1.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
Geometry
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Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Geometry
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An illustration of Desargues' theorem, an important result in Euclidean and projective geometry
Geometry
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Geometry lessons in the 20th century
Geometry
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A European and an Arab practicing geometry in the 15th century.
2.
Right triangle
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A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a triangle is the basis for trigonometry. The side opposite the angle is called the hypotenuse. The sides adjacent to the angle are called legs. Side a may be identified as the adjacent to angle B and opposed to angle A, while side b is the side adjacent to angle A. If the lengths of all three sides of a triangle are integers, the triangle is said to be a Pythagorean triangle. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the then the other is height. As a formula the area T is T =12 a b where a and b are the legs of the triangle and this formula only applies to right triangles. From this, The altitude to the hypotenuse is the mean of the two segments of the hypotenuse. Each leg of the triangle is the mean proportional of the hypotenuse, in equations, f 2 = d e, b 2 = c e, a 2 = c d where a, b, c, d, e, f are as shown in the diagram. Moreover, the altitude to the hypotenuse is related to the legs of the triangle by 1 a 2 +1 b 2 =1 f 2. For solutions of this equation in integer values of a, b, f, the altitude from either leg coincides with the other leg. Since these intersect at the vertex, the right triangles orthocenter—the intersection of its three altitudes—coincides with the right-angled vertex. The Pythagorean theorem states that, In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs. This can be stated in equation form as a 2 + b 2 = c 2 where c is the length of the hypotenuse, Pythagorean triples are integer values of a, b, c satisfying this equation. The radius of the incircle of a triangle with legs a and b. The radius of the circumcircle is half the length of the hypotenuse, thus the sum of the circumradius and the inradius is half the sum of the legs, R + r = a + b 2
Right triangle
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Right triangle
3.
Right angle
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In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two adjacent parts of a straight line. More precisely, if a ray is placed so that its endpoint is on a line, as a rotation, a right angle corresponds to a quarter turn. The presence of an angle in a triangle is the defining factor for right triangles. The term is a calque of Latin angulus rectus, here rectus means upright, in Unicode, the symbol for a right angle is U+221F ∟ Right angle. It should not be confused with the similarly shaped symbol U+231E ⌞ Bottom left corner, related symbols are U+22BE ⊾ Right angle with arc, U+299C ⦜ Right angle variant with square, and U+299D ⦝ Measured right angle with dot. The symbol for an angle, an arc, with a dot, is used in some European countries, including German-speaking countries and Poland. Right angles are fundamental in Euclids Elements and they are defined in Book 1, definition 10, which also defines perpendicular lines. Euclid uses right angles in definitions 11 and 12 to define acute angles, two angles are called complementary if their sum is a right angle. Book 1 Postulate 4 states that all angles are equal. Euclids commentator Proclus gave a proof of this using the previous postulates. Saccheri gave a proof as well but using a more explicit assumption, in Hilberts axiomatization of geometry this statement is given as a theorem, but only after much groundwork. A right angle may be expressed in different units, 1/4 turn, 90° π/2 radians 100 grad 8 points 6 hours Throughout history carpenters and masons have known a quick way to confirm if an angle is a true right angle. It is based on the most widely known Pythagorean triple and so called the Rule of 3-4-5 and this measurement can be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem, Thales theorem states that an angle inscribed in a semicircle is a right angle. Two application examples in which the angle and the Thales theorem are included. Cartesian coordinate system Orthogonality Perpendicular Rectangle Types of angles Wentworth, G. A, Euclid, commentary and trans. by T. L. Heath Elements Vol.1 Google Books
Right angle
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A right angle is equal to 90 degrees.
4.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
Pythagorean theorem
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The Plimpton 322 tablet records Pythagorean triples from Babylonian times.
Pythagorean theorem
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Pythagorean theorem The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Pythagorean theorem
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Geometric proof of the Pythagorean theorem from the Zhou Bi Suan Jing.
Pythagorean theorem
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Exhibit on the Pythagorean theorem at the Universum museum in Mexico City
5.
