AMathematical Temple 4:6:9 Ratio etc Dr Uday Dokras, MBA CALSTATE,USA, Phd Stockholm,SWEDEN There are around two million temples in India, and each year the number gets increased substantially. As of end 2020. Varanasi, the city of temples in India, is aptly named since it is home to over 2,000 temples. Varanasi, the home of thousands and hundreds of pilgrimage in and around India, is the oldest city in India as well. After an extraordinary confluence of religion and art, these temples have become a shining example of the cultural heritage and diversity of India. Historic dynasties were responsible for the creation of these and such temples have a large sizeand are beautiful. In India, for hundreds of years, the only form of tourism that existed was purely religious and devotional. Even to this day, millions of devotees flock to these incredible famous temples every year in a quest for moksha and inner peace. Indian temple architecture has been a subject that is widely studied and documented whereas analysis of the geometry of temples is still in its initial stages. Mathematics and architecture however have been related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings. In ancient Egypt, ancient Greece, India, and the Islamic world, buildings including pyramids, temples, mosques, palaces and mausoleums were laid out with specific proportions for religious reasons. In Islamic architecture, geometric shapes and geometric tiling patterns are used to decorate buildings, both inside and outside. Some Hindu temples have a fractal-like structure where parts resemble the whole, conveying a message about the infinite in Hindu cosmology. In Renaissance architecture, symmetry and proportion were deliberately emphasized by architects such as Leon Battista Alberti, Sebastiano Serlio and Andrea Palladio, influenced by Vitruvius's De architectura from ancient Rome and the arithmetic of the Pythagoreans from ancient Greece. At the end of the nineteenth century, Vladimir Shukhov in Russia and Antoni Gaudí in Barcelona pioneered the use of hyperboloid structures; in the Sagrada Família, Gaudí 1 also incorporated hyperbolic paraboloids, tessellations, catenary arches, catenoids, helicoids, and ruled surfaces. In the twentieth century, styles such as modern architecture and Deconstructivism explored different geometries to achieve desired effects. Minimal surfaces have been exploited in tent-like roof coverings as at Denver International Airport, while Richard Buckminster Fuller pioneered the use of the strong thinshell structures known as geodesic domes. Vaastu Shastra, the ancient Indian canons of architecture and town planning, employs symmetrical drawings called mandalas. Complex calculations are used to arrive at the dimensions of a building and its components. The designs are intended to integrate architecture with nature, the relative functions of various parts of the structure, and ancient beliefs utilizing geometric patterns (yantra), symmetry and directional alignments. However, early builders may have come upon mathematical proportions by accident. The mathematician Georges Ifrah notes that simple "tricks" with string and stakes can be used to lay out geometric shapes, such as ellipses and right angles. Plan of Meenakshi Amman Temple, Madurai, from 7th century onwards. The four gateways (numbered I-IV) are tall gopurams.right>>>Gopuram of the Hindu Virupaksha Temple has a fractal-like structure where the parts resemble the whole. The mathematics of fractals has been used to show that the reason why existing buildings have universal appeal and are visually satisfying is because they provide the viewer with a sense of scale at different viewing distances. For example, in the tall gopuram gatehouses of Hindu temples such as the Virupaksha Temple at Hampi built in the seventh century, and others such as the Kandariya Mahadev Temple at Khajuraho, the parts and the whole have the 2 same character, with fractal dimension in the range 1.7 to 1.8. The cluster of smaller towers (shikhara, lit. 'mountain') about the tallest, central, tower which represents the holy Mount Kailash, abode of Lord Shiva, depicts the endless repetition of universes in Hindu cosmology. The religious studies scholar William J. Jackson observed of the pattern of towers grouped among smaller towers, themselves grouped among still smaller towers, that: The ideal form gracefully artificed suggests the infinite rising levels of existence and consciousness, expanding sizes rising toward transcendence above, and at the same time housing the sacred deep within. The Meenakshi Amman Temple is a large complex with multiple shrines, with the streets of Madurai laid out concentrically around it according to the shastras. The four gateways are tall towers (gopurams) with fractal-like repetitive structure as at Hampi. The enclosures around each shrine are rectangular and surrounded by high stone walls. The 4:6:9 ratio has a calendrical, astronomical and cosmological significance, as in Angkor Wat. Archaeologists believe that the 4: 6: 9 ratio and tuning units have a calendar, ... The same is true of the ancient structures of Angkor Wat in Cambodia. The gates are adorned with Kala’s head carved on top of each and Makaras projecting from each side. This Kala-Makara motif is commonly found on the gates of Javanese temples. The positioning of the 72 ‘occupied’ Stupas in three circles of 32, 24 and 16 might give us an indication as to what specific aspects of the cosmos these bell-shaped structures point. The circular platforms on which the Stupas are situated seem to concern themselves with the stars. 