Five tiles
The five shapes of the tiles are:
a regular decagon with ten interior angles of 144°;
an elongated (irregular convex) hexagon with interior angles of 72°, 144°, 144°, 72°, 144°, 144°;
a bow tie (non-convex hexagon) with interior angles of 72°, 72°, 216°, 72°, 72°, 216°;
a rhombus with interior angles of 72°, 108°, 72°, 108°; and
a regular pentagon with five interior angles of 108°.
All sides of these figures have the same length; and all their angles are multiples of 36° (π/5 radians). All of them, except the pentagon, have bilateral (reflection) symmetry through two perpendicular lines. Some have additional symmetries. Specifically, the decagon has tenfold rotational symmetry (rotation by 36°); and the pentagon has fivefold rotational symmetry (rotation by 72°).
Girih
Girih are lines (strapwork) that decorate the tiles. The tiles are used to form girih patterns, from the Persian word گره, meaning "knot". In most cases, only the girih (and other minor decorations like flowers) are visible rather than the boundaries of the tiles themselves. The girih are piece-wise straight lines that cross the boundaries of the tiles at the center of an edge at 54° (3π/10) to the edge. Two intersecting girih cross each edge of a tile. Most tiles have a unique pattern of girih inside the tile that are continuous and follow the symmetry of the tile. However, the decagon has two possible girih patterns one of which has only fivefold rather than tenfold rotational symmetry.
Mathematics of girih tilings
In 2007, the physicists Peter J. Lu and Paul J. Steinhardt suggested that girih tilings possessed properties consistent with self-similar fractal quasicrystalline tilings such as Penrose tilings, predating them by five centuries.
This finding was supported both by analysis of patterns on surviving structures, and by examination of 15th-century Persian scrolls. However, we have no indication of how much more the architects may have known about the mathematics involved. It is generally believed that such designs were constructed by drafting zigzag outlines with only a straightedge and a compass. Templates found on scrolls such as the 97-foot- (29.5 metres) long Topkapi Scroll may have been consulted. Found in the Topkapi Palace in Istanbul, the administrative center of the Ottoman Empire and believed to date from the late 15th century, the scroll shows a succession of two- and three- dimensional geometric patterns. There is no text, but there is a grid pattern and color-coding used to highlight symmetries and distinguish three-dimensional projections. Drawings such as shown on this scroll would have served as pattern-books for the artisans who fabricated the tiles, and the shapes of the girih tiles dictated how they could be combined into large patterns. In this way, craftsmen could make highly complex designs without resorting to mathematics and without necessarily understanding their underlying principles.
This use of repeating patterns created from a limited number of geometric shapes available to craftsmen of the day is similar to the practice of contemporary European Gothic artisans. Designers of both styles were concerned with using their inventories of geometrical shapes to create the maximum diversity of forms. This demanded a skill and practice very different from mathematics.
Periodicity
Most of the entries used in Islamic architecture are periodic: there are repeating unit cells in the same direction in a lattice. The patterns found in some entrances can not be repeated to lay the entire plane. The patterns in the Darb-ı Imam tract built in Isfahan in 1453 are aperiodic, that is, they have a structure which is not regular and repetitive. [one]
Can be converted into Penrose Karones of Girih tiles. Peninsula Karoly of Girih was discovered about 5 centuries ago.
Self-similarity
However, in some constructions, the shapes that are used to decorate large entrance tiles are used which are smaller than the entrance tiles. Another feature of the Darb-i Imam designs is the resemblance in different dimensions: a similar look to the tomb is seen from a distance, and when you look closer to it, the detail in the large pattern is on the surface. The smaller tile division process of the tiles provides a generalization of the overall planar apex.
Arabesque Engineering
Girih is a model of complex geometric drawings that scientists call quasicrystal.
