Plato was an Athenian philosopher during the Classical period in Ancient Greece, founder of the Platonist school of thought, the Academy, the first institution of higher learning in the Western world. He is considered the pivotal figure in the history of Ancient Greek and Western philosophy, along with his teacher and his most famous student, Aristotle. Plato has often been cited as one of the founders of Western religion and spirituality; the so-called Neoplatonism of philosophers like Plotinus and Porphyry influenced Saint Augustine and thus Christianity. Alfred North Whitehead once noted: "the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato."Plato was the innovator of the written dialogue and dialectic forms in philosophy. Plato appears to have been the founder of Western political philosophy, his most famous contribution bears his name, the doctrine of the Forms known by pure reason to provide a realist solution to the problem of universals.
He is the namesake of Platonic love and the Platonic solids. His own most decisive philosophical influences are thought to have been along with Socrates, the pre-Socratics Pythagoras and Parmenides, although few of his predecessors' works remain extant and much of what we know about these figures today derives from Plato himself. Unlike the work of nearly all of his contemporaries, Plato's entire oeuvre is believed to have survived intact for over 2,400 years. Although their popularity has fluctuated over the years, the works of Plato have never been without readers since the time they were written. Due to a lack of surviving accounts, little is known about education. Plato belonged to an influential family. According to a disputed tradition, reported by doxographer Diogenes Laërtius, Plato's father Ariston traced his descent from the king of Athens and the king of Messenia, Melanthus. Plato's mother was Perictione, whose family boasted of a relationship with the famous Athenian lawmaker and lyric poet Solon, one of the seven sages, who repealed the laws of Draco.
Perictione was sister of Charmides and niece of Critias, both prominent figures of the Thirty Tyrants, known as the Thirty, the brief oligarchic regime, which followed on the collapse of Athens at the end of the Peloponnesian War. According to some accounts, Ariston tried to force his attentions on Perictione, but failed in his purpose; the exact time and place of Plato's birth are unknown. Based on ancient sources, most modern scholars believe that he was born in Athens or Aegina between 429 and 423 BC, not long after the start of the Peloponnesian War; the traditional date of Plato's birth during the 87th or 88th Olympiad, 428 or 427 BC, is based on a dubious interpretation of Diogenes Laërtius, who says, "When was gone, joined Cratylus the Heracleitean and Hermogenes, who philosophized in the manner of Parmenides. At twenty-eight, Hermodorus says, went to Euclides in Megara." However, as Debra Nails argues, the text does not state that Plato left for Megara after joining Cratylus and Hermogenes.
In his Seventh Letter, Plato notes that his coming of age coincided with the taking of power by the Thirty, remarking, "But a youth under the age of twenty made himself a laughingstock if he attempted to enter the political arena." Thus, Nails dates Plato's birth to 424/423. According to Neanthes, Plato was six years younger than Isocrates, therefore was born the same year the prominent Athenian statesman Pericles died. Jonathan Barnes regards 428 BC as the year of Plato's birth; the grammarian Apollodorus of Athens in his Chronicles argues that Plato was born in the 88th Olympiad. Both the Suda and Sir Thomas Browne claimed he was born during the 88th Olympiad. Another legend related that, when Plato was an infant, bees settled on his lips while he was sleeping: an augury of the sweetness of style in which he would discourse about philosophy. Besides Plato himself and Perictione had three other children; the brothers Adeimantus and Glaucon are mentioned in the Republic as sons of Ariston, brothers of Plato, though some have argued they were uncles.
In a scenario in the Memorabilia, Xenophon confused the issue by presenting a Glaucon much younger than Plato. Ariston appears to have died in Plato's childhood, although the precise dating of his death is difficult. Perictione married Pyrilampes, her mother's brother, who had served many times as an ambassador to the Persian court and was a friend of Pericles, the leader of the democratic faction in Athens. Pyrilampes had a son from a previous marriage, famous for his beauty. Perictione gave birth to Pyrilampes' second son, the half-brother of Plato, who appears in Parmenides. In contrast to his reticence about himself, Plato introduced his distinguished relatives into his dialogues, or referred to them with some precision. In addition to Adeimantus and Glaucon in the Republic, Charmides has a dialogue named after him; these and other references suggest a considerable amount of family pride and enable us to reconstruct Plato's family tree. According to Burnet, "the opening scene of the Ch
A church building or church house simply called a church, is a building used for Christian religious activities for Christian worship services. The term is used by Christians to refer to the physical buildings where they worship, but it is sometimes used to refer to buildings of other religions. In traditional Christian architecture, the church is arranged in the shape of a Christian cross; when viewed from plan view the longest part of a cross is represented by the aisle and the junction of the cross is located at the altar area. Towers or domes are added with the intention of directing the eye of the viewer towards the heavens and inspiring visitors. Modern church buildings have a variety of architectural layouts; the earliest identified Christian church building was a house church founded between 233 and 256. From the 11th through the 14th centuries, a wave of building of cathedrals and smaller parish churches were erected across Western Europe. A cathedral is a church building Roman Catholic, Eastern Orthodox, or Oriental Orthodox, housing a cathedra, the formal name for the seat or throne of a presiding bishop.