3 The ship reliefs on the East depict a ship underneath celestial objects, commemorating a voyage to Africa. This shows the Indonesians crossed the ocean without a compass, depending solely on the stars for navigation. In 2003–2004, a wooden replica of the Borobudur ship was made and sailed the Cinnamon Route from Jakarta to Accra in Ghana to demonstrate the trans-Indian Ocean trading links of medieval Indonesians and Africans. In the book edited by John M Mack, Bruce R Vogeli, Mathematics And Its Teaching In The Asia-pacific Region there are numerous articles pointing to the teaching of mathematics in Schools in the Asia Pacific Region that generally includes East Asia, South Asia, Southeast Asia, and Oceania.This tradition is an ancient one and mathematics formed an common and integral part of social thought and everyday interactions. Though this book provides the global mathematics education community with information on the recent and current status of the teaching of mathematics in a group of island nations in the Asia-Pacific region. Sri Lanka, Indonesia, Japan, the Philippines, Australia, Papua New Guinea, New Zealand, and twelve nations in the South Pacific Ocean.It is the third volume in a series conceived by Dr Bruce Vogeli of Columbia University Teachers College and published by WSP, aimed at producing contemporary accounts of mathematics teaching in a world-wide group of nations. Previous volumes have covered Central and South American nations and a selection of Muslim nations respectively, it is clear that historical traditions facilitate this phenomenon. HINDU MANDALAS Prambhanam Temple in Indonesia BUDDHIST MANDALAS- Borobudur below 4 5 Prambanan or Rara Jonggrang ( PICS ABOVE) an 8th-century Hindu temple compound in Special Region of Yogyakarta, Indonesia, dedicated to the Trimūrti, the expression of God as the Creator (Brahma), the Preserver (Vishnu) and the Destroyer (Shiva) is designed as a Mandala( HINDU). The temple compound is located approximately 17 kilometres (11 mi) 6 northeast of the city of Yogyakarta on the boundary between Central Java and Yogyakarta provinces. The temple compound, a UNESCO World Heritage Site, is the largest Hindu temple site in Indonesia and the second-largest in Southeast Asia after Angkor Wat.[1] It is characterized by its tall and pointed architecture, typical of Hindu architecture, and by the towering 47-metre-high (154 ft) central building inside a large complex of individual temples. Prambanan temple compounds originally consists of 240 temple structures; which represents the grandeur of ancient Java's Hindu art and architecture. History & Geography: The period in which the Javanese constructed Prambanan and neighboring temples is shrouded in legend and mystery. Strong cultural and religious influence arrived in what is now present-day Indonesia from the Indian subcontinent, beginning around the 1st century CE. This influence grew rapidly from c. 400 CE onwards. Hindu and Buddhist merchants and traders, settled in the region, intermarried with the local population, and facilitated long-distance trade relations between the indigenous Javanese, ancient India, and the rest of Southeast Asia. Over the centuries, the Javanese blended the culture and religions of ancient India with their own. Some historians contend that Prambanan's construction began under the orders of King Rakai Pikatan (r. 830-860 CE?) at some point between c. 840-850 CE. Rakai Pikatan oversaw the construction and design of the main temple complex, while other structures were built soon thereafter by later kings, including Rakai Kayuwangi (r. 850-898 CE), Balitung (r. 899-911 CE), Daksa (r. 910-919 CE), and Tulodong (r. 919-924 CE). At some point, locals diverted a nearby river to flow past the temple complex as well. Rakai Pikatan and his successors belonged to the Hindu Sanjaya dynasty, which was the rival of the Buddhist Sailendra dynasty for power in the “Medang” or Mataram kingdom in central Java. It is worth noting that the Sailendra dynasty oversaw the construction of the Buddhist temple of Borobudur. As Borobudur is located only 19 km (12 miles) from Prambanan, some scholars interpret Prambanan's genesis as a direct artistic, political, and religious response to that of Borobudur and in turn the competitive Sailendra dynasty. It is even said that Rakai Pikatan's wife, Princess Pramodhawardhani (fl. 820-860 CE), was the daughter of King Samaratungga (r. 812-833 CE) who may have overseen Borobudur's construction. (Despite all this, other Javanese historians see the “Sailendra” and “Sanjaya” dynasties as one and the same family, interpreting the religious patronage of Buddhism or Hinduism as the result of a ruler's personal belief.) 7 Prambanan Complex Model-Gunawan Kartapranata (CC BY-SA) According to ancient Khmer sources, King Jayavarman II (r. 802-835 CE), the founder of the Khmer Empire (802-1431 CE), spent much of his life in Java, and was appointed by Samaratungga as the governor of Indrapura, which later became the capital city of Champa around c. 875 CE. It is said that Jayavarman visited both Borobidur as well as Prambanan, which inspired him to build the city of Angkor Wat on a grand scale. This is quite possible as the Sailendra and Sanjaya dynasties exerted much political and cultural influence through their thalassocracies in what is present-day Java, Sumatra, Malaya, and southern Cambodia in the 8th, 9th, and 10th centuries CE. Prambanan's abandonment mirrors that of nearby Borobudur. As the center of political power shifted from central to eastern Java under the reign of King Mpu Sindok (fl. 928 CE), Prambanan became less important politically and culturally to the ancient Javanese. The Sanjaya dynasty had successfully usurped power from the Sailendra dynasty, leaving the island of Java under their near complete suzerainty. Volcanic eruptions from Mount Merapi in the 10th century CE and countless earthquakes may have impacted Prambanan's appeal as a place of worship and pilgrimage as well among the Hindu faithful. In What is the relationship of Mathematics and Art and why we should care? Hokky Situngkir reveals that there have been a wide range of any human activities concerning the term of “Art and Mathematics”. Regarding directly to the historical root, there are a great deal of discussions on art and mathematics and their connections. The paper elaborates the connection between the two discourses of art and mathematics and how they influence each other.Batik from Indonesia 8 for instance, also mirrors mathematical modellings and repetitive structures found in Prambanan and Borobudur temple construction. pattern of self-similarity in Prambanan Temple (https://www.researchgate.net/publication/28764583_What_is_the_Relatedness_of_Mathematics_and_Art_and_why _we_should_care/figures?lo=1) 1. The complex is laid out in the form of a mandala, and features the towering, broad spires that are typical of Hindu temple architecture, and represent Meru, the holy mountain where the gods live. 2. Originally there were 240 temples in the complex but many of them have deteriorated or been looted leaving just scattered stones. 