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Influence of the Star Pattern in Islamic Art

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A prominent feature of Islamic art is the abstract, non-figurative representations, which find their complete expression in geometric patterns. These regularly formed patterns can be found in all areas of Islamic art, starting with stone ornaments on buildings, artful wood decorations, coloured ornaments and arabesques on tiles, to name just a few artful uses in Islamic architecture and Oriental crafts.

Many Islamic Designs are built on squares and circles, typically repeated, overlapped and nested to form complicated and complex patterns. A recurring motif is an 8-pointed star, which is often seen in Islamic tiles. It consists of two squares, one of which is rotated by 45 degrees. The fourth basic shape is the Polygon, including pentagons and octagons. All of these can be combined and revised to form intricate patterns with a variety of symmetries including reflections and rotations. 

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Such patterns can be interpreted as mathematical tessellations, which can extend indefinitely and thus suggest Infinity. They are built on grids that only require a ruler and compass to draw. The artist and educator Roman Verostko argues that such constructions are, in fact, Algorithms that make Islamic geometric patterns of the precursors of the modern algorithmic art.

The concepts of ornamentation have always taken an important place in the Turkish Islamic Arts from the past to the present and have been shown on various building elements and objects. They are composed of figures, writing, geometric or herbal motifs and have found a place for themselves in examples of art areas such as carpet, tile, ceramics, sculpture, skin and lighting, primarily architectural works. These decorations are found in the works created during the Anatolian Seljuks which have a significant position among the Turkish-Islamic states.

Grünbaum and Shephard show how to specify star polygons using the{ n / d} notation, where n and d are integers, n as 3, and 1 as < n/2 as well.The star is built by placing points v1,. To vn at the vertices of a regular n-gon, and join each vi to vi+d, where the indices are moduled n. For ornamental design purposes, Lee[99] applies an analogous s parameter to the one described above, arriving at the final definition{ n / d}s.

Instead of relying on{ n / d}s notation, parameterizing stars by giving directly the contact angle allocation is more convenient. My implementation of Hankin’s method is basically based on the user’s choice of contact angle, and thus making this angle the basis for design elements allows for a smoother integration with the already given construction technique.

In Taprats a star is created from a direction that consists of a single line segment that acts effectively as a beam. The section begins at M, and is 2r long. A single degree of freedom, the touch point, parameterizes it with the contact angle θ. Examples of this definition is the form (10, 1, s, s) for a variety of choices from a wide variety of θ and s choices.

The parameterization, based on the tuple (n, r, s, θ), generalizes in a straightforward manner the original star notation. A little trigonometry, given a star{ n / d}s and a radius r, shows that this star can be reparameterized as (n, r, s, πd / n) As an extension, angle permitting non-integral values of d can take on any real value in the range (0, π/2) a similar extension can be performed on the original notation.

For a given normal n-gon of radius r and an angle of contact, we can now see that the infer-ence algorithm used to apply Hankin’s method will be generated (n, r, 1,). In fact, the special case shown in Figure 3.4, where a star receives an extra internal geometry plate, is simply (n, r, 2, possibly). Typically, an appropriate value of s can be immediately calculated for stars from n and / r: usually, s= 2 if n > 6 and > 2π/n, and s= 1 if not. Examples of this definition is the form (10, 1, s, s) for a variety of choices from a wide variety of θ and s choices.

The rosette is among Islamic art’s most characteristic motifs that they were gradually adapted to more general contexts through experimentation. A rosette can be seen as a star that has hexagons added to in the concavities between neighboring points (see Figure above). Every hexagon straddles the star’s reflection axis, and therefore has bilateral symmetry. 

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