Square (algebra)
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In mathematics, a square is the result of multiplying a number by itself. The verb to square is used to denote this operation, squaring is the same as raising to the power 2, and is denoted by a superscript 2, for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the adjective which corresponds to squaring is quadratic. The square of an integer may also be called a number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, for instance, the square of the linear polynomial x +1 is the quadratic polynomial x2 + 2x +1. One of the important properties of squaring, for numbers as well as in other mathematical systems, is that. That is, the function satisfies the identity x2 =2. This can also be expressed by saying that the function is an even function. The squaring function preserves the order of numbers, larger numbers have larger squares. In other words, squaring is a function on the interval. Hence, zero is its global minimum, the only cases where the square x2 of a number is less than x occur when 0 < x <1, that is, when x belongs to an open interval. This implies that the square of an integer is never less than the original number, every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of one number, itself. For this reason, it is possible to define the square root function, no square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. There are several uses of the squaring function in geometry. The name of the squaring function shows its importance in the definition of the area, the area depends quadratically on the size, the area of a shape n times larger is n2 times greater. The squaring function is related to distance through the Pythagorean theorem and its generalization, Euclidean distance is not a smooth function, the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance, which has a paraboloid as its graph, is a smooth, the dot product of a Euclidean vector with itself is equal to the square of its length, v⋅v = v2
Square (algebra)
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The composition of the tiling Image:ConformId.jpg (understood as a function on the complex plane) with the complex square function
Square (algebra)
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5⋅5, or 5 2 (5 squared), can be shown graphically using a square. Each block represents one unit, 1⋅1, and the entire square represents 5⋅5, or the area of the square.
6.
Square root
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In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square is a. For example,4 and −4 are square roots of 16 because 42 =2 =16, every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the square root of 9 is 3, denoted √9 =3. The term whose root is being considered is known as the radicand, the radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two roots, √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a, although the principal square root of a positive number is only one of its two square roots, the designation the square root is often used to refer to the principal square root. For positive a, the square root can also be written in exponent notation. Square roots of numbers can be discussed within the framework of complex numbers. In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, a method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the root of numbers having many digits. It was known to the ancient Greeks that square roots of positive numbers that are not perfect squares are always irrational numbers, numbers not expressible as a ratio of two integers. This is the theorem Euclid X,9 almost certainly due to Theaetetus dating back to circa 380 BC, the particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist, a symbol for square roots, written as an elaborate R, was invented by Regiomontanus. An R was also used for Radix to indicate square roots in Gerolamo Cardanos Ars Magna, according to historian of mathematics D. E. Smith, Aryabhatas method for finding the root was first introduced in Europe by Cataneo in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm, the letter jīm resembles the present square root shape. Its usage goes as far as the end of the century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol √ for the root was first used in print in 1525 in Christoph Rudolffs Coss
Square root
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First leaf of the complex square root
Square root
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The mathematical expression 'The (principal) square root of x"
7.
Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
Latin
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Latin inscription, in the Colosseum
Latin
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Julius Caesar 's Commentarii de Bello Gallico is one of the most famous classical Latin texts of the Golden Age of Latin. The unvarnished, journalistic style of this patrician general has long been taught as a model of the urbane Latin officially spoken and written in the floruit of the Roman republic.
Latin
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A multi-volume Latin dictionary in the University Library of Graz
Latin
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Latin and Ancient Greek Language - Culture - Linguistics at Duke University in 2014.
8.
Ancient Greek
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Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
Ancient Greek
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Inscription about the construction of the statue of Athena Parthenos in the Parthenon, 440/439 BC
Ancient Greek
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Ostracon bearing the name of Cimon, Stoa of Attalos
Ancient Greek
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The words ΜΟΛΩΝ ΛΑΒΕ as they are inscribed on the marble of the 1955 Leonidas Monument at Thermopylae
9.