3. The Prambanan temple complex consists of three zones; first the outer zone, second the middle zone that contains hundreds of small temples, and third the holiest inner zone that contains eight main temples and likewise, eight small shrines. 4. The three main inner shrines are dedicated to Brahma the Creator, Vishnu the Keeper and Shiva the Destroyer. 5. The three towers cut a striking figure in any conditions, but are perhaps most breathtaking when lit up at night. 9 10 Borobudur, Central Java, Indonesia This spectacular 8th century Buddhist Monument is a shrine for the Lord Buddha and is built as a massive stupa. The foundation is a square, approximately 118 meters on each side and has nine platforms, 504 Buddha statues and over 2,500 relief panels. The basic unit of measurement used in Borobudur’s construction was the tala, which is thought to be the length of a human face from hairline to chin. A survey conducted in the 1970’s revealed multiple ratios of 4:6:9, indicating that the architect had used fractal and self-similar geometry in the design. Researchers believe the recurring ratio and the use of the tala measurement have Buddhist cosmological and astronomical significance. 11 Northwest view of Borobudur, image via Wikipedia The archaeological excavation into Borobudur during reconstruction suggests that adherents of Hinduism or a pre-Indic faith had already begun to erect a large structure on Borobudur's hill before the site was appropriated by Buddhists. The foundations are unlike any Hindu or Buddhist shrine structures, and therefore, the initial structure is considered more indigenous Javanese than Hindu or Buddhist. Design Borobudur is built as a single large stupa and, when viewed from above, takes the form of a giant tantric Buddhist mandala, simultaneously representing the Buddhist cosmology and the nature of mind. The original foundation is a square, approximately 118 metres (387 ft) on each side. It has nine platforms, of which the lower six are square and the upper three are circular. The upper platform contains seventy-two small stupas surrounding one large central stupa. Each stupa is bell-shaped and pierced by numerous decorative openings. Statues of the Buddha sit inside the pierced enclosures. The design of Borobudur took the form of a step pyramid. Previously, the prehistoric Austronesian megalithic culture in Indonesia had constructed several earth mounds and stone step pyramid structures called punden berundak as discovered in Pangguyangan site near Cisolok and in Cipari near Kuningan. The construction of stone pyramids is based on native beliefs that mountains and high places are the abode of ancestral spirits or hyangs. The punden berundak step pyramid is the basic design in Borobudur, believed to be the continuation of older megalithic tradition incorporated with Mahayana Buddhist ideas and symbolism. The monument's three divisions symbolize the three "realms" of Buddhist cosmology, namely Kamadhatu (the world of desires), Rupadhatu (the world of forms), and finally Arupadhatu (the formless world). Ordinary sentient beings live out their lives on the lowest level, the realm of desire. Those who have burnt out all desire for continued existence leave the world of desire and live in the world on the level of form alone: they see forms but are not drawn to them. Finally, full Buddhas go beyond even form and experience reality at its purest, most fundamental level, the formless ocean of nirvana. The liberation from the cycle of Saṃsāra where the enlightened soul had no longer attached to worldly form corresponds to the concept of Śūnyatā, the complete voidness or the nonexistence of the self. Kāmadhātu is represented by the base, Rupadhatu by the five square platforms (the body), and Arupadhatu by the three circular platforms and the large topmost stupa. The architectural features between the three stages have metaphorical differences. For instance, square and detailed decorations in the Rupadhatu disappear into plain circular platforms in the Arupadhatu to represent how the world of forms—where men are still attached with forms and names—changes into the world of the formless. Congregational worship in Borobudur is performed in a walking pilgrimage. Pilgrims are guided by the system of staircases and corridors ascending to the top platform. Each platform represents 12 one stage of enlightenment. The path that guides pilgrims was designed to symbolize Buddhist cosmology. In 1885, a hidden structure under the base was accidentally discovered. The "hidden footing" contains reliefs, 160 of which are narratives describing the real Kāmadhātu. The remaining reliefs are panels with short inscriptions that apparently provide instructions for the sculptors, illustrating the scenes to be carved. The real base is hidden by an encasement base, the purpose of which remains a mystery. It was first thought that the real base had to be covered to prevent a disastrous subsidence of the monument into the hill. There is another theory that the encasement base was added because the original hidden footing was incorrectly designed, according to Vastu Shastra, the Indian ancient book