1.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, 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 for dealing with lengths, areas, volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since then, into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, curves, as well as the more advanced notions of manifolds and topology or metric. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense. The educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, analytic geometry. Euclidean geometry also has applications in computer science, various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry.
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 angles of a right triangle is the basis for trigonometry. The right angle is called the hypotenuse. The sides adjacent to the right angle are called legs. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. As a formula the T is T = 1 2 a b where a and b are the legs of the triangle. This formula only applies to right triangles. From this: The altitude to the hypotenuse is the geometric mean of the two segments of the hypotenuse. Each leg of the triangle is the mean proportional of the segment of the hypotenuse, adjacent to the leg. Thus f = a b c. For solutions of this equation in integer values of a, b, f, c, see here. The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the triangle's orthocenter -- the intersection of its three altitudes -- coincides with the right-angled vertex. Pythagorean triples are integer values of b, c satisfying this equation. The radius of the circumcircle is half the length of the hypotenuse, R = 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 the adjacent angles are equal, then they are right angles. As a rotation, a right angle corresponds to a turn. The presence of a right angle in a triangle is the defining factor for right triangles, making the right basic to trigonometry. The term is a calque of Latin rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line. 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 +22 BE ⊾ RIGHT ANGLE WITH ARC, U +299 D ⦝ MEASURED RIGHT ANGLE WITH DOT. Right angles are fundamental in Euclid's Elements. They are defined in definition 10, which also defines perpendicular lines. Euclid uses right angles in definitions 12 to define acute angles and obtuse angles. Two angles are called complementary if their sum is a right angle. Saccheri gave a proof as well but using a more explicit assumption. In Hilbert's 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.
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 Pythagoras's 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 other 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, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including algebraic proofs, with some dating back thousands of years. He may well have been the first to prove it. In any event, the proof is called a proof by rearrangement. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem, Q.E.D. That Pythagoras originated this very simple proof is sometimes inferred from the writings of mathematician Proclus. This is known as the Pythagorean one. If the length of b are known, then c can be calculated as c = a 2 + b 2. 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 Pythagorean Proposition contains 370 proofs.
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. The adjective which corresponds to squaring is quadratic. The square of an integer may also be called a perfect square. In algebra, the operation of squaring is often generalized in systems of mathematical values other than the numbers. For instance, the square of the polynomial x + 1 is the quadratic polynomial x2 + 2x + 1. That is, the square function satisfies the identity x2 = 2. This can also be expressed by saying that the squaring function is an even function. The squaring function preserves the order of positive numbers: larger numbers have larger squares. In other words, squaring is a monotonic function on the interval. Hence, zero is its global minimum. This implies that the square of an integer is never less than the original number. Every real number is the square of exactly two numbers, one of, strictly positive and the other of, strictly negative. Zero is the square of only one number, itself. No square root can be taken within the system of real numbers because squares of all real numbers are non-negative.
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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For example, 4 and 4 are square roots of 16 because 42 = 2 = 16. For example, the square root of 9 is 3, denoted √ 9 = 3, because 32 = 3 × 3 = 9 and 3 is non-negative. The term whose root is being considered is known as the radicand. The radicand is the expression underneath the radical sign, in this example 9. A has two square roots: √ a, positive, − √ a, negative. Together, these two roots are denoted ± √a. For positive a, the principal square root can also be written in exponent notation, as a1/2. Square roots of negative numbers can be discussed within the framework of complex numbers. A method for finding very good approximations to the square roots of 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the square root of numbers having many digits. This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to 380 BC. The particular case √ 2 is traditionally attributed to Hippasus. It is exactly the length of the diagonal of a square with length 1. 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.
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 Greek alphabets. Latin was originally spoken in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, initially in Italy and subsequently throughout the Roman Empire. Vulgar Latin developed such as Italian, Portuguese, Spanish, French, Romanian. Latin, Italian and French have contributed many words to the English language. Ancient Greek roots are used in theology, biology, medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin. Vulgar Latin was the colloquial form attested in inscriptions and the works of comic playwrights like Plautus and Terence. Later, Early Modern Latin and Modern Latin evolved. Latin was used until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the Roman Rite of the Catholic Church. Many students, scholars and members of the Catholic clergy speak Latin fluently. It is taught around the world. The language has been passed down through various forms.
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 roughly divided into the Archaic period, Hellenistic period. It is antedated by Mycenaean Greek. The language of the Hellenistic phase is known as Koine. Prior to the Koine period, Greek of earlier periods included several regional dialects. Ancient Greek was the language of Homer and of classical Athenian historians, philosophers. It has been a standard subject of study in educational institutions of the West since the Renaissance. This article primarily contains information of the language. Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Doric, many of them with several subdivisions. Some dialects are found in literary forms used in literature, while others are attested only in inscriptions. There are also historical forms. Homeric Greek is a literary form of Archaic Greek used by other authors. Homeric Greek had significant differences in pronunciation from Classical Attic and other Classical-era dialects. The early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence.