In Greek, the adjective kyriak-ós/-ē/-ón means "belonging, or pertaining, to a Kyrios", the usage was adopted by early Christians of the Eastern Mediterranean with regard to anything pertaining to the Lord Jesus Christ: hence "Kyriakós oíkos", "Kyriakē", or "Kyriakē proseukhē". In standard Greek usage, the older word "ecclesia" was retained to signify both a specific edifice of Christian worship, the overall community of the faithful; this usage was retained in Latin and the languages derived from Latin, as well as in the Celtic languages and in Turkish. In the Germanic and some Slavic languages, the word kyriak-ós/-ē/-ón was adopted instead and derivatives formed thereof. In Old English the sequence of derivation started as "cirice" Middle English "churche", "church" in its current pronunciation. German Kirche, Scots kirk, Russian церковь, etc. are all derived. According to the New Testament, the earliest Christians did not build church buildings. Instead, they synagogues; the earliest archeologically identified Christian church is a house church, the Dura-Europos church, founded between 233 and 256.
In the second half of the 3rd century AD, the first purpose-built halls for Christian worship began to be constructed. Although many of these were destroyed early in the next century during the Diocletianic Persecution larger and more elaborate church buildings began to appear during the reign of the Emperor Constantine the Great. From the 11th through the 14th centuries, a wave of building of cathedrals and smaller parish churches occurred across Western Europe. In addition to being a place of worship, the cathedral or the parish church was used by the community in other ways, it could serve as a hall for banquets. Mystery plays were sometimes performed in cathedrals, cathedrals might be used for fairs; the church could be used as a place to store grain. Between 1000 and 1200 the romanesque style became popular across Europe. While the name of the romanesque era refers to the tradition of Roman architecture, it was a West- and Central European trend. Romanesque buildings appear rather compact.
Typical features are circular arches, octagonal towers and cushion capitals on the pillars. In the early romanesque era, coffering on the ceiling was fashionable, while in the same era, groined vault was more popular; the rooms became the motivs of sculptures became more epic. The Gothic style emerged around 1140 in spread through all of Europe; the gothic buildings were less compact than they had been in the romanesque era and contained symbolic and allegoric features. For the first time, pointed arches, rib vaults and buttresses were used, with the result that massive walls were not longer needed to stabilise the building. Due to that advantage, the area of the windows became bigger, which resulted in a brighter and more friendly atmosphere inside the church; the nave so did the pillars and the church steeple. The amibition to test out the limits of the architectural possibilities resulted in the collapse of several towers. In Germany and the Netherlands, but in Spain, it became popular to build hall churches, in which every vault has the same height.
Cathedrals were built in a lavish way, as in the romanesque era. Examples for that are the Notre-Dame de Paris and the Notre-Dame de Reims in France, but the San Francesco d’Assisi in Palermo, the Salisbury Cathedral and the Wool Church in Lavenham, England. Many gothic churches contain features from the romanesque era; some of the most well-known gothic churches stayed unfinished for hundreds of years, after the gothic style was not popular anymore. About half of the Cologne Cathedral was for example build in the 19th century. In the 15th and 16th century, the change in e
Mathematics and art
Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, painting, architecture and textiles; this article focuses, however, on mathematics in the visual arts. Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione, illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, in his paintings; the engraver Albrecht Dürer made many references to mathematics in his work Melencolia I.
