AI – ‘Penrose tiles are a set of two shapes that can cover a plane without repeating, known as an aperiodic tiling. They were discovered by mathematician Roger Penrose and exhibit a unique type of symmetry and structure. Despite their lack of translational symmetry, they possess five-fold rotational symmetry and reflection symmetry.
Here’s a more detailed explanation:
- Aperiodic Tiling:
Penrose tilings are aperiodic, meaning they don’t have a repeating pattern that can be translated to cover the entire plane.
- Two Shapes:
The two basic shapes used in Penrose tilings are often called the “kite” and the “dart,” or sometimes “fat rhombus” and “thin rhombus”.
- Matching Rules:
These shapes have specific matching rules that dictate how they can be placed together to form the tiling. These rules ensure that the tiling remains aperiodic.
- Five-Fold Symmetry:
Although non-periodic, Penrose tilings often exhibit five-fold rotational symmetry around certain points, meaning the pattern looks the same when rotated by 72 degrees (360/5).
- Golden Ratio:
The proportions of the tiles are related to the golden ratio (approximately 1.618), a mathematical constant found in nature and art.
- Quasicrystals:
Penrose tilings are related to quasicrystals, which are materials that have ordered structures but lack the translational symmetry of typical crystals.
- Construction:
Penrose tilings can be constructed in various ways, including using a “pentagrid” or through substitution rules.
‘Are Penrose Tiles found in Islamic architecture?’
AI – ‘Yes, Penrose tiles, or rather, patterns highly similar to Penrose tilings, have been found in Islamic architecture, particularly in the intricate geometric designs known as “girih”. Studies suggest that these designs, predating Penrose’s work by centuries, demonstrate a sophisticated understanding of aperiodic tiling systems.
Elaboration:
Islamic artisans used “girih” tiles, geometric shapes with specific matching rules, to create complex, non-repeating patterns on buildings.
Centuries later, British mathematical physicist Roger Penrose described similar aperiodic tilings, which were initially believed to be a new mathematical concept.
Research by Peter Lu and others indicates that Islamic artisans had already mastered the principles of these tilings, using them in their architectural designs.
- Example:
The Darb-i Imam shrine in Iran, built in the 15th century, is one example where a near-perfect Penrose tiling was found.
- Implications:
This discovery suggests that Islamic mathematicians and artisans possessed a deeper understanding of geometry and tiling systems than previously acknowledged, with some designs even anticipating the concept of quasicrystals.
‘Are Penrose Tiles on the walls of the Alhambra?’
AI – ‘Yes, the walls of the Alhambra Palace in Granada, Spain, feature intricate geometric tiling patterns known as girih patterns
. Some of these patterns share similarities in their symmetries with the aperiodic Penrose tilings, a concept developed in the West much later, in the 1970s by Roger Penrose. This suggests a potentially deeper mathematical understanding among the medieval Islamic artisans than previously believed.
The geometric designs at the Darb-i Imam shrine in Iran, which predate Penrose’s work, are considered a “nearly perfect” example of Penrose tiling. Although not explicitly stated as being on the Alhambra’s walls, the discovery of such patterns in other medieval Islamic architecture suggests a connection to the sophisticated geometric designs in the Alhambra.
The artisans who created the Alhambra’s tilework mastered all 17 possible plane symmetry groups, which highlights their advanced mathematical knowledge. The Alhambra’s geometric patterns also symbolized the infinite nature of life and God, reflecting Islamic beliefs.’
‘Have Penrose Tiles been found in Kazakhstan and elsewhere?’
AI . ‘Yes, Penrose tiles, or rather patterns very similar to Penrose tiles, have been discovered in Islamic architecture, including in Uzbekistan (which was part of the Timurid Empire). These patterns, known as “girih” in Islamic art, predate the formal discovery of Penrose tiling by Roger Penrose in the 1970s. While Penrose is credited with mathematically defining and analyzing this aperiodic tiling, the concept was already present in the geometric designs of Islamic artisans centuries earlier.
More Details:
Islamic art, particularly in the Middle East and North Africa, has a rich history of using intricate geometric patterns. These patterns, often found in mosques and other religious buildings, sometimes incorporate Penrose-like tiling.
- Uzbekistan Discovery:
Peter Lou (or Lu), a physicist, famously discovered Penrose tiling on a 14th-century Madrasa (Islamic school) in Uzbekistan during a visit. He published his findings in 2007, showing that the pattern predated Penrose’s work by centuries.
- Girih Tiles:
The specific patterns found in Islamic art are often referred to as “girih” tiles. These tiles, like Penrose tiles, exhibit aperiodicity, meaning they can cover a surface without repeating.
- Beyond Uzbekistan:
Similar patterns have been found in other parts of the Islamic world, including Iran and Turkey, demonstrating the widespread use of these designs.
The Topkapi Palace Museum in Istanbul houses a 15th-century Timurid-Turkmen scroll with a complex pattern that can be broken down into the shapes found in Penrose tiles Science News.’
See also – A Mysterious Design That Appears Across Millennia | Terry Moore | TED: https://www.youtube.com/watch?v=XqUOgqlZ8bc
Roger Penrose – Forbidden crystal symmetry in mathematics and architecture: https://www.youtube.com/watch?v=th3YMEamzmw