Plato
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Plato was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world. He is widely considered the most pivotal figure in the development of philosophy, unlike nearly all of his philosophical contemporaries, Platos entire work is believed to have survived intact for over 2,400 years. Along with his teacher, Socrates, and his most famous student, Aristotle, Plato laid the foundations of Western philosophy. 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. In addition to being a figure for Western science, philosophy. Friedrich Nietzsche, amongst other scholars, called Christianity, Platonism for the people, Plato was the innovator of the written dialogue and dialectic forms in philosophy, which originate with him. He was not the first thinker or writer to whom the word “philosopher” should be applied, few other authors in the history of Western philosophy approximate him in depth and range, perhaps only Aristotle, Aquinas and Kant would be generally agreed to be of the same rank. Due to a lack of surviving accounts, little is known about Platos early life, the philosopher came from one of the wealthiest and most politically active families in Athens. Ancient sources describe him as a bright though modest boy who excelled in his studies, the exact time and place of Platos birth are unknown, but it is certain that he belonged to an aristocratic and influential family. Based on ancient sources, most modern scholars believe that he was born in Athens or Aegina between 429 and 423 BCE. According to a tradition, reported by Diogenes Laertius, Ariston traced his descent from the king of Athens, Codrus. Platos mother was Perictione, whose family boasted of a relationship with the famous Athenian lawmaker, besides Plato himself, Ariston and Perictione had three other children, these were two sons, Adeimantus and Glaucon, and a daughter Potone, the mother of Speusippus. The brothers Adeimantus and Glaucon are mentioned in the Republic as sons of Ariston, and presumably brothers of Plato, but in a scenario in the Memorabilia, Xenophon confused the issue by presenting a Glaucon much younger than Plato. Then, at twenty-eight, Hermodorus says, went to Euclides in Megara, as Debra Nails argues, The text itself gives no reason to infer that Plato left immediately for Megara and implies the very opposite. Thus, Nails dates Platos birth to 424/423, 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. Ariston appears to have died in Platos childhood, although the dating of his death is difficult. Perictione then married Pyrilampes, her mothers brother, who had served many times as an ambassador to the Persian court and was a friend of Pericles, Pyrilampes had a son from a previous marriage, Demus, who was famous for his beauty. Perictione gave birth to Pyrilampes second son, Antiphon, the half-brother of Plato and these and other references suggest a considerable amount of family pride and enable us to reconstruct Platos family tree
Plato
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Plato: copy of portrait bust by Silanion
Plato
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Plato from The School of Athens by Raphael, 1509
Plato
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Plato and Socrates in a medieval depiction
Plato
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Plato (left) and Aristotle (right), a detail of The School of Athens, a fresco by Raphael. Aristotle gestures to the earth, representing his belief in knowledge through empirical observation and experience, while holding a copy of his Nicomachean Ethics in his hand. Plato holds his Timaeus and gestures to the heavens, representing his belief in The Forms
10.
Timaeus (dialogue)
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Timaeus is one of Platos dialogues, mostly in the form of a long monologue given by the title character Timaeus of Locri, written c.360 BC. The work puts forward speculation on the nature of the physical world, participants in the dialogue include Socrates, Timaeus, Hermocrates, and Critias. Some scholars believe that it is not the Critias of the Thirty Tyrants who is appearing in this dialogue, but his grandfather and it has been suggested that Timaeus influenced a book about Pythagoras, written by Philolaus. The dialogue takes place the day after Socrates described his ideal state, in Platos works such a discussion occurs in the Republic. Hermocrates wishes to oblige Socrates and mentions that Critias knows just the account to do so, Critias proceeds to tell the story of Solons journey to Egypt where he hears the story of Atlantis, and how Athens used to be an ideal state that subsequently waged war against Atlantis. Critias believes that he is getting ahead of himself, and mentions that Timaeus will tell part of the account from the origin of the universe to man, the history of Atlantis is postponed to Critias. The main content of the dialogue, the exposition by Timaeus, Timaeus begins with a distinction between the physical world, and the eternal world. The physical one is the world changes and perishes, therefore it is the object of opinion. The eternal one never changes, therefore it is apprehended by reason, the speeches about the two worlds are conditioned by the different nature of their objects. Indeed, a description of what is changeless, fixed and clearly intelligible will be changeless and fixed, while a description of changes and is likely, will also change. As being is to becoming, so is truth to belief, therefore, in a description of the physical world, one should not look for anything more than a likely story. Timaeus suggests that since nothing becomes or changes without cause, then the cause of the universe must be a demiurge or a god, and since the universe is fair, the demiurge must have looked to the eternal model to make it, and not to the perishable one. Hence, using the eternal and perfect world of forms or ideals as a template, he set about creating our world, Timaeus continues with an explanation of the creation of the universe, which he ascribes to the handiwork of a divine craftsman. The demiurge, being good, wanted there to be as good as was the world. The demiurge is said to bring out of substance by imitating an unchanging. The ananke, often translated as necessity, was the only other co-existent element or presence in Platos cosmogony, later Platonists clarified that the eternal model existed in the mind of the Demiurge. Timaeus describes the substance as a lack of homogeneity or balance, in which the four elements were shapeless, mixed, considering that order is favourable over disorder, the essential act of the creator was to bring order and clarity to this substance. Therefore, all the properties of the world are to be explained by the choice of what is fair and good, or
Timaeus (dialogue)
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Plato from The School of Athens by Raphael, 1509
Timaeus (dialogue)
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Plato 's Atlantis described in Timaeus and Critias
Timaeus (dialogue)
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Medieval manuscript of Calcidius' Latin Timaeus translation.
11.
