1.
Hindu temple architecture
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The Hindu temple architecture is an open, symmetry driven structure, with many variations, on a square grid of padas, depicting perfect geometric shapes such as circles and squares. In ancient Indian texts, a temple is a place for Tirtha - pilgrimage and it is a sacred site whose ambience and design attempts to symbolically condense the ideal tenets of Hindu way of life. The architectural principles of Hindu temples in India are described in Shilpa Shastras, susan Lewandowski states that the underlying principle in a Hindu temple is built around the belief that all things are one, everything is connected. A Hindu temple is meant to encourage reflection, facilitate purification of one’s mind, the specific process is left to the devotee’s school of belief. The primary deity of different Hindu temples varies to reflect this spiritual spectrum and these harmonious places were recommended in these texts with the explanation that such are the places where gods play, and thus the best site for Hindu temples. While major Hindu Mandirs are recommended at sangams, river banks, lakes and seashore, Brhat Samhita, here too, they recommend that a pond be built preferably in front or to the left of the temple with water gardens. If water is present naturally nor by design, water is symbolically present at the consecration of temple or the deity. A Hindu temple design follows a design called vastu-purusha-mandala. The name is a composite Sanskrit word with three of the most important components of the plan, mandala means circle, Purusha is universal essence at the core of Hindu tradition, while Vastu means the dwelling structure. The design lays out a Hindu temple in a symmetrical, self-repeating structure derived from central beliefs, myths, cardinality, the four cardinal directions help create the axis of a Hindu temple, around which is formed a perfect square in the space available. The circle of mandala circumscribes the square, the square is considered divine for its perfection and as a symbolic product of knowledge and human thought, while circle is considered earthly, human and observed in everyday life. The square is divided into perfect square grids, in large temples, this is often a 8x8 or 64 grid structure. In ceremonial temple superstructures, this is an 81 sub-square grid, the square is symbolic and has Vedic origins from fire altar, Agni. The alignment along cardinal direction, similarly is an extension of Vedic rituals of three fires and this symbolism is also found among Greek and other ancient civilizations, through the gnomon. The second design of 4 padas has a central core at the diagonal intersection. The 9 pada design has a sacred surrounded center, and is the template for the smallest temple, older Hindu temple vastumandalas may use the 9 through 49 pada series, but 64 is considered the most sacred geometric grid in Hindu temples. It is also called Manduka, Bhekapada or Ajira in various ancient Sanskrit texts, each pada is conceptually assigned to a symbolic element, sometimes in the form of a deity or to a spirit or apasara. The central square of the 64 is dedicated to the Brahman, in a Hindu temple’s structure of symmetry and concentric squares, each concentric layer has significance
2.
Islamic geometric patterns
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Islamic decoration, which tends to avoid using figurative images, makes frequent use of geometric patterns which have developed over the centuries. These may constitute the entire decoration, may form a framework for floral or calligraphic embellishments, interest in Islamic geometric patterns is increasing in the West, both among craftsmen and artists including M. C. Islamic art mostly avoids figurative images to avoid becoming objects of worship, Islamic geometric patterns derived from simpler designs used in earlier cultures, Greek, Roman, and Sasanian. They are one of three forms of Islamic decoration, the others being the based on curving and branching plant forms. Geometric designs and arabesques are forms of Islamic interlace patterns, david Wade states that Much of the art of Islam, whether in architecture, ceramics, textiles or books, is the art of decoration – which is to say, of transformation. Wade argues that the aim is to transfigure, turning mosques into lightness and pattern and she argues that beauty, whether in poetry or in the visual arts, was enjoyed for its own sake, without commitment to religious or moral criteria. Many Islamic designs are built on squares and circles, typically repeated, overlapped and interlaced to form intricate, a recurring motif is the 8-pointed star, often seen in Islamic tilework, it is made of two squares, one rotated 45 degrees with respect to the other. The fourth basic shape is the polygon, including pentagons and octagons, all of these can be combined and reworked to form complicated patterns with a variety of symmetries including reflections and rotations. Such patterns can be seen as mathematical tessellations, which can extend indefinitely and they are constructed on grids that require only ruler and compass to draw. Artist and educator Roman Verostko argues that such constructions are in effect algorithms, the circle symbolizes unity and diversity in nature, and many Islamic patterns are drawn starting with a circle. On this basis is constructed a star surrounded by six smaller irregular hexagons to form a tessellating star pattern. This forms the basic design which is outlined in white on the wall of the mosque and that design, however, is overlaid with an intersecting tracery in blue around tiles of other colours, forming an elaborate pattern that partially conceals the original and underlying design. A similar design forms the logo of the Mohammed Ali Research Center and he observed that many different combinations of polygons can be used as long as the residual spaces between the polygons are reasonably symmetrical. For example, a grid of octagons in contact has squares as the residual spaces, every octagon is the basis for an 8-point star, as seen at Akbars tomb, Sikandra. Hankin considered the skill of the Arabian artists in discovering suitable combinations of polygons and he further records that if a star occurs in a corner, exactly one quarter of it should be shown, if along an edge, exactly one half of it. The mathematical properties of the tile and stucco patterns of the Alhambra palace in Granada. Some authors have claimed on dubious grounds to have found most or all of the 17 wallpaper groups there, the earliest geometrical forms in Islamic art were occasional isolated geometric shapes such as 8-pointed stars and lozenges containing squares. These date from 836 in the Great Mosque of Kairouan, Tunisia, the next development, marking the middle stage of Islamic geometric pattern usage, was of 6- and 8-point stars, which appear in 879 at the Ibn Tulun Mosque, Cairo, and then became widespread
3.