By analyzing the ornamental structure and widely used patterns, the researchers found a complex model, created from geometrical shapes such as stars, anchors and polygons. It was used in Islamic buildings in the 15th century. The design is advanced, yet it has a symmetry that does not repeat itself. Discovered by the West for the first time in the 1970s thanks to the description of the British mathematician and physicist Roger Pinrose.
"In the times of the Clash of Civilizations, this should be a subject of reflection, offering the West new motives to study the culture and history of the Islamic world, especially in the stage Geopolitical current, "and also says, if our work will contribute to highlighting the progress of science and mathematics in the Middle Ages in the Muslim world, I would be very proud. Perhaps a higher level of understanding between two cultures will not see much the same way.
The secret of Islamic architecture in the Middle Ages is the use of mathematical formulas of the twentieth century
An American scientific study has shown that in geometric ornamentation complex structures reveal sophisticated knowledge unknown in the West until the 1970s.
What is the common factor in the schools of Uzbekistan and Baghdad, the Isfahan Mosque in Iran, the holy buildings in Agra in India and Herat in Afghanistan ? Mastery of ceramic decorations, with a system capable of creating beautiful arabesque architecture with authentic symmetry. There is a logo or logo of Islam, with a constant existence from the Middle Ages, from Central Asia to the Middle East.
But behind what seems so far a craft school skill, hiding complex mathematical formulas, was understood by the West 500 years later, in 1970. It also supports an American study published in the scientific journal Science.
The secret of intricate Islamic tiling designs is what scientists call semi-crystalline engineering. The scheme determines the structure of the crystal without maintaining the exact symmetry face. So as to achieve very complex forms, involving very advanced mathematical knowledge.
For a long time it was believed that the geometric decoration that characterized the Islamic architecture was carried out thanks to calipers and ruler. But Peter J. Lu of Harvard University, along with Paul J. Steinhardt of Princeton University says that these tools are not sufficient to interpret the results of this perfection, which are devoid of any distortion and carried out on vast spaces.
History
An analysis of the infiltration patterns seen in the remaining structures, as well as the examination of 15th century Iranian documents, supported this finding. However, there is no indication that architects of that period knew the mathematical dimension of the subject. It is known that the girih designs before the Girih tiles were made with only a line and a compass. The earliest proof of the first use of the entrance tiles for Girih designs belongs to about 1200 years. [one]
At around 1200 years, the stars and polygons, which have 5- and 10-fold rotational symmetry, began to appear. It is also possible to draw these shapes with a compass and a line. However, around the 15th century, designs containing hexagonal (or pentagonal) stars were no longer periodic. These shapes were not made with compasses and lines but with planes that were able to cover space between them. When you lay the plane with these tiles with strip lines on it, an entrance from the strips came to the square. It is not yet known exactly when the use of the gods in place of the compass and line in the construction of the horny entrances has been passed. It was stated that it would be difficult to draw pegs because the designs on the walls of the Mama Hatun Kümbeti ( Tercan, Erzincan ) built in about 1200 years are not in the form of octagons, but it will be very easy to construct with girich karos. It was found in 1197 that the patterns in the walls of the Kümbed-i Kabud walls in Marage had an opening corresponding to the entrance tiles of the fine ornamentation between the strips. The entrant patterns in Kümbed-i Kabudi had a periodic structure, that is, a copy of the pattern was drawn in such a way that a certain distance could be shifted over itself. It is not known exactly when this was first used to replace them, in spite of these findings that the entrance tiles were being used in 1200. Two thousand and a half centuries after Kabul Kabud (1453) Darb-i Imam tomb built in Isfahan had a much more complex structure. As explained above, Darb-ı Imam tomb designs are aperiodic and self-made.
It is seen from the molds found in the Topkapi Parchment, which is thought to belong to the 15th century, that girih tiles were utilized in the design of Girih designs. It has been suggested that the shapes found in this parchment may have been used by artisans who made the entrance gates. Hence, the craftsmen might have made complex layouts without resorting to mathematics and without understanding the basic principles.
Source From Wikipedia
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