about architecture and town planning. Regardless of why it was commissioned, the encasement base was built with detailed and meticulous design and with aesthetic and religious consideration. Little is known about Gunadharma, the architect of the complex. His name is recounted from Javanese folk tales rather than from written inscriptions. As mentioned above the basic unit of measurement used during construction was the tala, defined as the length of a human face from the forehead's hairline to the tip of the chin or the distance from the tip of the thumb to the tip of the middle finger when both fingers are stretched at their maximum distance. The unit is thus relative from one individual to the next, but the monument has exact measurements. A survey conducted in 1977 revealed frequent findings of a ratio of 4:6:9 around the monument. The architect had used the formula to lay out the precise dimensions of the fractal and self-similar geometry in Borobudur's design. This ratio is also found in the designs of Pawon and Mendut, nearby Buddhist temples. Archeologists have conjectured that the 4:6:9 ratio and the tala have calendrical, astronomical and cosmological significance, as is the case with the temple of Angkor Wat in Cambodia. 13 Building structure The main structure can be divided into three components: base, body, and top The base is 123 m × 123 m (404 ft × 404 ft) in size with 4 metres (13 ft) walls. The body is composed of five square platforms, each of diminishing height. The first terrace is set back 7 metres (23 ft) from the edge of the base. Each subsequent terrace is set back 2 metres (6.6 ft), leaving a narrow corridor at each stage. The top consists of three circular platforms, with each stage supporting a row of perforated stupas, arranged in concentric circles. There is one main dome at the center, the top of which is the highest point of the monument, 35 metres (115 ft) above ground level. Stairways at the center of each of the four sides give access to the top, with a number of arched gates overlooked by 32 lion statues. The gates are adorned with Kala's head carved on top of each and Makaras projecting from each side. This Kala-Makara motif is commonly found on the gates of Javanese temples. The main entrance is on the eastern side, the location of the first narrative reliefs. Stairways on the slopes of the hill also link the monument to the low-lying plain. Approximately 55,000 cubic metres (72,000 cu yd) of andesite stones were taken from neighbouring stone quarries to build the monument. The stone was cut to size, transported to the site and laid without mortar. Knobs, indentations and dovetails were used to form joints between stones. The roof of stupas, niches and arched gateways were constructed in corbelling method. Reliefs were created in situ after the building had been completed. 14 The monument is equipped with a good drainage system to cater to the area's high stormwater run-off. To prevent flooding, 100 spouts are installed at each corner, each with a unique carved gargoyle in the shape of a giant or makara. Borobudur differs markedly from the general design of other structures built for this purpose. Instead of being built on a flat surface, Borobudur is built on a natural hill. However, construction technique is similar to other temples in Java. Without the inner spaces seen in other temples, and with a general design similar to the shape of pyramid, Borobudur was first thought more likely to have served as a stupa, instead of a temple. A stupa is intended as a shrine for the Buddha. Sometimes stupas were built only as devotional symbols of Buddhism. A temple, on the other hand, is used as a house of worship. The meticulous complexity of the monument's design suggests that Borobudur is in fact a temple. Angkor Wat, Siem Reap, Cambodia It has long been theorized that the architect of Cambodia’s magnificent 12th century temple complex based its design on themes of calendrical, historical and astrological significance.But in the 1990’s a professor from the University of Michigan discovered that some of Angkor Wat’s sections are extremely precise: the northern and southern outer corridors measure 202.14 meters long and the eastern and western measure 114.22 and 114.24, respectively. Using these measurements, the professor was able to work out the standard measurement used in the construction of Angkor Wat (a cubit) and decode numerous sacred and historical details. She also discovered that the architect had cleverly repeated numbers throughout the building, for example, each half of a split bridge connecting Angkor Wat to the outside world measures 216 cubits and is decorated with 216 columns. 15 In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0. The golden ratio is also called the golden mean or golden section (Latin: sectio aurea). Other names include extreme and mean ratio, medial section, divine proportion (Latin: proportio divina), divine section (Latin: sectio divina), golden proportion, golden cut, and golden number. . Velmurugan and Kalaivanan Raja studied the existence of the Golden Rule in in Brihadeeshwarar temple, Tanjavur, Tamil Nadu, India, built in 1010 AD. It is listed on the UNESCO’s world heritage site of the Chola temples in southern India. This temple represents an outstanding creative achievement in the architectural idea of the pure form of the Dravida temples. Golden ratio has great influence in architecture, mathematics and art. he researchers analyzed existence of the Golden ratio in structural design of Tanjavur 16 Brihadeeshwarar temple prakaram. They used the Phi Grid and Phi Spiral software to measure the golden ratio and verified their result.They concluded that this temple is one of mysterious gift of our ancient people which exposed architect, spirituality, tradition and science. An ancient Tamil architecture had more knowledge about the Golden ratio and they used to determine pleasing dimensional relationships between the length and width of Karpagragam, Nandhi pavilion and entire courtyard, so that Karpagragam (main place god statue) and Nandhi pavilion situated in inner courtyard of Brihadeeshwarar temple be acting on the major role to bring the Golden ratio for the entire courtyard. 