Buttress
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A buttress is an architectural structure built against or projecting from a wall which serves to support or reinforce the wall. Buttresses are fairly common on more ancient buildings, as a means of providing support to act against the forces arising out of the roof structures that lack adequate bracing. The term counterfort can be synonymous with buttress, and is used when referring to dams, retaining walls. Early examples of buttresses are found on the Eanna Temple, dating to as early as the 4th millennium BCE, in addition to flying and ordinary buttresses, brick and masonry buttresses that support wall corners can be classified according to their ground plan. The gallery below shows views of various types of buttress supporting the corner wall of a structure
Buttress
Buttress
Buttress
Buttress
12.
Cathetus
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In a right triangle, a cathetus, commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a side about the right angle, the side opposite the right angle is the hypotenuse. In the context of the hypotenuse, the catheti are sometimes referred to simply as the two sides. If the catheti of a triangle have equal lengths, the triangle is isosceles. If they have different lengths, a distinction can be made between the minor and major cathetus. In a right triangle, the length of a cathetus is the mean of the length of the adjacent segment cut by the altitude to the hypotenuse. By the Pythagorean theorem, the sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse, geographic Information Systems, An Introduction, 3rd ed. New York, Wiley, p.271,2002, Cathetus at Encyclopaedia of Mathematics Weisstein, Eric W. Cathetus
Cathetus
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A right-angled triangle where c 1 and c 2 are the catheti and h is the hypotenuse
13.
Law of cosines
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In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known. Though the notion of the cosine was not yet developed in his time, Euclids Elements, dating back to the 3rd century BC, the cases of obtuse triangles and acute triangles are treated separately, in Propositions 12 and 13 of Book 2. Using notation as in Fig.2, Euclids statement can be represented by the formula A B2 = C A2 + C B2 +2 and this formula may be transformed into the law of cosines by noting that CH = cos = − cos γ. Proposition 13 contains an analogous statement for acute triangles. In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi, the theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form, the theorem is used in triangulation, for solving a triangle or circle, i. e. These formulas produce high round-off errors in floating point calculations if the triangle is very acute and it is even possible to obtain a result slightly greater than one for the cosine of an angle. The third formula shown is the result of solving for a the quadratic equation a2 − 2ab cos γ + b2 − c2 =0 and this equation can have 2,1, or 0 positive solutions corresponding to the number of possible triangles given the data. It will have two positive solutions if b sin γ < c < b, only one positive solution if c = b sin γ and these different cases are also explained by the side-side-angle congruence ambiguity. Consider a triangle with sides of length a, b, c and this triangle can be placed on the Cartesian coordinate system by plotting the following points, as shown in Fig.4, A =, B =, and C =. By the distance formula, we have c =2 +2, an advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute vs. right vs. obtuse. Drop the perpendicular onto the c to get c = a cos β + b cos α. Multiply through by c to get c 2 = a c cos β + b c cos α. By considering the other perpendiculars obtain a 2 = a c cos β + a b cos γ, b 2 = b c cos α + a b cos γ. Adding the latter two equations gives a 2 + b 2 = a c cos β + b c cos α +2 a b cos γ and this proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right. It uses the fact that the cosine of an angle expresses the relation between the two sides enclosing that angle in any right triangle
Law of cosines
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Figure 1 – A triangle. The angles α (or A), β (or B), and γ (or C) are respectively opposite the sides a, b, and c.
14.
Rectangular coordinates
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
Rectangular coordinates
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The right hand rule.
Rectangular coordinates
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Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Rectangular coordinates
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3D Cartesian Coordinate Handedness
15.
Polar coordinates
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The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, the concepts of angle and radius were already used by ancient peoples of the first millennium BC. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle, the Greek work, however, did not extend to a full coordinate system. From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca —and its distance—from any location on the Earth, from the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. There are various accounts of the introduction of polar coordinates as part of a coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidges Origin of Polar Coordinates, grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs, in Method of Fluxions, Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the Seventh Manner, For Spirals, and nine other coordinate systems. In the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole, Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoullis work extended to finding the radius of curvature of curves expressed in these coordinates, the actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacocks 1816 translation of Lacroixs Differential and Integral Calculus, alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them. The radial coordinate is often denoted by r or ρ, the angular coordinate is specified as ϕ by ISO standard 31-11. Angles in polar notation are generally expressed in degrees or radians. Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics, in many contexts, a positive angular coordinate means that the angle ϕ is measured counterclockwise from the axis. In mathematical literature, the axis is often drawn horizontal. Adding any number of turns to the angular coordinate does not change the corresponding direction. Also, a radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the point can be expressed with an infinite number of different polar coordinates or
Polar coordinates
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Hipparchus
Polar coordinates
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Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°).