Selimiye Mosque
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The Selimiye Mosque is an Ottoman imperial mosque, which is located in the city of Edirne, Turkey. The mosque was commissioned by Sultan Selim II, and was built by architect Mimar Sinan between 1569 and 1575 and it was considered by Sinan to be his masterpiece and is one of the highest achievements of Islamic architecture. This grand mosque stands at the center of a külliye which comprises a medrese, a dar-ül hadis, a timekeepers room, in this mosque Sinan employed an octagonal supporting system that is created through eight pillars incised in a square shell of walls. The four semi domes at the corners of the square behind the arches spring from the pillars, are intermediary sections between the huge encompassing dome and the walls. While conventional mosques were limited by an interior, Sinans effort at Edirne was a structure that made it possible to see the mihrab from any location within the mosque. Surrounded by four minarets, the Mosque of Selim II has a grand dome atop it. Around the rest of the mosque were many additions, libraries, schools, hospices, baths, soup kitchens for the poor, markets, hospitals, and these annexes were aligned axially and grouped, if possible. In front of the mosque sits a rectangular court with an equal to that of the mosque. The innovation however, comes not in the size of the building, the mihrab is pushed back into an apse-like alcove with a space with enough depth to allow for window illumination from three sides. This has the effect of making the panels of its lower walls sparkle with natural light. The amalgamation of the hall forms a fused octagon with the dome-covered square. Formed by eight massive dome supports, the octagon is pierced by four half dome covered corners of the square, the beauty resulting from the conformity of geometric shapes engulfed in each other was the culmination of Sinans lifelong search for a unified interior space. At the Bulgarian siege of Edirne in 1913, the dome of the mosque was hit by Bulgarian artillery, owing to the domes extremely sturdy construction, the mosque survived the assault with only minor damage. On Mustafa Kemal Pashas order, it has not been restored since then, some damage can be seen on the image of the dome above, at and near the dark red calligraph to the immediate left of the central blue area. The mosque was depicted on the reverse of the Turkish 10,000 lira banknotes of 1982-1995, the mosque, together with its külliye, was included on UNESCOs World Heritage List in 2011. Selimiye Mosque was built at the peak of Ottoman military and cultural power, as the empire started to grow, the emperor had found an immediate urge to centralize the city. Sinan was asked to help to construct the Selimiye Mosque, making the mosque distinctive, like all other Ottoman mosques in the earlier periods, the Selimiye Mosque had a multitude of little domes and half domes. However, the limit in building Selimiye was to viewing the mosque as a unit from inside or outside rather than separate masses
4.
