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Tetractys
The tetractys ( el, τετρακτύς), or tetrad, or the tetractys of the decad is a triangular number, triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mysticism, mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary motions and music. Pythagorean symbol # The first four numbers symbolize the ''musica universalis'' and the Cosmos as: ## (1) Unity (Monad (philosophy), Monad) ## (2) Dyad (Greek philosophy), Dyad – Power – Limit/Unlimited (peras/Apeiron (cosmology), apeiron) ## (3) Harmony (Triad) ## (4) Kosmos (Tetrad) # The four rows add up to ten, which was unity of a higher order (The Dekad). # The Tetractys symbolizes the classical element, four classical elements—air, fire, water, and earth. # The Tetractys represented the organiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetractys
The tetractys ( el, τετρακτύς), or tetrad, or the tetractys of the decad is a triangular number, triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mysticism, mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary motions and music. Pythagorean symbol # The first four numbers symbolize the ''musica universalis'' and the Cosmos as: ## (1) Unity (Monad (philosophy), Monad) ## (2) Dyad (Greek philosophy), Dyad – Power – Limit/Unlimited (peras/Apeiron (cosmology), apeiron) ## (3) Harmony (Triad) ## (4) Kosmos (Tetrad) # The four rows add up to ten, which was unity of a higher order (The Dekad). # The Tetractys symbolizes the classical element, four classical elements—air, fire, water, and earth. # The Tetractys represented the organiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagoreanism
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek colony of Crotone, Kroton, in modern Calabria (Italy). Early Pythagorean communities spread throughout Magna Graecia. Pythagoras' death and disputes about his teachings led to the development of two philosophical traditions within Pythagoreanism. The ''akousmatikoi'' were superseded in the 4th century BC as a significant mendicant school of philosophy by the Cynicism (philosophy), Cynics. The ''mathēmatikoi'' philosophers were absorbed into the Platonic Academy, Platonic school in the 4th century BC. Following political instability in Magna Graecia, some Pythagorean philosophers fled to mainland Greece while others regrouped in Rhegium. By about 400 BC the majority of Pythagorean philosophers had left Italy. Pythagorean ideas exercised a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac a^2, *The perimeter is p=3a\,\! *The radius of the circumscribed circle is R = \frac *The radius of the inscribed circle is r=\frac a or r=\frac *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac a Denoting the radius of the circumscribed circle as ''R'', we can determine using trigonometry that: *The area of the triangle is \mathrm=\fracR^2 Many of these quantities have simple r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diapason (interval)
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." The interval between the first and second harmonics of the harmonic series is an octave. In Western music notation, notes separated by an octave (or multiple octaves) have the same name and are of the same pitch class. To emphasize that it is one of the perfect intervals (including unison, perfect fourth, and perfect fifth), the octave is designated P8. Other interval qualities are also possible, though rare. The octave above or below an indicated note is sometimes abbreviated ''8a'' or ''8va'' ( it, all'ottava), ''8va bassa'' ( it, all'ottava bassa, sometimes also ''8vb''), or simply ''8'' for the octave in the direction indicated by placing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five consecutive Musical note, notes in a diatonic scale. The perfect fifth (often abbreviated P5) spans seven semitones, while the Tritone, diminished fifth spans six and the augmented fifth spans eight semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. The perfect fifth may be derived from the Harmonic series (music), harmonic series as the interval between the second and third harmonics. In a diatonic scale, the dominant (music), dominant note is a perfect fifth above the tonic (music), tonic note. The perfect fifth is more consonance and dissonance, consonant, or stable, than any other interval except the unison and the octave. It occurs above the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicomachus
Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in Gerasa, in the Roman province of Syria (now Jerash, Jordan). He was a Neopythagorean, who wrote about the mystical properties of numbers.Eric Temple Bell (1940), ''The development of mathematics'', page 83.Frank J. Swetz (2013), ''The European Mathematical Awakening'', page 17, Courier Life Little is known about the life of Nicomachus except that he was a Pythagorean who came from Gerasa.} Historians consider him a Neopythagorean based on his tendency to view numbers as having mystical properties. The age in which he lived (c. 100 AD) is only known because he mentions Thrasyllus in his ''Manual of Harmonics'', and because his ''Introduction to Arithmetic'' was apparently translated into Latin in the mid 2nd century by Apuleius.Henrietta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Introduction To Arithmetic
The book ''Introduction to Arithmetic'' ( grc-gre, Ἀριθμητικὴ εἰσαγωγή, ''Arithmetike eisagoge'') is the only extant work on mathematics by Nicomachus (60–120 AD). Summary The work contains both philosophical prose and basic mathematical ideas. Nicomachus refers to Plato quite often, and writes that philosophy can only be possible if one knows enough about mathematics. Nicomachus also describes how natural numbers and basic mathematical ideas are eternal and unchanging, and in an abstract realm. It consists of two books, twenty-three and twenty-nine chapters, respectively. Although he was preceded by the Babylonians and the Chinese, Nicomachus provided one of the earliest Greco-Roman multiplication tables, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the British Museum). Influence The ''Introduction to Arithmetic'' of Nicomachus was a standard textbook in Neoplatonic schools an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initiation
Initiation is a rite of passage marking entrance or acceptance into a group or society. It could also be a formal admission to adulthood in a community or one of its formal components. In an extended sense, it can also signify a transformation in which the initiate is 'reborn' into a new role. Examples of initiation ceremonies might include Christian baptism or confirmation, Jewish bar or bat mitzvah, acceptance into a fraternal organization, secret society or religious order, or graduation from school or recruit training. A person taking the initiation ceremony in traditional rites, such as those depicted in these pictures, is called an ''initiate''. See also rite of passage. Characteristics William Ian Miller notes the role of ritual humiliation in comic ordering and testing. Mircea Eliade discussed initiation as a principal religious act by classical or traditional societies. He defined initiation as "a basic change in existential condition", which liberates man from p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prayer
Prayer is an invocation or act that seeks to activate a rapport with an object of worship through deliberate communication. In the narrow sense, the term refers to an act of supplication or intercession directed towards a deity or a deified ancestor. More generally, prayer can also have the purpose of thanksgiving or praise, and in comparative religion is closely associated with more abstract forms of meditation and with charms or spells. Prayer can take a variety of forms: it can be part of a set liturgy or ritual, and it can be performed alone or in groups. Prayer may take the form of a hymn, incantation, formal creedal statement, or a spontaneous utterance in the praying person. The act of prayer is attested in written sources as early as 5000 years ago. Today, most major religions involve prayer in one way or another; some ritualize the act, requiring a strict sequence of actions or placing a restriction on who is permitted to pray, while others teach that prayer may b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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