Polar coordinates
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A planimeter, which mechanically computes polar integrals
16.
Angle
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
Angle
–
An angle enclosed by rays emanating from a vertex.
17.
Atan2
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In a variety of computer languages, the function atan2 is the arctangent function with two arguments. For any real number x and y not both equal to zero, atan2 is the angle in radians between the positive x-axis of a plane and the point given by the coordinates on it. The angle is positive for angles, and negative for clockwise angles. The purpose of using two arguments instead of one, i. e and it also avoids the problem of division by zero, as atan2 will return a valid answer as long as y is non-zero. The atan2 function was first introduced in computer programming languages, but now it is common in other fields of science. It dates back at least as far as the FORTRAN programming language and is found in many modern programming languages. Among these languages are, Cs math. h standard library, the Java Math library. NETs System. Math, the Python math module, the Ruby Math module, in addition, many scripting languages, such as Perl, include the C-style atan2 function. The one-argument arctangent function cannot distinguish between diametrically opposite directions, for example, the anticlockwise angle from the x-axis to the vector, calculated in the usual way as arctan, is π/4, or 45°. However, the angle between the x-axis and the vector appears, by the method, to be arctan, again π/4, even though the answer clearly should be −3π/4. In addition, an attempt to find the angle between the x-axis and the vector requires evaluation of arctan, which fails on division by zero, the atan2 function calculates the arc tangent of the two variables y and x. It is similar to calculating the arc tangent of y/x, except that the signs of both arguments are used to determine the quadrant of the result. Thus, the function takes into account the signs of both vector components, and places the angle in the correct quadrant, e. g. atan2 = π/4. In addition, atan2 can produce an angle of ±π/2 while the ordinary arctangent method breaks down, the atan2 function is useful in many applications involving vectors in Euclidean space, such as finding the direction from one point to another. A principal use is in computer graphics rotations, for converting rotation matrix representations into Euler angles, the function atan2 computes the principal value of the argument function applied to the complex number x+iy. That is, atan2 = Pr arg = Arg, the argument can be changed by 2π without making any difference to the angle, but to define atan2 uniquely one uses the principal value in the range (−π, π]. On implementations without signed zero, or when given positive zero arguments and it will always return a value in the range rather than raising an error or returning a NaN. In Common Lisp, where optional arguments exist, the atan function allows one to supply the x coordinate. In Mathematica, the form ArcTan is used where the one parameter form supplies the normal arctangent, Mathematica classifies ArcTan as an indeterminate expression
Atan2
–
The derivation of the atan2(y,x) refers to this figure.
Atan2
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atan2 at selected points.
18.
Product (mathematics)
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In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance,6 is the product of 2 and 3, the order in which real or complex numbers are multiplied has no bearing on the product, this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, matrix multiplication, for example, and multiplication in other algebras is in general non-commutative. There are many different kinds of products in mathematics, besides being able to multiply just numbers, polynomials or matricies, an overview of these different kinds of products is given here. Placing several stones into a pattern with r rows and s columns gives r ⋅ s = ∑ i =1 s r = ∑ j =1 r s stones. Integers allow positive and negative numbers, the product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case, the product operator for the product of a sequence is denoted by the capital Greek letter Pi ∏. The product of a sequence consisting of one number is just that number itself. The product of no factors at all is known as the empty product, commutative rings have a product operation. Under the Fourier transform, convolution becomes point-wise function multiplication, others have very different names but convey essentially the same idea. A brief overview of these is given here, by the very definition of a vector space, one can form the product of any scalar with any vector, giving a map R × V → V. A scalar product is a map, ⋅, V × V → R with the following conditions. From the scalar product, one can define a norm by letting ∥ v ∥, = v ⋅ v, now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping f map V to W, and let the linear mapping g map W to U, then one can get g ∘ f = g = g j k f i j v i b U k. Or in matrix form, g ∘ f = G F v, in which the i-row, j-column element of F, denoted by Fij, is fji, the composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication. To see this, let r = dim, s = dim, let U = be a basis of U, V = be a basis of V and W = be a basis of W. Then B ⋅ A = M W U ∈ R s × t is the matrix representing g ∘ f, U → W, in other words, the matrix product is the description in coordinates of the composition of linear functions. For inifinite-dimensional vector spaces, one also has the, Tensor product of Hilbert spaces Topological tensor product, the tensor product, outer product and Kronecker product all convey the same general idea
Product (mathematics)
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3 by 4 is 12
19.