Edirne
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Edirne served as the third capital city of the Ottoman Empire from 1363 to 1453, before Constantinople became the empires fourth and final capital. At present, Edirne is the capital of Edirne Province in Turkish Thrace, the citys estimated population in 2014 was 165,979. The city was founded as Hadrianopolis, named for the Roman Emperor Hadrian and this name is still used in the Modern Greek. The name Adrianople was used in English, until the Turkish adoption of Latin alphabet in 1928 made Edirne the internationally recognized name. The Turkish, Edirne, Bulgarian, Одрин, Albanian, Edrêne, Slovene, Одрин and Serbian, Једрене / Jedrene are adapted forms of the name Hadrianopolis or of its Turkish version, see also its other names. The area around Edirne has been the site of no fewer than 16 major battles or sieges, military historian John Keegan identifies it as the most contested spot on the globe and attributes this to its geographical location. According to Greek mythology, Orestes, son of king Agamemnon, built this city as Orestias, at the confluence of the Tonsus and the Ardiscus with the Hebrus. The city was founded eponymously by the Roman Emperor Hadrian on the site of a previous Thracian settlement known as Uskadama, Uskudama, Uskodama or Uscudama and it was the capital of the Bessi, or of the Odrysians. Hadrian developed it, adorned it with monuments, changed its name to Hadrianopolis after himself, licinius was defeated there by Constantine I in 323, and Emperor Valens was killed by the Goths in 378 during the Battle of Adrianople. In 813, the city was seized by Khan Krum of Bulgaria who moved its inhabitants to the Bulgarian lands towards the north of the Danube. During the existence of the Latin Empire of Constantinople, the Crusaders were decisively defeated by the Bulgarian Emperor Kaloyan in the Battle of Adrianople. Later Theodore Komnenos, Despot of Epirus, took possession of it in 1227, in 1369, the city was conquered by the Ottoman sultan Murad I. The city remained the Ottoman capital for 90 years until 1453, Edirne is famed for its many mosques, domes, minarets, and palaces from the Ottoman period. Under Ottoman rule, Edirne was the city of the administrative unit, the eponymous Eyalet of Edirne, and after land reforms in 1867. Sultan Mehmed II, the conqueror of Constantinople, was born in Edirne, Sultan Mehmed IV left the palace in Constantinople and died in Edirne in 1693. During his exile in the Ottoman Empire, the Swedish king Charles XII stayed in the city during most of 1713, baháulláh, the founder of the Baháí Faith, lived in Edirne from 1863 to 1868. He was exiled there by the Ottoman Empire before being banished further to the Ottoman penal colony in Akka and he referred to Edirne in his writings as the Land of Mystery. Edirne was briefly occupied by imperial Russian troops in 1829 during the Greek War of Independence, the city suffered a fire in 1905
5.
Self-similarity
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In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the world, such as coastlines, are statistically self-similar. Self-similarity is a property of fractals. Scale invariance is a form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant, it can be continually magnified 3x without changing shape, the non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a line may resemble the whole. It happens if the quantity f exhibits dynamic scaling, the idea is just an extension of the idea of similarity of two triangles. Note that two triangles are similar if the values of their sides are different however the corresponding dimensionless quantities, such as their angles. In mathematics, self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x- and this means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic affine transformation. We call L = a self-similar structure, the homeomorphisms may be iterated, resulting in an iterated function system. The composition of functions creates the algebraic structure of a monoid, when the set S has only two elements, the monoid is known as the dyadic monoid. The dyadic monoid can be visualized as a binary tree, more generally, if the set S has p elements. The automorphisms of the monoid is the modular group, the automorphisms can be pictured as hyperbolic rotations of the binary tree. A more general notion than self-similarity is Self-affinity, the Mandelbrot set is also self-similar around Misiurewicz points. Self-similarity has important consequences for the design of networks, as typical network traffic has self-similar properties. For example, in engineering, packet switched data traffic patterns seem to be statistically self-similar. This property means that simple models using a Poisson distribution are inaccurate, andrew Lo describes stock market log return self-similarity in econometrics. Finite subdivision rules are a technique for building self-similar sets, including the Cantor set
6.
Apophysis (software)
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Apophysis is an open source fractal flame editor and renderer for Microsoft Windows and Macintosh. Users can export fractalflames to other fractal flame rendering programs, such as FLAM3, there is a separate version of Apophysis that has support for 3D. There are numerous clones, ports, and forks of it, scott Draves invented Fractal Flames and published an open source implementation written in C in the early 90s. In 2001 Ronald Hordijk translated his code into Delphi and created a non-animated screensaver, and in 2003 or 2004 Mark Townsend took Ronalds code and added a graphical user interface to create Apophysis. It has since improved and updated by Peter Sdobnov, Piotr Borys. Since 2009, there is a version of Apophysis called Apophysis 7X, originally, it was targeting to provide support for modern Microsoft Windows operating systems like Windows Vista and 7. A strong feedback from the Apophysis users encouranged the developer Georg Kiehne to provide updates which made 7X the most popular and advanced version of Apophysis so far. Apophysis uses the Scripter Studio scripting library to allow users to write scripts which run and either create a new flame, edit the existing flames, one such instance is rendering an entire batch of fractals. Apophysis supports a plugin API so that new variations can be developed and distributed. There are numerous plugins available from the user communities. Fractal flame Fractal-generating software Fractal art Ultra Fractal Chaotica Official website Apophysis on SourceForge. net
7.