In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesburg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, cross-stitch, embroidery, weaving and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as linear perspective, the analysis of symmetry, mathematical objects such as polyhedra and the Möbius strip. Magnus Wenninger creates colourful stellated polyhedra as models for teaching. Mathematical concepts such as recursion and logical paradox can be seen in paintings by Rene Magritte and in engravings by M. C. Escher. Computer art makes use of fractals including the Mandelbrot set, sometimes explores other mathematical objects such as cellular automata.
Controversially, the artist David Hockney has argued that artists from the Renaissance onwards made use of the camera lucida to draw precise representations of scenes. Other relationships include the algorithmic analysis of artworks by X-ray fluorescence spectroscopy, the finding that traditional batiks from different regions of Java have distinct fractal dimensions, stimuli to mathematics research Filippo Brunelleschi's theory of perspective, which led to Girard Desargues's projective geometry. A persistent view, based on the Pythagorean notion of harmony in music, holds that everything was arranged by Number, that God is the geometer of the world, that therefore the world's geometry is sacred, as seen in artworks such as William Blake's The Ancient of Days. Polykleitos the elder was a Greek sculptor from the school of Argos, a contemporary of Phidias, his works and statues consisted of bronze and were of athletes. According to the philosopher and mathematician Xenocrates, Polykleitos is ranked as one of the most important sculptors of classical antiquity for his work on the Doryphorus and the statue of Hera in the Heraion of Argos.
While his sculptures may not be as famous as those of Phidias, they are much admired. In the Canon of Polykleitos, a treatise he wrote designed to document the "perfect" anatomical proportions of the male nude, Polykleitos gives us a mathematical approach towards sculpturing the human body. Polykleitos uses the distal phalanx of the little finger as the basic module for determining the proportions of the human body. Polykleitos multiplies the length of the distal phalanx by the square root of two to get the distance of the second phalanges and multiplies the length again by √2 to get the length of the third phalanges. Next, he takes the finger length and multiplies that by √2 to get the length of the palm from the base of the finger to the ulna; this geometric series of measurements progresses until Polykleitos has formed the arm, body, so on. The influence of the Canon of Polykleitos is immense in Classical Greek and Renaissance sculpture, many sculptors following Polykleitos's prescription.
While none of Polykleitos's original works survive, Roman copies demonstrate his ideal of physical perfection and mathematical precision. Some scholars argue; the Canon applies the basic mathematical concepts of Greek geometry, such as the ratio and symmetria and turns it into a system capable of describing the human form through a series of continuous geometric progressions. In classical times, rather than making distant figures smaller with linear perspective, painters sized objects and figures according to their thematic importance. In the Middle Ages, some artists used reverse perspective for special emphasis; the Muslim mathematician Alhazen described a theory of optics in his Book of Optics in 1021, but never applied it to art. The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts. Two major motives drove artists in the Renaissance towards mathematics. First, painters needed to figure out how to depict three-dimensional scenes on a two-dimensional canvas.
Second and artists alike were convinced that mathematics was the true essence of the physical world and that the entire
A temple is a structure reserved for religious or spiritual rituals and activities such as prayer and sacrifice. It is used for such buildings belonging to all faiths where a more specific term such as church, mosque or synagogue is not used in English; these include Hinduism and Jainism among religions with many modern followers, as well as other ancient religions such as Ancient Egyptian religion. The form and function of temples is thus variable, though they are considered by believers to be in some sense the "house" of one or more deities. Offerings of some sort are made to the deity, other rituals enacted, a special group of clergy maintain, operate the temple; the degree to which the whole population of believers can access the building varies significantly. Temples have a main building and a larger precinct, which may contain many other buildings, or may be a dome shaped structure, much like an igloo; the word comes from Ancient Rome, where a templum constituted a sacred precinct as defined by a priest, or augur.
It has the same root as the word "template", a plan in preparation of the building, marked out on the ground by the augur. Templa became associated with the dwelling places of a god or gods. Despite the specific set of meanings associated with the word, it has now become used to describe a house of worship for any number of religions and is used for time periods prior to the Romans; the temple-building tradition of Mesopotamia derived from the cults of gods and deities in the Mesopotamian religion. It spanned several civilizations; the most common temple architecture of Mesopotamia is the structure of sun-baked bricks called a Ziggurat, having the form of a terraced step pyramid with a flat upper terrace where the shrine or temple stood. Ancient Egyptian temples were meant as places for the deities to reside on earth. Indeed, the term the Egyptians most used to describe the temple building, ḥwt-nṯr, means "mansion of a god". A god's presence in the temple linked the human and divine realms and allowed humans to interact with the god through ritual.