Trigonometry
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Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies, Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles, thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees, the ancient Nubians used a similar method. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
Trigonometry
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Hipparchus, credited with compiling the first trigonometric table, is known as "the father of trigonometry".
Trigonometry
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All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
Trigonometry
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Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.
20.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
Triangle
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The Flatiron Building in New York is shaped like a triangular prism
Triangle
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A triangle
21.
Space diagonal
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In geometry a space diagonal of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with face diagonals, which connect vertices on the face as each other. An axial diagonal is a diagonal that passes through the center of a polyhedron. For example, in a cube with edge length a, all four space diagonals are axial diagonals, more generally, a cuboid with edge lengths a, b, and c has all four space diagonals axial, with common length a 2 + b 2 + c 2. A regular octahedron has 3 axial diagonals, of length a 2, a regular icosahedron has 6 axial diagonals of length a 1 + φ2, where φ is the golden ratio /2. For the cube to be considered magic, these four lines must sum correctly, the word triagonal is derived from the fact that as a variable point travels down the line, three coordinates change. The equivalent in a square is diagonal, because two coordinates change, in a tesseract it is quadragonal because 4 coordinates change, etc. This section applies particularly to magic hypercubes, the magic hypercube community has started to recognize an abbreviated expression for these space diagonals. By using r as a variable to describe the various agonals, If r =2 then we have a diagonal. 3 coordinates change 4 = a quadragonal,4 coordinates change n = the dimension of the hypercube, the 2n-1 agonals are required to sum correctly for the hypercube to be considered magic. By extension, if r =1, the line is parallel to a face, a 1-agonal may be called a monagonal, in keeping with a diagonal, a triagonal, etc. Lines parallel to the faces of the hypercube have, in the past, because the prefix pan indicates all, we can concisely state the characteristics or a magic hypercube. For example, If pan-r-agonals sum correctly for r =1 and 2, If pan-r-agonals sum correctly for r =1 and 3, we have a pantriagonal magic cube. If the r-agonals sum correctly for r =1 and n, the length of an r-agonal of a hypercube with side length a is a r. Face diagonal Magic cube Magic hypercube Magic cube classes Hypotenuse John R. Hendricks, The Pan-3-Agonal Magic Cube, Journal of Recreational Mathematics 5,1,1972, unsolved Problems in Number Theory, 2nd ed. New York, Springer-Verlag, p.173,1994, mathWorld. de Winkel Magic Encyclopedia Heinz - Basic cube parts John Hendricks Hypercubes
Space diagonal
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The above picture demonstrates how to graphically build a space diagonal and mathematically calculate it with Pythagoras' Theorem.
Space diagonal
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AC' (shown in blue) is a space diagonal while AC (shown in red) is a face diagonal
22.
Taxicab geometry
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Taxicab distance depends on the rotation of the coordinate system, but does not depend on its reflection about a coordinate axis or its translation. A circle is a set of points with a distance, called the radius. In taxicab geometry, distance is determined by a different metric than in Euclidean geometry, Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. The image to the right shows why this is true, by showing in red the set of all points with a distance from a center. As the size of the city blocks diminishes, the points become more numerous, while each side would have length √2r using a Euclidean metric, where r is the circles radius, its length in taxicab geometry is 2r. Thus, a circles circumference is 8r, thus, the value of a geometric analog to π is 4 in this geometry. The formula for the circle in taxicab geometry is | x | + | y | =1 in Cartesian coordinates. A circle of radius 1 is the von Neumann neighborhood of its center, however, this equivalence between L1 and L∞ metrics does not generalize to higher dimensions. e. With its diagonals as coordinate axes, to reach from one square to another, only kings require the number of moves equal to the distance, rooks, queens and bishops require one or two moves. In solving a system of linear equations, the regularisation term for the parameter vector is expressed in terms of the ℓ1 -norm of the vector. This approach appears in the signal recovery framework called compressed sensing, Taxicab geometry can be used to assess the differences in discrete frequency distributions. For example, in RNA splicing positional distributions of hexamers, which plot the probability of each hexamer appearing at each given nucleotide near a splice site, each position distribution can be represented as a vector where each entry represents the likelihood of the hexamer starting at a certain nucleotide. A large L1-distance between the two vectors indicates a significant difference in the nature of the distributions while a small distance denotes similarly shaped distributions. This is equivalent to measuring the area between the two distribution curves because the area of each segment is the difference between the two curves likelihoods at that point. When summed together for all segments, it provides the same measure as L1-distance, normed vector space Metric Orthogonal convex hull Hamming distance Akritean distance Fifteen puzzle Random walk Manhattan wiring Eugene F. Krause. City-block metric on PlanetMath Weisstein, Eric W. Taxicab Metric, paul E. Black, Dictionary of Algorithms and Data Structures, NIST Taxi. - AMS column about Taxicab geometry TaxicabGeometry. net - a website dedicated to taxicab geometry research and information
Taxicab geometry
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Taxicab geometry versus Euclidean distance: In taxicab geometry, the red, yellow, and blue paths all have the shortest length of 12. In Euclidean geometry, the green line has length, and is the unique shortest path.