Multibrot set
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The name is a portmanteau of multiple and Mandelbrot set. Z ↦ z d + c. where d ≥2, the exponent d may be further generalized to negative and fractional values. The case of d =2 is the classic Mandelbrot set from which the name is derived, the sets for other values of d also show fractal images when they are plotted on the complex plane. Each of the examples of various powers d shown below is plotted to the same scale, values of c belonging to the set are black. The example d =2 is the original Mandelbrot set, the examples for d >2 are often called Multibrot sets. These sets include the origin and have fractal perimeters, with rotational symmetry. When d is negative the set surrounds but does not include the origin, there is interesting complex behaviour in the contours between the set and the origin, in a star-shaped area with -fold rotational symmetry. An alternative method is to render the exponent along the vertical axis and this requires either fixing the real or the imaginary value, and rendering the remaining value along the horizontal axis. The resulting set rises vertically from the origin in a column to infinity. The first prominent bump or spike is seen at an exponent of 2, the third image here renders on a plane that is fixed at a 45-degree angle between the real and imaginary axes. All the above images are rendered using an Escape Time algorithm that identifies points outside the set in a simple way, much greater fractal detail is revealed by plotting the Lyapunov exponent, as shown by the example below. The Lyapunov exponent is the error growth-rate of a given sequence, first calculate the iteration sequence with N iterations, then calculate the exponent as λ =1 N ln | z | and if the exponent is negative the sequence is stable. The white pixels in the picture are the parameters c for which the exponent is positive aka unstable, the colours show the periods of the cycles which the orbits are attracted to. All points colored dark-blue are attracted by a point, all points in the middle are attracted by a period 2 cycle. ESCAPE TIME ALGORITHM For each pixel on the screen do, The complex value z has coordinates on the plane and is raised to various powers inside the iteration loop by codes shown in this table. Powers not shown in the table can be obtained by concatenating the codes shown
8.
Algorithmic art
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Algorithmic art, also known as algorithm art, is art, mostly visual art, of which the design is generated by an algorithm. Algorithmic artists are sometimes called algorists, algorithmic art, also known as computer-generated art, is a subset of generative art and is related to systems art. Fractal art is an example of algorithmic art, the final output is typically displayed on a computer monitor, printed with a raster-type printer, or drawn using a plotter. Variability can be introduced by using pseudo-random numbers, there is no consensus as to whether the product of an algorithm that operates on an existing image can still be considered computer-generated art, as opposed to computer-assisted art. Some of the earliest known examples of computer-generated algorithmic art were created by Georg Nees, Frieder Nake, A. Michael Noll, Manfred Mohr and these artworks were executed by a plotter controlled by a computer, and were therefore computer-generated art but not digital art. The act of creation lay in writing the program, which specified the sequence of actions to be performed by the plotter and her early work with copier and telematic art focused on the differences between the human hand and the algorithm. Aside from the work of Roman Verostko and his fellow algorists. These are important here because they use a different means of execution, whereas the earliest algorithmic art was drawn by a plotter, fractal art simply creates an image in computer memory, it is therefore digital art. The native form of an artwork is an image stored on a computer –this is also true of very nearly all equation art. However, in a stricter sense fractal art is not considered algorithmic art, from one point of view, for a work of art to be considered algorithmic art, its creation must include a process based on an algorithm devised by the artist. This input may be mathematical, computational, or generative in nature, inasmuch as algorithms tend to be deterministic, meaning that their repeated execution would always result in the production of identical artworks, some external factor is usually introduced. This can either be a number generator of some sort. By this definition, fractals made by a program are not art. However, defined differently, algorithmic art can be seen to include fractal art, the artist Kerry Mitchell stated in his 1999 Fractal Art Manifesto, Fractal Art is not. Computer Art, in the sense that the computer does all the work. The work is executed on a computer, but only at the direction of the artist, turn a computer on and leave it alone for an hour. When you come back, no art will have been generated, algorist is a term used for digital artists who create algorithmic art. Algorists formally began correspondence and establishing their identity as artists following a panel titled Art, the co-founders were Roman Verostko and Jean-Pierre Hébert. Fractal art consists of varieties of computer-generated fractals with colouring chosen to give an attractive effect, especially in the western world, it is not drawn or painted by hand
9.
Fractal
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A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry, if the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge, Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set, Fractals also include the idea of a detailed pattern that repeats itself. Fractals are different from other geometric figures because of the way in which they scale, doubling the edge lengths of a polygon multiplies its area by four, which is two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, but if a fractals one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the dimension of the fractal. As mathematical equations, fractals are usually nowhere differentiable, the term fractal was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning broken or fractured, there is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as beautiful, damn hard, increasingly useful, Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal. The word fractal often has different connotations for laypeople than for mathematicians, the mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background. If this is done on fractals, however, no new detail appears, nothing changes, self-similarity itself is not necessarily counter-intuitive. The difference for fractals is that the pattern reproduced must be detailed, a regular line, for instance, is conventionally understood to be 1-dimensional, if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake and it is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. This also leads to understanding a third feature, that fractals as mathematical equations are nowhere differentiable, in a concrete sense, this means fractals cannot be measured in traditional ways. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, according to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity. In his writings, Leibniz used the term fractional exponents, also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called self-inverse fractals