These rituals, it was believed, sustained the god and allowed it to continue to play its proper role in nature. They were therefore a key part of the maintenance of maat, the ideal order of nature and of human society in Egyptian belief. Maintaining maat was the entire purpose of Egyptian religion, thus it was the purpose of a temple as well. Ancient Egyptian temples were of economic significance to Egyptian society; the temples stored and redistributed grain and came to own large portions of the nation's arable land. In addition, many of these Egyptian temples utilized the Tripartite Floor Plan in order to draw visitors to the center room. Though today we call most Greek religious buildings "temples," the ancient Greeks would have referred to a temenos, or sacred precinct, its sacredness connected with a holy grove, was more important than the building itself, as it contained the open air altar on which the sacrifices were made. The building which housed the cult statue in its naos was a rather simple structure, but by the middle of the 6th century BCE had become elaborate.
Greek temple architecture had a profound influence on ancient architectural traditions. The rituals that located and sited Roman temples were performed by an augur through the observation of the flight of birds or other natural phenomenon. Roman temples faced east or toward the rising sun, but the specifics of the orientation are not known today. In ancient Rome only the native deities of Roman mythology had a templum; the Romans referred to a holy place of a pagan religion as fanum. Medieval Latin writers sometimes used the word templum reserved for temples of the ancient Roman religion. In some cases it is hard to determine whether a temple was an outdoor shrine. For temple buildings of the Vikings, the Old Norse term hof is used. A Zoroastrian temple may be called a Dar-e-mehr and a Atashkadeh. A fire temple in Zoroastrianism is the place of worship for Zoroastrians. Zoroastrians revere fire in any form, their temples contains an eternal flame, with Atash Behram as the highest grade of all, as it combines 16 different types of fire gathered in elaborate rituals.
In the Zoroastrian religion, together with clean water, are agents of ritual purity. Clean, white "ash for the purification ceremonies is regarded as the basis of ritual life," which, "are the rites proper to the tending of a domestic fire, for the temple fire is that of the hearth fire raised to a new solemnity". Hindu temples are known by many different names, varying on region and language, including Alayam, Mandira, Gudi, Koil, Kovil, Déul, Devasthana, Deva Mandiraya and Devalaya. A Hindu temple is the seat and dwelling of Hindu gods, it is a structure designed to bring human gods together according to Hindu faith. Inside its Garbhagriha innermost sanctum, a Hindu temple contains a Hindu god's image. Hindu temples are magnificent with a rich history. There is evidence of use of sacred ground as far back as the Bronze Age and during the Indus Valley Civilization. Outside of the Indian subcontinent (India
Marcus Vitruvius Pollio known as Vitruvius, was a Roman author, civil engineer and military engineer during the 1st century BC, known for his multi-volume work entitled De architectura. His discussion of perfect proportion in architecture and the human body led to the famous Renaissance drawing by Leonardo da Vinci of Vitruvian Man. By his own description Vitruvius served as an artilleryman, the third class of arms in the military offices, he served as a senior officer of artillery in charge of doctores ballistarum and libratores who operated the machines. Little is known about Vitruvius' life. Most inferences about him are extracted from his only surviving work De Architectura, his first name Marcus and his cognomen Pollio are uncertain. Marcus Cetius Faventinus writes of "Vitruvius Polio aliique auctores". An inscription in Verona, which names a Lucius Vitruvius Cordo, an inscription from Thilbilis in North Africa, which names a Marcus Vitruvius Mamurra have been suggested as evidence that Vitruvius and Mamurra were from the same family.
Neither association, however, is borne out by De Architectura, nor by the little, known of Mamurra. Vitruvius was a military engineer, or a praefect architectus armamentarius of the apparitor status group, he is mentioned in Pliny the Elder's table of contents for Naturalis Historia, in the heading for mosaic techniques. Frontinus refers to "Vitruvius the architect" in his late 1st-century work De aquaeductu. Born a free Roman citizen, by his own account, Vitruvius served in the Roman army under Caesar with the otherwise poorly identified Marcus Aurelius, Publius Minidius, Gnaeus Cornelius; these names vary depending on the edition of De architectura. Publius Minidius is written as Publius Numidicus and Publius Numidius, speculated as the same Publius Numisius inscribed on the Roman Theatre at Heraclea; as an army engineer he specialized in the construction of ballista and scorpio artillery war machines for sieges. It is speculated; the locations where he served can be reconstructed from, for example, descriptions of the building methods of various "foreign tribes".