23.
Special right triangles
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A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a triangle may have angles that form simple relationships. This is called a right triangle. A side-based right triangle is one in which the lengths of the sides form ratios of numbers, such as 3,4,5. Angle-based special right triangles are specified by the relationships of the angles of which the triangle is composed, the angles of these triangles are such that the larger angle, which is 90 degrees or π/2 radians, is equal to the sum of the other two angles. The side lengths are generally deduced from the basis of the circle or other geometric methods. This approach may be used to reproduce the values of trigonometric functions for the angles 30°, 45°. In plane geometry, constructing the diagonal of a results in a triangle whose three angles are in the ratio 1,1,2, adding up to 180° or π radians. Hence, the angles respectively measure 45°, 45°, and 90°, the sides in this triangle are in the ratio 1,1, √2, which follows immediately from the Pythagorean theorem. Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, of all right triangles, the 45°–45°–90° degree triangle has the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely √2/4. Triangles with these angles are the only right triangles that are also isosceles triangles in Euclidean geometry. However, in geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles. This is a triangle whose three angles are in the ratio 1,2,3 and respectively measure 30°, 60°, the sides are in the ratio 1, √3,2. The proof of this fact is clear using trigonometry, the geometric proof is, Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D, then ABD is a 30°–60°–90° triangle with hypotenuse of length 2, and base BD of length 1. The fact that the remaining leg AD has length √3 follows immediately from the Pythagorean theorem, the 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α + δ, α + 2δ are the angles in the then the sum of the angles 3α + 3δ = 180°. After dividing by 3, the angle α + δ must be 60°, the right angle is 90°, leaving the remaining angle to be 30°
Special right triangles
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Special angle-based triangles inscribed in a unit circle are handy for visualizing and remembering trigonometric functions of multiples of 30 and 45 degrees.
24.
Pythagoras
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Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written centuries after he lived. He was born on the island of Samos, and travelled, visiting Egypt and Greece, around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild. In 520 BC, he returned to Samos, Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a mathematician and scientist and is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues, some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Burkert states that Aristoxenus and Dicaearchus are the most important accounts, Aristotle had written a separate work On the Pythagoreans, which is no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans and his disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while adding some legendary accounts. Herodotus, Isocrates, and other writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos. His father is said to have been a gem-engraver or a wealthy merchant, a late source gives his mothers name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, around 530 BC he arrived in the Greek colony of Croton in what was then Magna Graecia. There he founded his own school the members of which he engaged to a disciplined. He furthermore aquired some political influence, on Greeks and non-Greeks of the region, following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum, after his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population. In ancient sources there was disagreement and inconsistency about the late life of Pythagoras. His tomb was shown at Metapontum in the time of Cicero, according to Walter Burkert, Most obvious is the contradiction between Aristoxenus and Dicaearchus, regarding the catastrophe that overwhelmed the Pythagorean society
Pythagoras
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Bust of Pythagoras of Samos in the Capitoline Museums, Rome.
Pythagoras
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Bust of Pythagoras, Vatican
Pythagoras
–
A scene at the Chartres Cathedral shows a philosopher, on one of the archivolts over the right door of the west portal at Chartres, which has been attributed to depict Pythagoras.
Pythagoras
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Croton on the southern coast of Magna Graecia (Southern Italy), to which Pythagoras ventured after feeling overburdened in Samos.
25.
Online Etymology Dictionary
–
The Online Etymology Dictionary is a free online dictionary that describes the origins of English-language words. Douglas Harper compiled the etymology dictionary to record the history and evolution of more than 30,000 words, including slang, the core body of its etymology information stems from Ernest Weekleys An Etymological Dictionary of Modern English. Other sources include the Middle English Dictionary and the Barnhart Dictionary of Etymology, in producing his large dictionary, Douglas Harper says that he is essentially and for the most part a compiler, an evaluator of etymology reports which others have made. Harper works as a Copy editor/Page designer for LNP Media Group, as of June 2015, there were nearly 50,000 entries in the dictionary. It is cited in articles as a source for explaining the history
Online Etymology Dictionary
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Online Etymology Dictionary
Online Etymology Dictionary
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Screenshot of etymonline.com
26.