17 From this study,it was felt that one of the secret of beauty of the Brihadeeshwarar temple is Golden proportion influenced its plan of designs and constructions.  Thanjavur periya kovil built by Raja Raja cholan and his sister kundavai, both argent devote of Lord shiva. It was constructed by the King at the height of the Chola reign to signify his power and strength. Here are a few facts not known about this Chola temple of Thanjavur: The main attraction here is the huge tower above the sanctum which is about 216 feet(66 meter). The tower stands tall and one can see this fabulous structure even as one enters Tanjore. MYSTERY OF THE TEMPLE: The whole construction procedure of theTanjore big temple is that of a great mystery to the archeologists. The great legends who made research on the Tanjore big temple couldn't find out 18 what is the actual fact behind the construction of the Tanjore big temple. Some of the edible mysteries of the Tanjore big temple are • Underground passages in Tanjore big temple • Huge cap stone at the top of Tanjore big temple • Painting in Tanjore big temple • Granite stones used for the temple construction UNDERGROUND PASSAGES: The Tanjore big temple contains more than 100 underground passages to various places. The underground passages also contain some secret passage that leads to various places like thePalace of Raja Raja chola and also to other important places. The underground passages lead to the other temples and also to the other places in and around Tanjore. Most of the underground passages were sealed and if the people choose the wrong passage then there are possibilities that the path may also lead to the dangers. This can also be a trap for the kingdom safety of Raja Raja Chola. 19 HUGE CAP STONE (See PIC ABOVE) The biggest mystery of all the other mystery in Tanjore big temple is the huge cap stone in the top of the Tanjore big temple. The weight of the cap stone at the top of Tanjore big templeweighs 80 tons. The main exclamation about the cap stone is that how come the builders of Tanjore big temple were able to place the cap stone at the top of the Gopuram in Tanjore big temple. There were no cranes or any high end equipments used in those days to do these works. The only thing that can help is the elephants. The huge cap of Tanjore big temple is constructed in such a way that the shadow of the Tanjore big temple Gopuram will not fall on the ground. It will just fall on itself. This particular planning and type of construction is not an easy task to go with. PAINTING: Painting in Tanjore Big temple There are lots of high quality paintings in the Tanjore big temple that explain many things about the kingdom of Raja Raja Chola and also the greatness of the Raja Raja Chola. Some paintings also explain about the Raja Raja Chola's favorite God Shiva. There were also some stories present in the painting present in Tanjore big temple. In the secret passage from Tanjore big temple there were great paintings present that explain about many great things. The most admiring part of thepainting in Tanjore big temple is the painting of Raja Raja Chola offering his respect to his Guru. The colors chosen for the painting are great such that the paintings are still good and healthy to see. 20 GRANITE STONES USED FOR CONSTRUCTION: 21 Granite stones are one of the strongest stones in the world. With the strength of the granite stone the stone is also equally heavy enough that it cannot be carried from one place to other place in the easy way. The stones that are used in Tanjore big temple were brought from a place that is 50 miles away from the Tanjore big temple. Without elephants the work might not be possible for the transportation of the rocks from that place. It is said that more than 1000 elephants were used for the construction of the Tanjore big temple. The ancestors of Chola kingdom were good enough to build temples with the granite stones. . a) The original name of the deity was Rajarajeshwar. It was the Marathas who gave it the name Brihadeeshwara or the Great Ishwara. b) The main temple is entirely built of granite. More than 130,000 tons of granite is said to have been used to build it. c) A long associated myth with this temple temple is that the shadow of the main structure does not fall on the ground. However, this has been proven wrong by scientists. d) The statue of Nandhi at the entrance of the temple is carved out of a single stone. e) The main Vimanam, which is at about 200 feet is often called Dakshin Meru or Southern Meru. f) The inspiration to build the temple came to Raja Raja Cholan during his visit to Sri Lanka and is a result of a dream he had. g) The temple has a portrait of Raja Raja Cholan paying obeisance to Lord Natarajar. This is undoubtedly, the first ever instance of a royal portrait. h) Inscriptions in the temple point towards Kunjara Mallan Raja Raja Perunthachan as the chief architect of the temple. His successors survive to this day and practice the art of Vastu or Vastu Shastra. i) Depictions of nartakis or dancers showing eighty one of hundred and eight karanas (synchronised movements of hands and feet) in Bharata Natyam are carved here. These karanas are a part of karanas mentioned in the Natya Shastra of Bharata Muni or Sage Bharata. There is also evidence that the temple was a platform for talented dancers to showcase their talent. These depictions are first of their kind. j) The inscriptions also mention the different kinds of jewels used in the period. Each of these jewels are mentioned in detail. A total of twenty three different types of pearls, eleven varieties of diamonds and rubies are mentioned in these inscriptions. What astounds historians is that there was not