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:
Penrose tilings
are aperiodic, meaning they don’t have a repeating pattern that can be
translated to cover the entire plane.
The two basic
shapes used in Penrose tilings are often called the “kite” and the
“dart,” or sometimes “fat rhombus” and “thin
rhombus”.
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.
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).
The proportions
of the tiles are related to the golden ratio (approximately
1.618), a mathematical constant found in nature and art.
Penrose tilings
are related to quasicrystals, which are materials that have ordered structures
but lack the translational symmetry of typical crystals.
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.
The Darb-i Imam
shrine in Iran, built in the 15th century, is one example where a near-perfect
Penrose tiling was found.
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.
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.
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.
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