Penrose tiling

Definition

Penrose tiling refers to a non-periodic tessellation created by Roger Penrose that uses a set of shapes to cover a plane without repeating patterns. This unique arrangement of shapes produces a structure that exhibits a form of order while avoiding periodicity, making it an interesting subject in the study of symmetry and mathematics.

5 Must Know Facts For Your Next Test

1. Penrose tiling can be created using two shapes: a kite and a dart, which fit together to form complex patterns without repeating.
2. The discovery of Penrose tiling challenged traditional notions of symmetry in mathematics, as it showed that non-periodic patterns could still exhibit local symmetries.
3. One of the fascinating features of Penrose tiling is that it can fill a space indefinitely without ever repeating the same pattern.
4. Penrose tilings have applications in various fields, including material science, where they help understand quasicrystals and their unique properties.
5. The tiling has been used as a model for understanding how certain natural structures, like crystals, can form in non-periodic arrangements.

Review Questions

• How does Penrose tiling demonstrate the concept of aperiodicity in mathematical patterns?
  • Penrose tiling exemplifies aperiodicity through its unique arrangement of shapes that never forms a repeating pattern. The use of specific tiles, such as the kite and dart, allows for infinite coverage of a plane while maintaining local symmetries. This characteristic highlights how non-repeating patterns can still possess order, showcasing the concept of aperiodicity in mathematics.
• What are the implications of Penrose tiling on our understanding of symmetry in mathematics?
  • The implications of Penrose tiling on the understanding of symmetry are significant, as it introduced the idea that symmetrical structures can exist without periodicity. Traditional symmetry concepts often relied on repetition; however, Penrose’s work illustrated that patterns could maintain local symmetries while being globally non-repeating. This challenges mathematicians to broaden their perspective on symmetry beyond just periodic arrangements.
• Evaluate how the principles observed in Penrose tiling relate to the study and creation of quasicrystals in materials science.
  • The principles observed in Penrose tiling relate closely to the study and creation of quasicrystals by providing a mathematical framework for understanding non-periodic structures. Quasicrystals exhibit ordered arrangements similar to Penrose tilings but lack translational symmetry. This connection has allowed researchers to explore new materials with unique properties based on the insights gained from Penrose's work, ultimately influencing advances in material science and technology.

Related terms

Tessellation:

A pattern made up of one or more geometric shapes that fit together without any gaps or overlaps, covering a surface completely.

Quasicrystal: A structure that is ordered but not periodic, which shares similarities with Penrose tiling in its non-repeating pattern and symmetry properties.

Aperiodicity: The characteristic of a pattern that does not repeat itself at regular intervals, which is a fundamental aspect of Penrose tiling.