a single granite quarry in about 100 km radius of the temple. This means that transporting these stones would have been a herculean task. But Raja 22 Raja Cholan insisted on the use of these stones. All of these features make this Chola temple of Tanjore, a magnum opus of the Chola reign. Existence of the golden ratio in Tanjavur Brihadeeshwarar Temple C. Velmurugan, R. Kalaivanan Department of Mathematics, Vivekananda College, Madurai-635 234, Tamil Nadu, India.; (Open Journal of Mathematical Sciences-Vol. 4 (2020), Issue 1, pp. 211 – 219 Abstract In this study, we discussed the existence of golden ratio in Brihadeeshwarar temple, Tanjavur, Tamil Nadu, India, built in 1010 AD. It is listed on the UNESCO’s world heritage site of the Chola temples in southern India. This temple represents an outstanding creative achievement in the architectural idea of the pure form of the Dravida temples. Golden ratio has great influence in architecture, mathematics and art. We analyzed existence of the Golden ratio in structural design of Tanjavur Brihadeeshwarar temple prakaram. We used the Phi Grid and Phi Spiral software to measure the golden ratio and verified our result. The Golden ratio, mathematics and aesthetics are intricately associated between each other. The Golden ratio is an irrational number that is approximately equal to 1.618, which is represented by the Greek symbol known as phi (ϕϕ). This ratio is considered as the human optical sense and not limited to aesthetic beauty but also found its existence in natural world through the body proportions of living beings, the growth patterns of many plants, insects and also in the model of enigmatic universe. The properties of Golden section can be instituted in the pattern of mathematical series and geometrical patterns. Because of its unique and mystifying properties, many researchers and mathematicians have studied the Golden ratio. Akhtaruzzaman and Shafie discussed the geometrical substantiation of ϕϕ, the Golden ratio and the Baroque of nature, architecture, design and engineering. Golden ratio is key to the secret of beauty Saraf et alThapa et al., studied the relation of Golden ratio, mathematics and aesthetics. Craciunet et al., analyzed the generalized Golden ratio as a fixed point of an operator defined by an arbitrary mean satisfying certain conditions. Iosa et al., discussed Phi in physiology, psychology and biomechanics: The golden ratio between myth and science. Sen and Agarwal investigated Golden ratio in science, as random sequence source. Proportions in architecture are discussed by Singh. Fett studied an in-depth investigation of the Divine ratio. Recently, many authors discussed golden ratio. . Keeping this in mind, in present work we investigated existence of the Golden ratio in structural design of Tanjavur Brihadeeshwarar temple prakaram. We used the Phi Grid and Phi Spiral software to measure the golden ratio. Golden ratio analysis: Tanjavur Brihadeeshwarar temple The Brihadeeshwarar temple is a Hindu temple dedicated to god Shiva located in Tanjavur. This temple plan and improvement make practical and effective use of the axial and symmetrical geometry rules. The temple wall consists of an entrance with a double row of pillars. The perimeter wall forms a rectangular monastery which could be divided into 22 squares. The centre of the first square contains the Nandhi pavilion and second contains the god Shiva statue. In this temple main tower (Gopuram) height is 60m. The tower of the main gateways is 30 m high. In 23 this space there are five main sections: the sanctum with the towering superstructure (srivimana), the Nandhi pavilion in front (Nandi-mandapam) and in between these the main community hall (mukhamandapam), the great gathering hall (mahamandapam) and the pavilion that connects the great hall with the sanctum (ardhamandapam). 2.1.Working rules The following rules are helpful to measure the Golden ratio in the Brihadeeshwarar temple courtyards 1. Let the inner courtyards as rectangular [AEFJ] area. 2. Divide the rectangle in two part by dividing line[CH] in middle of the rectangle. 3. Draw line BI and DG in rectangle[AEFJ] passing through middle of karpagraham [L] and middle of Nandhi pavilion [M]. 4. Now we get three equal portions are: o (a) Middle of the Nandhi pavilion to Middle of the karpagraham [BDGI]. o (b) Entrance of the temple to Middle of the temple [EFHC]. o (c) Middle of temple to Back end of temple [ACHJ]. “We concluded that this temple is one of mysterious gift of our ancient people which exposed architect, spirituality, tradition and science. An ancient Tamil architecture had more knowledge about the Golden ratio and they used to determine pleasing dimensional relationships between the length and width of Karpagragam, Nandhi pavilion and entire courtyard, so that Karpagragam (main place god statue) and Nandhi pavilion situated in inner courtyard of Brihadeeshwarar temple be acting on the major role to bring the Golden ratio for the entire courtyard. From this study, felt that one of the secret of beauty of the Brihadeeshwarar temple is Golden proportion influenced its plan of designs and constructions.” Hokky Situngkir showed that Borobudur was Built Algorithmically. Elaboration from the proposal of Atmadi (1988) on the ratio 4:6:9 employed in Borobudur architecture, could hypothetically prove via a computational experiment using algorithmic rule that the temple is built from the beginning – and in advance, see it’s relation with the form of stupa as the element of building. “The self-similarity of Indonesian Borobudur Temple is observed through the dimensionality of stupa that is hypothetically closely related to whole architectural body. Fractal dimension is calculated by using the cube counting method and found that the dimension is 2.325, which is laid between the two-dimensional plane and three dimensional space. The applied fractal geometry and self-similarity of the building is emerged as the