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, especially the Western tradition. Unlike nearly all of his philosophical contemporaries, Plato's entire œuvre is believed to have survived intact for over 2,400 years. Along with his teacher, his most famous student, Aristotle, Plato laid the very foundations of Western philosophy and science. Alfred North Whitehead once noted: "the safest general characterization of the philosophical tradition is that it consists of a series of footnotes to Plato." Friedrich Nietzsche, amongst other scholars, called Christianity, "Platonism for the people." Plato was the innovator of dialectic forms in philosophy, which originate with him. ... ... He was not the first writer to whom the word "philosopher" should be applied. Due to a lack of surviving accounts, little is known about education. The philosopher came in Athens. Ancient sources describe him as a bright though modest boy who excelled in his studies. It is certain that he belonged to an aristocratic and influential family. Based on ancient sources, most modern scholars believe that he was born between 429 and 423 BCE. His father was Ariston.
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 Plato's 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 and human beings and is followed by the dialogue Critias. Participants in the dialogue include Socrates, Timaeus, Hermocrates, Critias. 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 Plato's works such a discussion occurs in the Republic. Hermocrates wishes to oblige Socrates and mentions that Critias knows just the account to do so. The history of Atlantis is postponed to Critias. The main content of the dialogue, the exposition by Timaeus, follows. Timaeus begins with a distinction between the physical world, the eternal world. The physical one is the world which perishes: therefore it is the object of opinion and sensation. 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. "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 (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. The counterfort can be synonymous with buttress, is often used when referring to dams, retaining walls and other structures holding back earth. Early examples of buttresses are found on the Eanna Temple, dating to early as the 4th millennium BCE. In addition to ordinary buttresses, brick and masonry buttresses that support wall corners can be classified according to their ground plan. The gallery below shows top-down views of various types of buttress supporting the wall of a structure. Retaining wall Cathedral architecture
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 other two sides". If the catheti of a right triangle have equal lengths, the triangle is isosceles. If they have different lengths, a distinction can be made between the minor and major cathetus. 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. Bernhardsen, T. Geographic Information Systems: An Introduction, 3rd ed. New York: Wiley, p. 271, 2002. Cathetus at Encyclopaedia of Mathematics Weisstein, Eric W. "Cathetus". MathWorld.
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 cases of obtuse triangles and acute triangles are treated separately, in Propositions 12 and 13 of Book 2. This formula may be transformed into the law of cosines by noting that CH = cos = − cos γ. Proposition 13 contains an entirely analogous statement for acute triangles. In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a form 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. 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 2ab γ + b2 − c2 = 0. This equation can have 2, 1, or 0 positive solutions corresponding to the number of possible triangles given the data. These different cases are also explained by the side-side-angle congruence ambiguity. Consider a triangle with sides of length a, b, c, where θ is the measurement of the angle opposite the side of length 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.
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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In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. 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 systematic link between Euclidean 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. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal including astronomy, physics, many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing. The Cartesian refers to Philosopher René Descartes who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors have a variable length measured to this axis. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. The cylindrical coordinates for three-dimensional space. The development of the coordinate system would play a fundamental role in the development of the Calculus by Gottfried Wilhelm Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces. A line with a Cartesian system is called a line.
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, 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 angle is called the angular coordinate, polar angle, or azimuth. 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 onward, astronomers developed methods for calculating the direction 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 formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's 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, while Cavalieri published his in 1635 with a corrected version appearing in 1653. 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 the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis.
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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This plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined at the point of intersection. Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is delimited by the sides. The angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ἀγκύλος, meaning "crooked, curved," and the English word "ankle". Both are connected with * ank -, meaning "to bend" or "bow". According to Proclus an angle must be a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle. Lower 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 and AC is denoted B A C ^.
Angle
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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. It also avoids the problems of division by zero. The angle is negative for clockwise angles. Now it is also common in other fields of science and engineering. Many scripting languages, such as Perl, include the atan2 function. In mathematical terms, atan2 computes the principal value of the function applied to the complex number x + iy. That is, atan2 = Pr arg = Arg. That is, − π < ≤ π. The atan2 function is useful in many applications involving vectors such as finding the direction from one point to another. A principal use is for converting rotation matrix representations into Euler angles. In some computer programming languages, a different name is used for the function. On scientific calculators the function can often be calculated given when is converted from rectangular coordinates to polar coordinates. The one-argument function can not 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 45 °. The atan2 function places the angle in the correct quadrant.