Henry Liddell
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Lewis Carroll wrote Alices Adventures in Wonderland for Henry Liddells daughter Alice. Liddell received his education at Charterhouse and Christ Church, Oxford and he gained a double first degree in 1833, then became a college tutor, and was ordained in 1838. Liddell was Headmaster of Westminster School from 1846 to 1855, meanwhile, his life work, the great lexicon, which he and Robert Scott began as early as 1834, had made good progress, and the first edition of Liddell and Scotts Lexicon appeared in 1843. It immediately became the standard Greek–English dictionary, with the 8th edition published in 1897, as Headmaster of Westminster Liddell enjoyed a period of great success, followed by trouble due to the outbreak of fever and cholera in the school. In 1855 he accepted the deanery of Christ Church, Oxford, in the same year he brought out his History of Ancient Rome and took a very active part in the first Oxford University Commission. His tall figure, fine presence and aristocratic mien were for years associated with all that was characteristic of Oxford life. In 1859 Liddell welcomed the then Prince of Wales when he matriculated at Christ Church, while he was Dean of Christ Church, he arranged for the building of a new choir school and classrooms for the staff and pupils of Christ Church Cathedral School on its present site. Before then the school was housed within Christ Church itself, in July 1846, Liddell married Miss Lorina Reeve, with whom he had several children, including Alice Liddell of Lewis Carroll fame. In conjunction with Sir Henry Acland, Liddell did much to encourage the study of art at Oxford, in 1891, owing to advancing years, he resigned the deanery. The last years of his life were spent at Ascot, where he died on 18 January 1898, Two roads in Ascot, Liddell Way and Carroll Crescent honour the relationship between Henry Liddell and Lewis Carroll. Liddell was an Oxford character in later years and he figures in contemporary undergraduate doggerel, I am the Dean, this Mrs Liddell. She plays first, I, second fiddle and she is the Broad, I am the High – We are the University. The Victorian journalist, George W. E and his father was Henry Liddell, Rector of Easington, the younger son of Sir Henry Liddell, 5th Baronet and the former Elizabeth Steele. His fathers elder brother, Sir Thomas Liddell, 6th Baronet, was raised to the Peerage as Baron Ravensworth in 1821 and his mother was the former Charlotte Lyon, a daughter of Thomas Lyon and the former Mary Wren. On 2 July 1846, Henry married Lorina Reeve and they were parents of ten children, Edward Harry Liddell. Alice Pleasance Liddell, for whom the story of the childrens classic Alices Adventures in Wonderland was originally told, rhoda Caroline Anne Liddell, she was invested as an Officer, Order of the British Empire in 1920. Albert Edward Arthur Liddell, he died in infancy, violet Constance Liddell, she was invested as a Member, Order of the British Empire in 1920. Sir Frederick Francis Liddell, First Parliamentary Counsel and Ecclesiastical Commissioner, lionel Charles Liddell, he was British Consul to Lyons and Copenhagen
Henry Liddell
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Henry Liddell, in an 1891 portrait by Sir Hubert von Herkomer.
Henry Liddell
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Caricature of Rev. Henry Liddell by 'Ape' from Vanity Fair (1875).
27.
Hypotenuse
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In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. For example, if one of the sides has a length of 3. The length of the hypotenuse is the root of 25. The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timaeus 54d, a folk etymology says that tenuse means side, so hypotenuse means a support like a prop or buttress, but this is inaccurate. The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle are a and b and that of the hypotenuse is c, many computer languages support the ISO C standard function hypot, which returns the value above. The function is designed not to fail where the straightforward calculation might overflow or underflow, some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. This gives both the length of the hypotenuse and the angle the hypotenuse makes with the line at the same time when given x and y. The angle returned will normally be given by atan2. Orthographic projections, The length of the hypotenuse equals the sum of the lengths of the projections of both catheti. And The square of the length of a cathetus equals the product of the lengths of its projection on the hypotenuse times the length of this. Given the length of the c and of a cathetus b. The adjacent angle of the b, will be α = 90° – β One may also obtain the value of the angle β by the equation. Cathetus Triangle Space diagonal Nonhypotenuse number Taxicab geometry Trigonometry Special right triangles Pythagoras Hypotenuse at Encyclopaedia of Mathematics Weisstein, Eric W. Hypotenuse
Hypotenuse
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A right-angled triangle and its hypotenuse.