building process implement the metric rules, since there is no universal metric standard known in ancient traditional Javanese culture thus the architecture is not based on final master plan. The paper also proposes how the hypothetical algorithmic architecture might be applied computationally in order to see some experimental generations of similar building. The paper ends with some conjectures for further challenge and insights related to fractal geometry in Javanese traditional cultural heritages. The 24 traditional Javanese metric system (tala) can vary from person to person. While Borobudur must be built by incorporating lots of workers, the architect must apply a rule in order to build such mega-structures did not turn into a fiasco. “ In Some architectural design principles of temples in Java: A study through the buildings projection on the reliefs of Borobudur Temple, Parmono Atmadi (Author)1988 presented the mathematically oriented specifications not only in the structure itself but also the reliefs dotting the walls in the 4:6:9 principle Angkor Wat's simple 4:5 and 6:7 rectangles are multiples of 216 metres. The side lengths directly map to precessional years (top) or seconds of the Earth's rotation (bottom). One could say that the island is one hour wide and its perimeter is 216 minutes. 25 Application of the Sri yantra to the temple plan and elevation for understanding the relationship between the philosophy underlying Hindu temple architecture and its practical applications is the aim of a paper. The existing theories and concepts, like the mandala, Sri yantra and diagonal grid need to be reevaluated for their effective use in temple design. The published Sri yantra drawings need to be modified to be made fit for direct application on temple plans and elevations. The study establishes relationship between Sri yantra and temple plan and elevation. The ancient diagram yantra usually because of its complex nature has been limited to its role as a device for meditation. The Lakshamana temple, dedicated to the Vaikuntha form of Vishnu, having four subsidiary shrines located at the four corners. Lakshamana has the essential elements of a Khajuraho temple’s plan- front hall (mukhamandapa), closed hall (gudhamandapa), vestibule in front of the sanctum door (antrala) and sanctum (garbhagriha). These elements are all aligned on a single axis. The east facing temple has the overall dimensions of 85’ x 44’. The height of the temple is 80’. Building material is sandstone that was brought from the neighbouring quarries, of varying shades of pink, buff or pale yellow. It was built by Chandela Ruler Yashovarman (930950 A.D). Unlike other temples in Khajuraho, its sanctum is Pancha-ratha on plan. Its shikhara is clustered with minor shikharas. The wall portion is studded with balconied windows with ornate balustrades. PROPORTIONS IN ARCHITECTURE Ar. Minakshi Rajput Singh,Associate Professor Amity School of Architecture and Planning, Amity University Chhattisgarh, Raipur, India International Journal of Research -GRANTHAALAYAH Huge faces of goddesses and finely sculpted figurines of yoginis greet visitors at the entrance. 26 Sri Yantra Maha Meru Temple, Amarkantak, by Chitra Ramaswamy is another example having an interface of interlocking triangles, snake hoods and more. It is a vast palette of blue, green and brown – in that order. The clear and azure blue sky appears punctuated by greens in enchanting hues as they spike heavenwards from atop the mountain ranges they inhabit. A tier below the emerald vegetation, it is all a monochrome in brown, a stupendous structure rising out of a basin-like valley. The Maha Meru Sri Yantra Temple is encircled by forest on two sides, the ancient and sacred Batte Krishna Kund, a pond on its western side, and a water reservoir to its north. The temple is constructed in the middle of the Maikal, Satpuda and Vindhyachal range of mountains, 3500 ft. above sea level in Amarkantak, Madhya Pradesh. Amarkantak enjoys the distinction of being the source of three rivers – the Juhila, the Sonbhadra and the Narmada. The borderline dividing north and south India also passes through this spot. The entrance to the sprawling temple precincts is through an impressive, sculpturally rich towering gate, the four sides of which are crowned with the faces of goddesses Saraswati, Kali, Bhuvaneshwari and Lakshmi.The lower segments of these sides are embellished with finely sculpted figurines of the 64 yoginis associated with the Tantrik cult, 16 on each side. In addition, Ganesh and Kartik are also featured on one of the sides. Elevated platform The temple is constructed on an elevated square platform measuring 90,000 sq.ft. and is structured in the form of a mandal of uniform length, breadth and height, each 27 measuring 52 feet. The temple, we are told, has been built on the tenets of temple architecture as laid down by sage Agasthya. The design, layout and plan of the temple adhere to ancient traditional knowledge and methods and are aligned with astrological aspects. The construction is in sync with stellar and planetary movements as per tantrik calendar. In keeping with this style of architecture, the Padambandh or foundation of the temple gives solid protection to the edifice. The next stage, the Sarpbandh, features a pair of sculpted sarps or serpents with tails entwined in each other. The serpents rise from the base of the outer walls of the temple and run through its perimeter in a circumambulatory mode till they appear with their hoods raised, at the entrance to the temple.Above the Devbandh, where one would normally see the tower or shikhar, is the Maha Meru Sri Yantra. Images of Goddess Maha Tripura Sundari with her consorts are sculpted on all its four sides. Three-dimensional projection The Maha Meru temple is the three-dimensional projection of the two-dimensional Sri Yantra or Sri Chakra which forms the core of Sri Vidya worship in Hinduism. The temple with its complicated architectural pattern that requires perfect mathematical calculations and accurately proportioned dimensions, is perhaps the first of its kind to be built anywhere in the world. It is formed by nine interlocking isosceles triangles of varying dimensions that surround and radiate from a focal or central point that is referred to as the bindu. 