Atan2
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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, x ⋅ is the product of x and. The order in which complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, multiplication in other algebras is in general non-commutative. An overview of these different kinds of products is given here. Integers allow 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, a ⋅ b ⋅ a are in general different. The operator for the product of a sequence is denoted by the capital Greek letter Pi ∏. The product of a sequence consisting of only one number is just that number itself. The product of no factors at all is equal to 1. Commutative rings have a operation. Under the Fourier transform, convolution becomes point-wise multiplication. Others convey essentially the same 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, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a separate 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 the 2nd AD, the Greco-Egyptian Ptolemy printed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta, its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata. These Indian works were expanded by Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, were applying them to problems in spherical geometry.
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 basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine 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 the same length. An equilateral triangle is also a regular 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 isosceles triangles. The 45 -- 45 -- 90 right triangle, which appears in the 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 and geometric figures to identify sides of equal lengths. In a triangle, the pattern is usually no more than 3 ticks.
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 same face as each other. An axial diagonal is a diagonal that passes through the center of a polyhedron. In a cube with edge length a, all four space diagonals are axial diagonals, of common length a 3. A regular octahedron has 3 axial diagonals, of length a 2, with length a. A regular icosahedron has 6 axial diagonals of a 1 + φ 2, where φ is the golden ratio / 2. For the cube to be considered magic, these four lines must sum correctly. The 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 community has started to recognize an abbreviated expression for these space diagonals. By using r as a variable to describe the various agonals, a concise notation is possible. If r = 2 then we have a diagonal. 2 coordinates change.
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 does not depend about a coordinate axis or its translation. A circle is a set of points with a fixed distance, called the radius, from a point called the center. In geometry, distance is determined by the shape of circles changes as well. Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. As the size of the city blocks diminishes, the points become a rotated square in a continuous geometry. While each side would have √ 2r using a Euclidean metric, where r is the circle's radius, its length in geometry is 2r. Thus, a circle's circumference is 8r. Thus, the value of a geometric analog to π is 4 in this geometry. 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. This approach appears in the framework called compressed sensing. Taxicab geometry can be used to assess the differences in discrete frequency distributions. 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. When summed together for all segments, it provides the same measure as L1-distance.
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 right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. "Angle-based" special right triangles are specified by the relationships of the angles of which the triangle is composed. The side lengths are generally deduced from the basis of geometric methods. This approach may be used to rapidly reproduce the values of trigonometric functions for 60 °. Hence, the angles respectively measure 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 ° -- 90 ° triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely √ 2/2. Of all right triangles, the ° -- 45 ° -- 90 ° 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 possible right triangles that are also isosceles triangles in Euclidean geometry. However, in hyperbolic geometry, there are infinitely 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°, 90°. The sides are in the ratio 1: √3: 2. The proof of this fact is clear using trigonometry.
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, the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so little reliable information is known about him. He travelled, visiting Egypt and Greece, maybe India. Around 530 BC, there established some kind of school or guild. In 520 BC, he returned to Samos. Pythagoras made influential contributions in the late 6th century BC. He 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 successors. Some accounts mention that numbers were important. Burkert states that Aristoxenus and Dicaearchus are the most important accounts. Aristotle had written a separate work On the Pythagoreans, no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans. Dicaearchus, Aristoxenus, Heraclides Ponticus had written on the same subject. According to Clement of Alexandria, Pythagoras was a disciple of Soches, Plato of Sechnuphis of Heliopolis. Herodotus, other early writers agree that Pythagoras was the son of Mnesarchus, born on a Greek island in the eastern Aegean called Samos.
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
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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
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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 evolution of more than 30,000 words, including technical terms. The core body of its etymology information stems from Ernest Weekley's An Etymological Dictionary of Modern English. Other sources include the Middle English Dictionary and the Barnhart Dictionary of Etymology. 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 as a source for explaining the evolution of words. Official website
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 Alice's Adventures in Wonderland for Henry Liddell's daughter Alice. Liddell received his education at Charterhouse and Christ Church, Oxford. He was ordained in 1838. Liddell was Headmaster of Westminster School from 1846 to 1855. 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 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. Aristocratic mien were for many years associated with all, characteristic of Oxford life. 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 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 Carroll Crescent honour the relationship between Henry Liddell and Lewis Carroll. Liddell was an Oxford “character” in later years.
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. The length of the hypotenuse is the square root of 25, 5. The ὑποτείνουσα was used by many other ancient authors. 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. 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 and can be slightly more accurate. Some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. The angle returned will normally be that given by atan2. Orthographic projections: The length of the hypotenuse equals the sum of the lengths of the orthographic projections of both catheti. Cathetus Triangle Space diagonal Nonhypotenuse number Taxicab geometry Trigonometry Special right triangles Pythagoras Hypotenuse at Encyclopaedia of Mathematics Weisstein, Eric W. "Hypotenuse". MathWorld.
Hypotenuse
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A right-angled triangle and its hypotenuse.