Although he describes places throughout De Architectura, he does not say. His service included north Africa, Hispania and Pontus. To place the role of Vitruvius the military engineer in context, a description of "The Prefect of the camp" or army engineer is quoted here as given by Flavius Vegetius Renatus in The Military Institutions of the Romans: The Prefect of the camp, though inferior in rank to the, had a post of no small importance; the position of the camp, the direction of the entrenchments, the inspection of the tents or huts of the soldiers and the baggage were comprehended in his province. His authority extended over the sick, the physicians who had the care of them, he had the charge of providing carriages and the proper tools for sawing and cutting wood, digging trenches, raising parapets, sinking wells and bringing water into the camp. He had the care of furnishing the troops with wood and straw, as well as the rams, onagri and all the other engines of war under his direction; this post was always conferred on an officer of great skill and long service, and, capable of instructing others in those branches of the profession in which he had distinguished himself.
At various locations described by Vitruvius and sieges occurred. He is the only source for the siege of Larignum in 56 BC. Of the battlegrounds of the Gallic War there are references to: the siege and massacre of the 40,000 residents at Avaricum in 52 BC; the broken siege at Gergovia in 52 BC. The circumvallation and Battle of Alesia in 52 BC, and the siege of Uxellodunum in 51 BC. These are all sieges of large Gallic oppida. Of the sites involved in Caesar's civil war, we find the Siege of Massilia in 49 BC, the Battle of Dyrrhachium of 48 BC, the Battle of Pharsalus in 48 BC, the Battle of Zela of 47 BC and the Battle of Thapsus in 46 BC in Caesar's African campaign. A legion that fits the same sequence of locations is the Legio VI Ferrata, of which ballista would be an auxiliary unit. Known for his writings, Vitruvius was himself an architect. In Roman times architecture was a broader subject than at present including the modern fields of architecture, construction management, construction engineering, chemical engineering, civil engineering, materials engineering, mechanical engineering, military engineering and urban planning.
Frontinus mentions him in connection with the standard sizes of pipes. He is credited as father of architectural acoustics for describing the technique of echeas placement in theaters; the only building, that we know Vitruvius to have worked on is one he tells us about, a basi
A sacred grove or sacred woods are any grove of trees that are of special religious importance to a particular culture. Sacred groves feature in various cultures throughout the world, they were important features of the mythological landscape and cult practice of Celtic, Germanic, ancient Greek, Near Eastern and Slavic polytheism, were used in India and West Africa. Examples of sacred groves include the Greco-Roman temenos, the Norse hörgr, the Celtic nemeton, but not associated with Druidic practice. During the Northern Crusades, there was a common practice of building churches on the sites of sacred groves; the Lakota and various other North American tribes consider particular forests or other natural landmarks to be sacred. Ancient holy trees remain in the English and Estonian countryside and are mentioned in folklore and fairytales. There are two mentions on this tradition in the Bible: Abraham planted a grove in Beersheba, called there the name of God. —Genesis 21:33 and where the women wove hangings for the grove.
—II Kings 23:7 Excavations at Labraunda have revealed a large shrine assumed to be that of Zeus Stratios mentioned by Herodotus as a large sacred grove of plane trees sacred to Carians. In Syria, there was a grove sacred to Adonis at Afqa; the most famous sacred groves in mainland Greece was the oak grove at Dodona. Outside the walls of Athens, the site of the Platonic Academy was a sacred grove of olive trees, still recalled in the phrase "the groves of Academe". In central Italy, the town of Nemi recalls the Latin nemus Aricinum, or "grove of Ariccia", a small town a quarter of the way around the lake. In Antiquity the area had no town, but the grove was the site of one of the most famous of Roman cults and temples: that of Diana Nemorensis, a study of which served as the seed for Sir James Frazer's seminal work on the anthropology of religion, The Golden Bough. A sacred grove behind the House of the Vestal Virgins on the edge of the Roman Forum lingered until its last vestiges were burnt in the Great Fire of Rome in 64 CE.