28 The triangles are so interfaced as to form forty three smaller triangles in a maze that symbolises the cosmos with its multitude of creations. The triangles themselves are inscribed by two rows of lotus petals, eight and sixteen, respectively, and an earth square, symbolic of a temple with four doors. Each of the levels of the Sri Chakra, also known as the nava chakra for its nine levels, is associated with a yogini, a mudra and a specific form of the principal deity Tripura Sundari who is worshipped by followers of the Sri Chakra. The sanctum sanctorum has a 62-inch-tall idol of the Divine Mother made of ashtadhatu, an alloy of eight metals. In keeping with the principles of Vaastu Shastra, Kaal Nritya or ‘the dance of time’ is portrayed on the external walls of the temple. On the four edges on top are four paws of a powerful lion, symbolising actions performed. Stuck in its paws are spherical balls, below which are depicted the poisonous serpents, symbolic of individuals trapped in the cycle of birth and death. While construction of the main temple is complete, work is still in progress as several smaller shrines are slated to be added over a period of time. To conclude: “ The interesting thing about mathematical concepts is that we can trace their development or discoveries throughout history. Most cultures of the ancient world had some form of mathematics, and these basic skills developed into what we now call modern mathematics. The divine ratio is similar in that it was used in many different sections of history. The divine ratio, sometimes called the golden ratio or golden section, has been found in very diverse areas. The mathematical concepts of the golden ration have been found throughout nature, in architecture, music as well as in art. Phi is an astonishing number because it has inspired thinkers in many disciplines, more-so than any other number has in the history of mathematics. This paper investigates how the golden ratio has influenced civilizations 29 throughout history and has intrigued mathematicians and others by its prevalence.” Phi could be the world's most astonishing number. It can be found in nature, throughout history, in art, music, and architecture. Many conflicting theories exist about the origins of phi (φ); however we cannot deny the principles that accompany it. Whether it is the mathematical relationships that seem to form around the number or the sheer aesthetics of the proportion, we must be aware that φ is all around us and rightly called the Divine Ratio .( An In-depth Investigation of the Divine Ratio Birch Fett Follow this and additional works at: Fett, Birch (2006) "An Indepth Investigation of the Divine Ratio," The Mathematics Enthusiast: Vol. 3 : No. 2 , Article 4. Available at: https://scholarworks.umt.edu/tme/vol3/iss2/4 https://scholarworks.umt.edu/tme) Fibonacci Sequence (FS) and Golden Ratio(GR) are all part of the esoteric art was also used to design Apple logo, Mac OS Lion logo, iPhone 4 and recently logo iCloud. Designing logo using FS and GR is a skillful art. Let us take Golden Ratio which is related closely to Fibonacci Sequence (a series of numbers formed by adding up two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth). And Golden Ratio is what a perfect symmetry of relationship comes to represent between two proportions which makes it aesthetically relevant. 30 What exactly is Golden Ratio? Say there is a large rectangle (called golden rectangle) consisting of a square (with sides equal in length to the shortest length of the rectangle) and also a smaller rectangle. The side lengths of the rectangle are said to be in Golden Ratio, that is, if you remove this square from the rectangle there will be another, smaller rectangle and it goes on infinitely like Fibonacci numbers (which work in reverse). Now adding a square equal to the length of the longest side of the rectangle will approach closer to a Golden Rectangle and the Golden Ratio. Golden Ratio is evident from arts and design when human civilization had just begun about 6000 years ago. Ancient Egyptians had mastered this art in building pyramids some 3,000 years ago when human art and design had turned into an accomplished art. Today we can see such dexterity in such complexity be it contemporary art or design. In ancient Greek architecture the width of a building and its height were built in Golden Ratio for its stunning aesthetic dimensional look that was made to look imposing in its grandeur. Even the portico size and the position of the columns supporting the structure were examples of GR. Experts say Leonardo da Vinci, like many artists throughout the ages, made use of the Golden Ratio to create pleasing compositions and one of which was the “Last Supper of Jesus and the Twelve Apostles” as the position of Jesus is perfectly plotted by arranging golden rectangles across the canvas while the figures (13 people said to be present at the supper) are arranged in the lower two thirds (the larger of the two parts of the Golden Ratio). Why talk of art and design created by human hand when the best example of Golden Ratio is Nature itself? If one cares to look around Nature has done this beautifully in its myriad creations be it flora and fauna or the human body itself. Imagine human body having a heart as big as the head. Would that be an example of golden ratio? Absolutely not. See how beautifully each flower, honeycomb, fruit comes to blossom. Their shapes come in all sizes exhibiting the same golden ratio principle which leaves one stunned at the marvel of God’s creation. No wonder Nature has proved itself to be the biggest artist of all. 31 32