiosities—they have practical implications:

• Quasicrystals: In 1984, physicist Dan Shechtman discovered materials with aperiodic atomic structures resembling Penrose tilings. This led to the Nobel Prize in Chemistry (2011).
• Computer Science: Penrose tilings are used in algorithms for generating non-repetitive textures and data encryption.
• Art and Design: Penrose patterns appear in architecture and visual design, including works by M.C. Escher.
• Mathematical Physics: Penrose tilings relate to models of aperiodic order and are used to understand wave diffraction patterns.

7. Mathematical Questions and Open Problems

Several deep mathematical questions remain regarding Penrose tilings:

• Tilings in Higher Dimensions: What happens when you extend Penrose tilings to three or more dimensions?
• Uniqueness of Patterns: How many distinct Penrose tilings exist?
• Dynamical Systems: How do Penrose tilings connect with ergodic theory and symbolic dynamics?
• Spectral Properties: What are the spectral characteristics of physical systems modeled by Penrose tilings?

8. Penrose Tilings and Tiling Theory

Penrose tilings belong to the broader study of tiling theory, which explores how shapes can fill space. Key concepts include:

• Aperiodic Sets of Prototiles: Penrose tilings demonstrate that a finite set of shapes can enforce non-periodicity.
• Combinatorial Properties: The combinatorics of Penrose tilings provide insight into geometric group theory and topological spaces.
• Decidability Issues: Determining whether a given patch extends to a full tiling is undecidable in the general case.

9. Summary

• Penrose tilings are aperiodic coverings of the plane with unique local patterns that never repeat globally.
• They are constructed using substitution rules, cut-and-project methods, or matching conditions.
• The golden ratio governs many geometric and combinatorial properties of Penrose tilings.
• Penrose tilings have applications in quasicrystals, computer science, and mathematical physics.
• They raise deep questions in geometry, combinatorics, and dynamical systems.

10. References

• Wikipedia: Penrose Tiling
• Penrose, Roger. The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (1989).
• Grünbaum, Branko & Shephard, Geoffrey C. Tilings and Patterns (1987).
• AoPS Wiki: Penrose Tiling
• Shechtman, Dan. Quasicrystals: The Nobel Prize in Chemistry (2011).