In the town of Spoleto, two stones from the late third century BCE, inscribed in archaic Latin, established punishments for the profanation of the woods dedicated to Jupiter have survived. The Bosco Sacro in the garden of Bomarzo, lends its associations to the uncanny atmosphere. Lucus Pisaurensis, the Sacred Grove of Pesaro, Italy was discovered by Patrician Annibale degli Abati Olivieri in 1737 on property he owned along the'Forbidden Road', just outside Pesaro; this Sacred Grove is the site of the Votive Stones of Pesaro and was dedicated to Salus, the ancient Roman demi-goddess of well-being. The city of Massilia, a Greek colony, had a sacred grove so close by it that Julius Caesar had it cut down to facilitate his siege. In Pharsalia, the poet Lucan dramatized it as a place where sunlight could not reach through the branches, where no animal or bird lived, where the wind did not blow, but branches moved on their own, where human sacrifice was practiced, in a clear attempt to dramatize the situation and distract from the sacrilege entailed in its destruction.
Sacred groves have survived in the Baltic states longer than in other parts of Europe. The main Baltic Prussian sanctuary, considered a sacred grove was Romowe. An important wave of destruction of sacred groves was carried out in the lands of present-day Lithuania after its Christianization in 1387, in Samogitia in 1413. However, some groves, such as in Šventybrastis, still survive in Lithuania. A sacred grove is known as svētbirzs in Latvian. Conversely, in Estonia numerous sacred groves have survived to the present day and have been protected by the government of the country; the Celts used sacred groves, called nemeton in Gaulish, for performing rituals, based on Celtic mythology. The deity involved was Nemetona – a Celtic goddess. Druids oversaw such rituals. Existence of such groves have been found in Germany, Czech Republic and Hungary in Central Europe, in many sites of ancient Gaul in France, as well as England and Northern Ireland. Sacred groves had been plentiful up until the 1st century BC, when the Romans attacked and conquered Gaul.
One of the best known nemeton sites is that in the Nevet forest near Locronan in France. Gournay-sur-Aronde, a village in the Oise department of France houses the remains of a nemeton. Nemetons were fenced off by enclosures, as indicated by the German term Viereckschanze – meaning a quadrangular space surrounded by a ditch enclosed by wooden palisades. Many of these groves, like the sacred grove at Didyma, Turkey are thought to be nemetons, sacred groves protected by druids based on Celtic mythology. In fact, according to Strabo, the central shrine at Galatia was called Drunemeton; some of these were sacred groves in Greek times, but were based on a different or changed mythology. In the animistic native Filipino religions worshiping anito spirits, balete trees known as nonok or nunuk, were regarded as abodes of spirits or gateways to the spirit world. Cutting them down was taboo, a superstition, still followed today. Outdoor shrines or altars known as dambana and tambara among other names were built near the trees during shaman rituals.
Aside from individual trees, natural formations, bodies of water, rocks and entire forests commonly became sacred places to various communities. Sacred groves feature prominently in Scandinavia; the most famous sacred grove of Northern Europe was at the Temple at Upps
In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria: Geometers have studied the Platonic solids for thousands of years, they are named for the ancient Greek philosopher Plato who hypothesized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, it has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes. The ancient Greeks studied the Platonic solids extensively; some sources credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.
The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing. Air is made of the octahedron. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a nonspherical solid, the hexahedron represents "earth"; these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven".
Aristotle added a fifth element, aithēr and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid. Euclid mathematically described the Platonic solids in the Elements, the last book of, devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, cube and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the Elements. Much of the information in Book XIII is derived from the work of Theaetetus. In the 16th century, the German astronomer Johannes Kepler attempted to relate the five extraterrestrial planets known at that time to the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler proposed a model of the Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres.
Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of Saturn. The six spheres each corresponded to one of the planets; the solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron and the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of, that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy, he discovered the Kepler solids. In the 20th century, attempts to link Platonic solids to the physical world were expanded to the electron shell model in chemistry by Robert Moon in a theory known as the "Moon model". For Platonic solids centered at the origin, simple Cartesian coordinates of the vertices are given below.
The Greek letter φ is used to represent the golden ratio 1 + √5/2 ≈ 1.6180. The coordinates for the tetrahedron and dodecahedron are given in two orientation sets, each containing half of the sign and position permutation of coordinates; these coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as or, one of two sets of 4 vertices in dual positions, as h or. Both tetrahedral positions make the compound stellated octahedron; the coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t or called a snub octahedron, as s or, seen in the compound of two icosahedra. Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientat