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Triangle (geometry)

Triangle (geometry)

:For alternative meanings, such as the musical instrument, see triangle (disambiguation). A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments.

Types of triangles

Triangles can be classified according to the relative lengths of their sides:
- In an equilateral triangle all sides are of equal length. An equilateral triangle is also equiangular, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon
- In an isosceles triangle two sides are of equal length. An isosceles triangle also has two equal internal angles.
- In a scalene triangle all sides have different lengths. The internal angles in a scalene triangle are all different.

Equilateral triangle	Isosceles triangle	Scalene triangle
Equilateral	Isosceles	Scalene

Triangles can also be classified according to the size of their largest internal angle, described below using degrees of arc.
- A right triangle (or right-angled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs of the triangle.
- An obtuse triangle has one internal angle larger than 90° (an obtuse angle).
- An acute triangle has internal angles that are all smaller than 90° (three acute angles).

Right triangle	Obtuse triangle	Acute triangle
Right	Obtuse	Acute

Basic facts

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE. A triangle is a polygon and a 2-simplex (see polytope). Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry. In the remainder we will consider a triangle with vertices A, B and C, angles α, β and γ and sides a, b and c. The side a is opposite to the vertex A and angle α and analogously for the other sides. trigonometry In Euclidean geometry, the sum of the angles α + β + γ is equal to two right angles (180° or π radians). This allows determination of the third angle of any triangle as soon as two angles are known. trigonometry A central theorem is the Pythagorean theorem stating that in any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. If side C is the hypotenuse, we can write this as :

c^2 = a^2 + b^2 \,

This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third—something unique to right triangles. The Pythagorean theorem can be generalized to the law of cosines: :

c^2 = a^2 + b^2 - 2ab \cos\gamma \,

which is valid for all triangles, even if γ is not a right angle. The law of cosines can be used to compute the side lengths and angles of a triangle as soon as all three sides or two sides and an enclosed angle are known. The law of sines states :

\fraca=\fracb=\fracc=\frac1d

where d is the diameter of the circumcircle (the circle which passes through all three points of the triangle). The law of sines can be used to compute the side lengths for a triangle as soon as two angles and one side are known. If two sides and an unenclosed angle is known, the law of sines may also be used; however, in this case there may be zero, one or two solutions. There are two special right triangles that appear commonly in geometry. The so-called "45-45-90 triangle" has angles with those angle measures and the ratio of its sides is :

;1:1:\sqrt

. The "30-60-90 triangle" has sides in the ratio of

1:\sqrt:2

Points, lines and circles associated with a triangle

There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained. Menelaus' theorem A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle can be found from the law of sines stated above. Thales' theorem states that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. Thales' theorem An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is not obtuse. The three vertices together with the orthocenter are said to form an orthocentric system. orthocentric system An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.
orthocentric system A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid. This is also the triangle's center of gravity: if the triangle were made out of wood, say, you could balance it on its centroid, or on any line through the centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice as large as the distance between the centroid and the midpoint of the opposite side. center of gravity The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles.
excircle The centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red point) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. The center of the incircle is not in general located on Euler's line. If one reflects a median at the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle.

Computing the area of a triangle

Calculating the area of a triangle is an elementary problem encountered often in many different situations. Various approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle.

Using geometry

The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base. This can be shown with the following geometric construction. area To find the area of a given triangle (green), first make an exact copy of the triangle (blue), rotate it 180°, and join it to the given triangle along one side to obtain a parallelogram. Cut off a part and join it at the other side of the parallelogram to form a rectangle. Because the area of the rectangle is bh, the area of the given triangle must be ½bh. parallelogram

Using vectors

The area of a parallelogram can also be calculated by the use of vectors. If AB and AC are vectors pointing from A to B and from A to C, respectively, the area of parallelogram ABDC is |AB × AC|, the magnitude of the cross product of vectors AB and AC. |AB × AC| is also equal to |h × AC|, where h represents the altitude h as a vector. The area of triangle ABC is half of this, or S = ½|AB × AC|. cross product

Using trigonometry

The altitude of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as S = ½ab sin γ. It is of course no coincidence that the area of a parallelogram is ab sin γ.

Using coordinates

If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (x₁, y₁) and C = (x₂, y₂), then the area S can be computed as 1/2 times the absolute value of the determinant :

\beginx_1 & x_2 \\ y_1 & y_2 \end

or S = ½ |x₁y₂ − x₂y₁|.

Using Heron's formula

Yet another way to compute S is Heron's formula: :

S = \sqrt

where s = ½ (a + b + c) is the semiperimeter, or one half of the triangle's perimeter.

Using the side lengths and a numerically stable formula

Heron's formula is numerically unstable for triangles with a very small angle. A stable alternative involves arranging the lengths of the sides so that: :a ≥ b ≥ c and computing :

S = \frac\sqrt

The brackets in the above formula are required in order to prevent numerical instability in the evaluation.

Non-planar triangles

If any four of a triangle's elements (vertices, and/or elements of its sides) are plane to each other, the triangle is called plane. Geometers also study non-planar triangles in noneuclidean geometries, such as spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry. While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, he would find that its angles would be greater than 180°.

External links

- [http://ostermiller.org/calc/triangle.html Triangle Calculator] - solves for remaining sides and angles when given three sides or angles, supports degrees and radians.
- [http://agutie.homestead.com/files/Napoleon0.htm Napoleon's theorem] A triangle with three equilateral triangles. A purely geometric proof. It uses the Fermat point to prove Napoleon's theorem without transformations by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
- William Kahan: [http://http.cs.berkeley.edu/~wkahan/Triangle.pdf Miscalculating Area and Angles of a Needle-like Triangle].
- Clark Kimberling: [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of triangle centers]. Lists some 1600 interesting points associated with any triangle.
- Christian Obrecht: [http://perso.wanadoo.fr/obrecht/ Eukleides]. Software package for creating illustrations of facts about triangles and other theorems in Euclidean geometry.
- [http://www.cut-the-knot.org/triangle Triangle constructions, remarkable points and lines, and metric relations in a triangle] at cut-the-knot
- [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1469&CurriculumID=4 Printable Worksheet on Types of Triangles]
- [http://www.vias.org/comp_geometry/geom_triangle.html Compendium Geometry] Analytical Geometry of Triangles Category:Polygons ko:삼각형 ja:三角形 th:รูปสามเหลี่ยม

Triangle (instrument)

The triangle is an idiophonic musical instrument of the percussion family. It is a bar of metal, usually steel in modern instruments, bent into a triangle shape. One of the angles is left open, with the ends of the bar not quite touching - this causes the instrument to be of indeterminate pitch. It is usually suspended from one of the other corners by a piece of thin wire or gut, leaving it free to vibrate. It is usually struck with a metal beater, giving a high-pitched, ringing tone. In folk music it is more often hooked over the hand so that one side can be damped by the fingers to vary the tone. The pitch can also be modulated slightly by varying the area struck and more subtle damping. The exact origins of the instrument are unknown, but a number of paintings from the Middle Ages depict the instrument being played by angels, which has led to the belief that it played some part in church services at that time. Other paintings show it being used in folk bands. Some triangles have jingling rings along the lower side. Although the instrument is nowadays generally in the form of an equilateral triangle, these early instruments were often isosceles triangles. The triangle has been used in the western classical orchestra since around the middle of the 18th century. Wolfgang Amadeus Mozart, Joseph Haydn and Ludwig van Beethoven all used it, though sparingly, usually in imitation of Janissary bands. The first piece to make the triangle really prominent was Franz Liszt's Piano Concerto No. 1, where it is used as a solo instrument in the second movement. The triangle appears to require no specialist ability to play and is often used in jokes and one liners as an archetypal instrument that even an idiot can play (see also Drummer jokes). The Martin Short sketch comedy character Ed Grimley is the best known example. However, triangle parts in classical music can be very demanding, and James Blades in the Grove Dictionary of Music and Musicians writes that "the triangle is by no means a simple instrument to play". In the hands of an expert it can be a surprisingly subtle and expressive instrument. Most difficulties in playing the triangle come from the complex rhythms which are sometimes written for it, although it can also be quite difficult to control the level of volume. Very quiet notes can be obtained by using a much lighter beater — knitting needles are sometimes used for the quietest notes. Composers sometimes call for a wooden beater to be used instead of a metal one, which gives a rather "duller" and quieter tone. Category:Idiophones Category:Latin percussion Category:Orchestral percussion ja:トライアングル

Geometry

Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematics and algebraic geometry.)

The earliest geometry

The earliest recorded beginnings of geometry may be traced to Ancient Egypt (see geometry in Egypt) and Ancient Babylon (see Babylonian mathematics) around 3000 B.C. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras; the Egyptians had a correct formula for the volume of a frustum of a square pyramid; the Babylonians had a trigonometry table. Chinese culture at this same time period was equally advanced, so it is likely that they had an equally advanced mathematics, but no artifacts have survived from which we could learn about it. This may be partly due to their early use of paper, rather than clay tablets or stone, to record their achievements.

The Greek period (c. 600 B.C. – 600 A.D.)

The Greek Period must be considered in detail, since geometry, for most of its history, was what the Greeks made it. For the Ancient Greeks, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies “eternal forms”, or abstractions, of which physical objects are only approximations; and they developed the idea of an “axiomatic theory”, which, for more than 2000 years, was regarded to be the ideal paradigm for all scientific theories.

Thales and Pythagoras

Thales (635-543 B.C.) of Ionia (now southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 B.C.) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and probably traveled to Babylon and Egypt. The theorem that bears his name was not his discovery, but he was the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers.

Plato

Plato (427-347 B.C.), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, “Let none enter here who are ignorant of geometry.” Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but a compass and straight edge – never measuring instruments such as a marked ruler or a protractor, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of the possible ruler and compass constructions, and three classic ruler-and-compass problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 B.C.), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.

Euclid

Euclid (365?-275? B.C.), probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in the ideal axiomatic form. The treatise is not a compendium of all that the Greeks knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt. The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read. # Any two points can be joined by a straight line. # Any finite straight line can be extended in a straight line. # A circle can be drawn with any center and any radius. # All right angles are equal to each other. # If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate). It was soon observed, and no doubt Euclid himself knew, that his fifth axiom could be replaced by the shorter statement “Given a line and a point not on the line, there is only one line through the given point and in the same plane with the given line that does not intersect the given line.” This is called Playfair’s Axiom, after the British teacher who proposed to make the replacement in all the school textbooks. The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof. Euclid’s first four axioms meet this criterion, but the fifth, even if replaced by Playfair’s Axiom, is not simple, and most would say not self-evident like the first four. The fifth resembled more the theorems that Euclid proved from the axioms. Furthermore, Euclid developed a substantial part of his theory of triangles without using the Fifth Axiom. The speculation arose, probably during Euclid’s lifetime, that the Fifth Axiom can and should be proved as a theorem from the first four, and thus is unnecessary as an axiom. Thus began many centuries of attempts to prove the Fifth Axiom, and the question was not settled until the 19th century.

Archimedes

Archimedes (287-212 B.C.), of Syracuse, Sicily, when it was a Greek city-state, was the greatest of the Greek mathematicians, and often named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.

After Archimedes

After Archimedes, Greek mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Greek geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The Middle Ages, Renaissance, and Reformation

The great library of Alexandria was burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later. Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign. The Islamic ascendency in the Middle East, north Africa, and Spain began about 640 A.D. Original Arab mathematics during this period was primarily algebraic rather than geometric, though there were important commentaries on geometry. Omar Khayyám, for example, was a geometer as well as a poet. Scholarship in Europe declined until even the great works of antiquity were lost to them, and survived only in the Islamic centers of learning. When Europe started to emerge from the intellectual darkness of the Middle Ages, the writers of Ancient Greece and Rome were rediscovered in Islamic libraries and translated from Arabic into Latin. Euclid’s Elements of Geometry was recovered, and the rigorous deductive methods of geometry were relearned. Development of geometry in the style of Euclid resumed, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.

The 17th and early 18th centuries

In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by Rene Descartes (1596-1650) and Pierre de Fermat (1601-1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591-1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Greek geometers, notably Pappus (c. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788-1867). In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton (1642-1727) and Gottfried Wilhelm von Leibniz (1646-1716). This was the beginning of a new field of mathematics now called analysis. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.

The late 18th and 19th centuries

Non-Euclidean geometry

The old problem of proving Euclid’s Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri, Lambert, and Legendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity. It remained to prove mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry. While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

Introduction of mathematical rigor

All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms were given by David Hilbert in 1894 in his dissertation Grundlagen der Geometrie (Foundations of Geometry). Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.

Analysis situs, or topology

In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as topology. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

The 20th century

External links

- [http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml What Is Geometry?] at cut-the-knot
- [http://www.elvenkids.com/tools/geometria/Geometria.php Geometria] An online tool to compute lines, surfaces and volumes of the main plane and solid figures, through direct and indirect formulas.
- [http://www.geogebra.at/ Geogebra] A free dynamic geometry tool, useful for exploring geometry.
- [http://agutie.homestead.com Geometry Step by Step from the Land of the Incas] by Antonio Gutierrez.
- [http://www.cut-the-knot.org/geometry.shtml Geometry] at cut-the-knot
- [http://www.islamicarchitecture.org/art/islamic-geometry-and-floral-patterns.html Islamic Geometry]
- Stanford Encyclopedia of Philosophy:
- [http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry]
- [http://plato.stanford.edu/entries/geometry-19th/ Geometry in the 19th Century]
- [http://www.egwald.com/geometry/index.php Online Interactive Geometric Objects] by Elmer G. Wiens Category:Geometry ko:기하학 ja:幾何学 simple:Geometry zh-min-nan:Kí-hô-ha̍k

Line segment

:Line (mathematics)#Line segment

Degree (angle)

:This article describes "degree" as a unit of angle. For alternative meanings, see degree (disambiguation). ---- A degree (in full, a degree of arc, arc degree, or arcdegree), usually symbolized °, is a measurement of plane angle, representing 1／360 of a full rotation. When that angle is with respect to a reference meridian, it indicates a location along a great circle of a sphere (such as Earth, Mars, or the celestial sphere).

History

The number 360 was probably adopted because of the number of days in a year. Primitive calendars, such as the Persian Calendar used 360 days for a year. This was most likely due to watching stars revolve around the North Star forming a circle one degree per day. Its application to measuring angles in geometry can possibly be traced to Thales who popularized geometry among the Greeks and lived in Anatolia (modern western Turkey) among people who had dealings with Egypt and Babylon. The degree and its subdivisions are the only units in use which are written without a separating space between the number and unit symbol (e.g. 15° 30', not 15 ° 30 ').

Further justification

360 is also readily divisible: 360 has 24 divisors (including 1 and 360), including every number from 1 to 10 except 7. For the number of degrees in a circle to be divisible by every number from 1 to 10, there would need to be 2520 degrees in a circle, which is a much less convenient number. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for latitudes and longitudes on the Earth, degree measurements may be written with decimal places, but the traditional sexagesimal unit subdivision is commonly seen. One degree is divided into 60 minutes (of arc), and one minute into 60 seconds (of arc). These units, also called the arcminute and arcsecond, are respectively represented as a single and double prime, or if necessary by a single and double closing quotation mark: for example, 40.1875° = 40° 11' 15". If still more accuracy is required, decimal divisions of the second are normally used, rather than thirds of 1／60 second, fourths of 1／60 of a third, and so on. These (rarely used) subdivisions were noted by writing the Roman numeral for the number of sixtieths in superscript: 1^I for a "prime" (minute of arc), 1^II for a second, 1^III for a third, 1^IV for a fourth, etc. Hence the modern symbols for the minute and second of arc.

Radians

In mathematics, angles in degrees are rarely used, as the convenient divisibility of the number 360 is not so important. For various reasons, mathematicians typically prefer to use the radian (symbol rad, an angle corresponding to an arc of a circle whose length equals the circle's radius (as opposed to its radius of curvature, or arcradius). Thus 180° = π rad, 1° ≈ 0.0174533 rad, and 1 rad ≈ 57.29578°. The radian is also the SI unit of angle.

Metric "decimal" degree

With the invention of the metric system, based on powers of ten, there was an attempt to define a "decimal degree" (grad or gon), so that the number of decimal degrees in a right angle would be 100, and there would be 400 degrees in a circle. Whilst this idea did not gain much momentum, most scientific calculators still support it.

Angle

This article is about angles in geometry. For other articles, see Angle (disambiguation) ---- An Angle (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Greek (angulοs) crooked, curved; both connected with the Aryan or Indo-European root ank-, to bend) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles provide a means of expressing the difference in slope between two rays meeting at a vertex without the need to explicitly define the slopes of the two rays. Angles are studied in geometry and trigonometry. Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.

Units of measure for angles

In order to measure an angle, a circle centered at the vertex is drawn. Since the circumference of a circle is always directly proportional to the length of its radius, the measure of the angle is independent of the size of the circle. Note that angles are dimensionless, since they are defined as the ratio of lengths.
- The radian measure of the angle is the length of the arc cut out by the angle, divided by the circle's radius. The SI system of units uses radians as the (derived) unit for angles. Because of the relationship to arc length, radians are a special unit. Sines and cosines whose argument is in radians have particular analytic properties, just as do exponential functions in the base e. (As we've discovered, this is no coincidence).
- The degree measure of the angle is the length of the arc, divided by the circumference of the circle, and multiplied by 360. The symbol for degrees is a small superscript circle, as in 360°. 2π radians is equal to 360° (a full circle), so one radian is about 57° and one degree is π/180 radians. Degrees are further broken down into minutes of arc and seconds of arc, which are 1/60th and 1/3600th of a degree, respectively. Minutes of arc are commonly encountered in discussions of external ballistics, as a minute of arc covers almost exactly 1 inch at 100 yards (1 m at 1200 m). A rifle capable of shooting "1 MOA", one minute of arc, can place all shots within 1 inch at 100 yards, 2 inches at 200 yards, etc. Minutes of arc were also used in navigation, and a nautical mile is roughly defined as one minute of arc of the earth's surface.
- The grad, also called grade, gradian or gon, is an angular measure where the arc is divided by the circumference, and multiplied by 400. It is used mostly in triangulation.
- The point is used in navigation, and is defined as 1/32 of a circle, or exactly 11.25°.
- The full circle or full turns represents the number or fraction of complete full turns. For example, π/2 radians = 90° = 1/4 full circle

Conventions on measurement

A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 is north-east. Negative bearings are not used in navigation, so north-west is 315. In mathematics radians are assumed unless specified otherwise because this removes the arbitrariness of the number 360 in the degree system and because the trigonometric functions can be developed into particularly simple Taylor series if their arguments are specified in radians.

Types of angles

An angle of π/2 radians or 90°, one-quarter of the full circle is called a right angle. Two line segments, rays, or lines (or any combination) which form a right angle are said to be either perpendicular or orthogonal:
- Angles smaller than a right angle are called acute angles
- Angles larger than a right angle are called obtuse angles.
- Angles equal to two right angles are called straight angles.
- Angles larger than two right angles are called reflex angles.
- The difference between an acute angle and a right angle is termed the complement of the angle
- The difference between an angle and two right angles is termed the supplement of the angle.

Some facts

In Euclidean geometry, the inner angles of a triangle add up to π radians or 180°; the inner angles of a quadrilateral add up to 2π radians or 360°. In general, the inner angles of a simple polygon with n sides add up to (n − 2) × π radians or (n − 2) × 180°. If two straight lines intersect, four angles are formed. Each one has an equal measure to the angle across from it; these congruent angles are called vertical angles. If a straight transversal line intersects two parallel lines, corresponding (alternate) angles at the two points of intersection are equal; adjacent angles are supplementary, that is they add to π radians or 180°.

A formal definition

A Euclidean angle is completely determined by the corresponding right triangle. In particular, if

\theta

is a Euclidean angle, it is true that :

\cos \theta = \frac

and :

\sin \theta = \frac

for two numbers

x

and

y

. So an angle can be legitimately given by two numbers

x

and

y

, or by a ratio

\frac

. What's more, to any such ratio there corresponds exactly one angle, since :

\frac = \frac = \frac

(i.e., changing the ratio will necessarily change the sin and cos, which in the geometric range

0 < \theta < 2\pi

are one-to-one - one sin or cos corresponds to one

\theta

Angles in different contexts

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula :

\mathbf \cdot \mathbf = \cos(\theta)\ \|\mathbf\|\ \|\mathbf\|.

This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>. The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave. Two intersecting planes form an angle, called their dihedral angle. It is defined as the angle between two lines normal to the planes. Also a plane and an intersecting line form an angle. This angle is equal to π/2 radians minus the angle between the intersecting line and the line that goes through the point of intersection and is perpendicular to the plane.

Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and g_ij are the components of the metric tensor G, :

\cos \theta = \frac
.

Angles in astronomy

In astronomy, one can measure the angular separation of two stars by imagining two lines through the Earth, each one intersecting one of the stars. Then the angle between those lines can be measured; this is the angular separation between the two stars. Astronomers also measure the apparent size of objects. For example, the full moon has an angular measurement of approximately 0.5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

Angles in maritime navigation

The modern format of angle used to indicate longitude or latitude is hemisphere degree minute.decimal, where there are 60 minutes in a degree, for instance N 51 23.438 or E 090 58.928. The obsolete (but still commonly used) format of angle used to indicate longitude or latitude is hemisphere degree minute' second", where there are 60 minutes in a degree and 60 seconds in a minute, for instance N 51 23′26″ or E 090 58′57″

External links

- [http://www.unitconversion.org/unit_converter/angle.html Online Angle Converter - convert between various units of angle, such as degree, radian, grad, gon, minute, second, sign, mil, and so on]
- [http://www.unitconversion.org/unit_converter/angle-v.html Interactive Angle Conversion Table - convert selected unit to all other units of angle]
- [http://www.cut-the-knot.org/triangle/ABisector.shtml Angle Bisectors] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/PerpBiInQuadri.shtml Angle Bisectors and Perpendiculars in a Quadrilateral] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/CyQuadri.shtml Angle Bisectors in a Quadrilateral] at cut-the-knot
- [http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml Constructing a triangle from its angle bisectors] at cut-the-knot
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units] Category:Elementary geometry Category:Trigonometry
- ko:각도 ja:角度 simple:Angle

300 BCE

Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC 297 BC 296 BC 295 BC ---- Births
- Deaths of children
- Events
- End of Jomon era and beginning of Yayoi era in Japan.
- Euclid's Elements written.
- Antioch was founded by Seleucus I Nicator (approximate date) Category:300s BC

Polygon

:For other use please see Polygon (disambiguation) A polygon (literally "many angle", see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. The straight line segments that make up the polygon are called its sides or edges and the points where the sides meet are the polygon's vertices. If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary.

Names and types

boundary boundary

Polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle and quadrilateral are exceptions. For larger numbers, mathematicians write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.

Naming polygons

To construct the name of a polygon with more than 20 and less than 100 sides, combine the prefixes as follows That is, a 42-sided figure would be named as follows: and a 50-sided figure But beyond nonagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons).

Taxonomic classification

The taxonomic classification of polygons is illustrated by the following tree:


                                       Polygon
                                      /       \
                                  Simple     Complex
                                 /     \
                            Convex     Concave
                             /
                        Cyclic 
                        /    
                   Regular

- A polygon is called simple if it is described by a single, non-intersecting boundary (hence has an inside and an outside); otherwise it is called complex.
- A simple polygon is called convex if it has no internal angles greater than 180°; otherwise it is called concave.
- A convex polygon is called concyclic or a cyclic polygon if all the vertices lie on a single circle.
- A cyclic polygon is called regular if all its sides are of equal length and all its angles are equal; for each number of sides, all regular polygons with the same number of sides are similar. The regular polygons most commonly found include:
- equilateral triangle
- square
- regular pentagon
- regular hexagon
- regular octagon Somewhat less common are:
- regular decagon
- regular dodecagon See also tilings of regular polygons.

Properties

We will assume Euclidean geometry throughout. An n-gon has 2n degrees of freedom, including 2 for position and 1 for rotational orientation, and 1 for over-all size, so 2n-4 for shape. In the case of a line of symmetry the latter reduces to n-2. Let k≥2. For an nk-gon with k-fold rotational symmetry (C_k), there are 2n-2 degrees of freedom for the shape. With additional mirror-image symmetry (D_k) there are n-1 degrees of freedom.

Angles

Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner angles of a simple n-gon is (n−2)π radians (or (n−2)180°), and the inner angle of a regular n-gon is (n−2)π/n radians (or (n−2)180°/n, or (n−2)/(2n) turns). This can be seen in two different ways:
- Moving around a simple n-gon (like a car on a road), the amount one "turns" at a vertex is 180° minus the inner angle. "Driving around" the polygon, one makes one full turn, so the sum of these turns must be 360°, from which the formula follows easily. The reasoning also applies if some inner angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between -1/2 and 1/2 winding.)
- Any simple n-gon can be considered to be made up of (n−2) triangles, each of which has an angle sum of π radians or 180°. Moving around an n-gon in general, the total amount one "turns" at the vertices can be any integer times 360°, e.g. 720° for a pentagram and 0° for an angular "eight". See also orbit (dynamics).

Area

orbit (dynamics)] Several formulae give the area of a regular polygon: :

A=\frac

: half the perimeter multiplied by the length of the apothem (the line drawn from the centre of the polygon perpendicular to a side) The area A of a simple polygon can be computed if the cartesian coordinates (x₁, y₁), (x₂, y₂), ..., (x_n, y_n) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is :A = ½ · (x₁y₂ − x₂y₁ + x₂y₃ − x₃y₂ + ... + x_ny₁ − x₁y_n) : = ½ · (x₁(y₂ − y_n) + x₂(y₃ − y₁) + x₃(y₄ − y₂) + ... + x_n(y₁ − y_n−1)) The formula was described by Meister in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem. If the polygon can be drawn on an equally-spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points. If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem.

Construction

All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle). A regular n-sided polygon can be constructed with ruler and compass if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.

Point in polygon test

In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x₀,y₀) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test.

Special cases

Some special cases are:
- angle of 0° or 180° (degenerate case)
- two non-adjacent sides are on the same line
- equilateral polygon: a polygon whose sides are equal [http://mathworld.wolfram.com/EquilateralPolygon.html (Williams 1979, pp. 31-32)]
- equiangular polygon: a polygon whose vertex angles are equal [http://mathworld.wolfram.com/EquiangularPolygon.html (Williams 1979, p. 32)] A triangle is equilateral iff it is equiangular. An equilateral quadrilateral is a rhombus, an equiangular quadrilateral is a rectangle or an "angular eight" with vertices on a rectangle.

Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher (i.e., a set of points such that no m-plane contains more than (m + 1) of them; such points are said to be in general position). For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron (in each case with interior). A regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length. The convex hull of any m of the n points is also a simplex, called an m-face. The 0-faces are called the vertices, the 1-faces are called the edges, the (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the binomial coefficient C(n + 1, m + 1). Consequently, the number of m-faces of an n-simplex may be found in column (m + 1) of row (n + 1) of Pascal's triangle.

The standard simplex

The standard n-simplex is the subset of Rⁿ⁺¹ given by :

\Delta^n = \

Removing the restriction t_i ≥ 0 in the above gives an n-dimensional affine subspace of Rⁿ⁺¹ containing the standard n-simplex. The vertices of the standard n-simplex are the points :e₀ = (1, 0, 0, …, 0), :e₁ = (0, 1, 0, …, 0), :

\vdots

:e_n = (0, 0, 0, …, 1). There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v₀, …, v_n) given by :

(t_0,\cdots,t_n) \mapsto \Sigma_i t_i v_i

The coefficients t_i are called the barycentric coordinates of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving or reversing.

Geometric properties

The volume of an n-simplex in n-dimensional space with vertices (v₀, ..., v_n) is :

\det
 \begin
  v_0-v_1 & v_1-v_2& \dots & v_-v_
 \end

where each column of the n × n determinant is the difference between two vertices. Any determinant which involves taking the difference between pairs of vertices, where the pairs connect the vertices as a simply connected graph will also give the (same) volume. Without the 1/n! it is the formula for the volume of an n-parallelepiped. One way to understand the 1/n! factor is as follows. If the coordinates of a point in a unit n-box are sorted, together with 0 and 1, and successive differences are taken, then since the results add to one, the result is a point in an n simplex spanned by the origin and the closest n vertices of the box. The taking of differences was an orthogonal (volume-preserving) transformation, but sorting compressed the space by a factor of n!. The volume under a standard n-simplex (i.e. between the origin and the simplex) is :

1 \over (n+1)!

The volume of a regular n-simplex with unit side length is :

\frac

as can be seen by multiplying the previous formula by xⁿ⁺¹, to get the volume under the n-simplex as a function of its vertex distance x from the origin, differentiating w.r.t.

x\,

, at

x=1/\sqrt

(where the n-simplex side length is 1), and normalizing by the length

dx/\sqrt\,

of the increment ( dx/(n+1),....dx/(n+1) ) along the normal vector.

Topology

Topologically, an n-simplex is equivalent to an n-ball. Every n-simplex is therefore an n-dimensional manifold with boundary. In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology. A finite set of k-simplexes embedded in an open subset of Rⁿ is called an affine k-chain. The simplexes in a chain need not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients. Note that each face of an n-simplex is an affine n-1-simplex, and thus the boundary of an n-simplex is an affine n-1-chain. Thus, if we denote one positively-oriented affine simplex as :

\sigma=[v_0,v_1,v_2,...v_n]

with the

v_j

denoting the vertices, then the boundary

\partial\sigma

of σ is the chain :

\partial\sigma = \sum_^n 
(-1)^j [v_0,...,v_,v_,...,v_n]

. More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map

f:\mathbb^n\rightarrow M

. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is, :

f(\sum_i a_i \sigma_i) = \sum_i a_i f(\sigma_i)

where the

a_i

are the integers denoting orientation and multiplicity. For the boundary operator

\partial

, one has: :

\partial f(\phi) = f (\partial \phi)

where φ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation (by definition of a map). A continuous map

f:\sigma\rightarrow X

to a topological space X is frequently referred to as a singular n-simplex.

References

- Walter Rudin, Principles of Mathematical Analysis (Third Edition), (1976) McGraw-Hill, New York, ISBN 0-07-054235-X (See chapter 10). Category:Polytopes Category:Topology ja:単体_(数学)

Polytope

In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. Beyond that, the term is used for a variety of related mathematical concepts. This is analogous to the way the term square may be used to refer to a square-shaped region of the plane, or just to its boundary, or even to a mere list of its vertices and edges along with some information about the way they are connected. The term was coined by Alicia Boole, the daughter of logician George Boole. The Platonic solids, or regular polytopes in three dimensions, were a major focus of study of ancient Greek mathematicians (most notably Euclid's Elements), probably because of their intrinsic aesthetic qualities. In modern times, polytopes and related concepts have found important applications in Computer graphics, Optimization, and numerous other fields.

Convex polytopes

One special kind of polytope is a convex polytope, which is the convex hull of a finite set of points. Convex polytopes can also be represented as the intersection of half-spaces. This intersection can be written as the matrix inequality: :

Ax \leq b

where A is an n by m matrix, n being the number of bounding half-spaces and m being the number of dimensions of the polytope. b is an n by 1 column vector. The coefficients of each row of A and b correspond with the coefficients of the linear inequality defining the respective half-space (see hyperplane for an explanation). Hence, each row in the matrix corresponds with a bounding hyperplane of the polytope. An n-dimensional convex polytope is bounded by a number of (n-1)-dimensional facets. Each pair of facets meet at an (n-2)-dimensional ridge. Ridges, in turn, meet at (n-3)-dimensional boundaries, and so on. These bounding sub-polytopes are referred to as faces (although the term may also refer specifically to the 2-dimensional case). A 0-dimensional face is a vertex; and a 1-dimensional face is an edge. A 3-dimensional face is called a cell. A facet consists of the points on the polytope that also satisfies the equality form of the matrix representation where only one row in A is present. Similarly, a ridge satisfies the equality form of the matrix representation where two rows in A are present. In general, an (n-j)-dimensional face satisfies the equality relation with j rows of A. These rows form the basis of the face. Geometrically speaking, this means that the face is the set of points on the polytope that lie in intersection of j of the polytope's bounding hyperplanes. The faces of a convex polytope thus form a lattice called its face lattice, where the subset relation is defined between basis hyperplanes. (The polytope itself is considered a 'face' in the face lattice, and is the maximum of the lattice.) Note that this terminology is not yet fully standardized. The term face is sometimes used to refer only to 2-dimensional subpolytopes, and other times used in place of facet. The word edge is sometimes used to refer to a ridge.

Simplicial decomposition

Now given any convex hull in r-dimensional space (but not in any (r-1)-plane, say) we can take linearly independent subsets of the vertices, and define r-simplices with them. In fact, you can choose several simplices in this way such that their union as sets is the original hull, and the intersection of any two is either empty or an s-simplex (for some s < r). For example, in the plane a square (convex hull of its corners) is the union of the two triangles (2-simplices), defined by a diagonal 1-simplex which is their intersection. In general, the definition (attributed to Alexandrov) is that an r-polytope is defined as a set with an r-simplicial decomposition with some additional properties. If a set has an r-simplicial decomposition this means it is a union of s-simplices for values of s with s at most r, that is closed under intersection, and such that the only time that one of simplices is contained in another is as a face. (For a more abstract treatment, see simplicial complex). What does this let us build? Let's start with the 1-simplex, or line segment. Then we have the line segment, of course, and anything that you can get by joining line segments end-to-end: center If two segments meet at each vertex (so not the case for the final one), we get a topological curve, called a polygonal curve. You can categorize these as open or closed, depending on whether the ends match up, and as simple or complex, depending on whether they intersect themselves. Closed polygonal curves are called polygons. Simple polygons in the plane are Jordan curves: they have an interior that is a topological disk. So does a 2-polytope (as you can see in the third example above), and these are often treated interchangeably with their boundary, the word polygon referring to either. Now the process can be repeated. Joining polygons along edges (1-faces) gives a polyhedral surface, called a skew polygon when open and a polyhedron when closed. Simple polyhedra are interchangeable with their interiors, which are 3-polytopes that can be used to build 4-dimensional forms (sometimes called polychora), and so on to higher polytopes. Other definitions (equivalent and otherwise) are possible and occur in the mathematical literature. Polytopes may be regarded as a tessellation of some sort of the manifold of their surface. The theory of abstract polytopes attempts to detach polytopes from the space containing them, considering their purely combinatorial properties. This allows the definition of the term to be extended to include objects for which it is difficult to define clearly a natural underlying space.
Uses
In the study of optimization, linear programming studies the maxima and minima of linear functions constricted to the boundary of an $n$ -dimensional polytope.
References

- Grünbaum, Branko, Convex polytopes, New York ; London : Springer, c2003. ISBN 0387004246.
Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.
See also

- list of regular polytopes
- abstract polytope
- Coxeter group
- discrete oriented polytope
- polychoron
- polyform
- polygon
- regular polytope
- Schläfli symbol
- simplex
- tiling Category:Geometry Category:Polytopes

Sine
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below. In modern usage, there are six basic trigonometric functions, which are tabulated below along with equations relating them to one another. (Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can equally define them geometrically or by other means and derive the relations.) A few other functions were common historically (and appeared in the earliest tables), but are now little-used, such as:
- versed sine (versin = 1 − cos)
- exsecant (exsec = sec − 1). Many more relations between these functions are listed in the article about trigonometric identities.
History
The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy also derived the equivalent of the half-angle formula sin²(A/2) = (1 − cos(A))/2, allowing him to create tables with any desired accuracy. Neither the tables of Hipparchus nor of Ptolemy have survived to the present day. The next significant development of trigonometry was in India, in the works known as the Siddhantas (4th–5th century), which first defined the sine as the modern relationship between half an angle and half a chord. The Siddhantas also contained the earliest surviving tables of sine values (along with 1 − cos values), in 3.75-degree intervals from 0 to 90 degrees. The Hindu works were later translated and expanded by the Arabs, who by the 10th century (in the work of Abu'l-Wefa) were using all six trigonometric functions, and had sine tables in 0.25-degree increments, to 8 decimal places of accuracy, as well as tables of tangent values. Our modern word sine comes, via sinus ("bay" or "fold") in Latin, from a mistranslation of the Sanskrit jiva (or jya). jiva (originally called ardha-jiva, "half-chord", in the 6th century Aryabhata) was transliterated by the Arabs as jiba (جب), but was confused for another word, jaib (جب) ("bay"), by European translators such as Robert of Chester and Gherardo of Cremona in Toledo in the 12th century, probably because jiba (جب) and jaib (جب) are written the same in Arabic (many vowels are excluded from words written in the Arabic alphabet). All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by the De triangulis omnimodus (1464) of Regiomontanus (1436–1476), as well as his later Tabulae directionum (which included the tangent function, unnamed). The Opus palatinum de triangulis of Rheticus, a student of Copernicus, was the first to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596. The Introductio in analysin infinitorum (1748) of Euler was primarily responsible for establishing the analytic treatment of trigonometric functions, defining them as infinite series and presenting "Euler's formula" e^ix = cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec..
Right triangle definitions
Euler's formula In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A: We use the following names for the sides of the triangle:
- The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
- The opposite side is the side opposite to the angle we are interested in, in this case a.
- The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b. All triangles are taken to exist in the Euclidean plane so that the inside angles of each triangle sum to π radians (or 180°). Then, 1) The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case : $\sin A = \frac = \frac$ . Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar. The set of zeroes of sine is $\left\$ . 2) The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case : $\cos A = \frac = \frac$ . The set of zeroes of cosine is $\left\$ . 3) The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case : $\tan A = \frac = \frac$ . The set of zeroes of tangent is $\left\$ . The remaining three functions are best defined using the above three functions. 4) The cosecant csc(A) is the multiplicative inverse of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side: : $\csc A = \frac = \frac$ . 5) The secant sec(A) is the multiplicative inverse of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side: : $\sec A = \frac = \frac$ . 6) The cotangent cot(A) is the multiplicative inverse of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side: : $\cot A = \frac = \frac$ .
Mnemonics
There are a number of mnemonics for the above definitions, for example SOHCAHTOA (sounds like "soak a toe-a" or "sock-a toe-a" depending upon which side of the Atlantic you hail from. Can also be read as "soccer tour"). It means:
- SOH ... sin = opposite/hypotenuse
- CAH ... cos = adjacent/hypotenuse
- TOA ... tan = opposite/adjacent. Many other such words and phrases have been contrived. For more see: trigonometry mnemonics.
Slope definitions
Equivalent to the right-triangle definitions, the trigonometric functions can be defined in terms of the rise, run, and slope of a line segment relative to some horizontal line. The slope is commonly taught as "rise over run" or rise/run. The three main trigonometric functions are commonly taught in the order sine, cosine, tangent. With a unit circle, this gives rise to the following matchings: # Sine is first, rise is first. Sine takes an angle and tells the rise. # Cosine is second, run is second. Cosine takes an angle and tells the run. # Tangent is the slope formula that combines the rise and run. Tangent takes an angle and tells the slope. This shows the main use of tangent and arctangent, which is converting between the two ways of telling how slanted a line is: angles and slopes. While the radius of the circle makes no difference for the slope (the slope doesn't depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run, just multiply the sine and cosine by the radius. For instance, if the circle has radius 5, the run at an angle of 1 is 5 cos(1).
Unit-circle definitions
unit circle] The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trig functions for all positive and negative arguments, not just for angles between 0 and π/2 radians. It also provides a single visual picture that encapsulates at once all the important triangles used so far. The equation for the unit circle is: : $x^2 + y^2 = 1 \,$ In the picture, some common angles, measured in radians, are given. Measurements in the counter clockwise direction are positive angles and measurements in the clockwise direction are negative angles. Let a line making an angle of θ with the positive half of the x-axis intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1. circle For angles greater than 2π or less than −2π, simply continue to rotate around the circle. In this way, sine and cosine become periodic functions with period 2π: : $\sin\theta = \sin\left(\theta + 2\pi k \right)$ : $\cos\theta = \cos\left(\theta + 2\pi k \right)$ for any angle θ and any integer k. The smallest positive period of a periodic function is called the primitive period of the function. The primitive period of the sine, cosine, secant, or cosecant is a full circle, i.e. 2π radians or 360 degrees; the primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees. Above, only sine and cosine were defined directly by the unit circle, but the other four trig functions can be defined by: : $\tan\theta = \frac \quad \sec\theta = \frac$ : $\csc\theta = \frac \quad \cot\theta = \frac$ integer Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (shown at right), and similar such geometric definitions were used historically. In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India (see below). cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD. tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(θ) is another tangent segment, AF. sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively. DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle). From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.)
Series definitions
exsec Please note: Here, and generally in calculus, all angles are measured in radians. (See also below). Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is the opposite of sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x: : $\sin x = x - \frac + \frac - \frac + \cdots = \sum_^\infty \frac$ : $\cos x = 1 - \frac + \frac - \frac + \cdots = \sum_^\infty \frac$ These identities are often taken as the definitions of the sine and cosine function. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (e.g. in Fourier series), since the theory of infinite series can be developed from the foundations of the real number system, independent of any geometric considerations. The differentiability and continuity of these functions are then established from the series definitions alone. Other series can be found: : $\tan x = x + \frac + \frac + \frac + \cdots = \sum_^\infty \frac, \quad \left | x \right | < \frac$ : $\csc x = \frac + \frac + \frac + \frac + \cdots = \frac + \sum_^\infty \frac, \quad 0 < \left | x \right | < \frac$ : $\sec x = 1 + \frac + \frac + \frac + \cdots = 1+ \sum_^\infty \frac, \quad \left | x \right | < \frac$ : $\cot x = \frac - \frac - \frac - \frac - \cdots = \frac - \sum_^\infty \frac, \quad 0 < \left | x \right | < \frac$ where : $E_n \,$ is the nth Euler number, and : $U_n \,$ is the nth up/down number.
Relationship to exponential function
It can be shown from the series definitions that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary: : $e^ = \cos\theta + i\sin\theta \,.$ This relationship was first noted by Euler and the identity is called Euler's formula. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in the complex plane, defined by e^ix, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent. Furthermore, this allows for the definition of the trigonometric functions for complex arguments z: : $\sin z \, = \, \sum_^\fracz^ \, = \, = -\imath \sinh \left( \imath z\right)$ : $\cos z \, = \, \sum_^\fracz^ \, = \, = \cosh \left(\imath z\right)$ where i^{2 = −1. Also, for purely real x,

: $\cos x \, = \, \mbox (e^)$

: $\sin x \, = \, \mbox (e^)$

It is also shown that exponential processes are intimately linked to periodic behavior.

Definitions via differential equations

Both the sine and cosine functions satisfy the differential equation

: $y\,$ =-y

i.e. each is the additive inverse of its own second derivative. Within the 2-dimensional vector space V consisting of all solutions of this equation, the sine function is the unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and the cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly independent, together they form a basis of V. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See linear differential equation.) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions. See the trigonometric identity article for this technique.

The tangent function is the unique solution of the nonlinear differential equation

: $y\,'=1+y^2$

satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation; see [http://www.usfca.edu/vca/PDF/vca-preface.pdf].

The significance of radians
Radians constitute a special argument to the sine and cosine functions. In particular, only those sines and cosines which map radians to ratios satisfy the differential equations which classically describe them. If an argument to sine or cosine in radians is scaled by frequency,
: $f(x) = \sin(kx); k \ne 0, k \ne 1 \,$
then the derivatives will scale by amplitude.
: $f'(x) = k\cos(kx) \,$ .

Here, k is a constant that represents a mapping between units. If x is in degrees, then
: $k = \frac.$

This means that the second derivative of a sine in degrees satisfies not the differential equation
: $y$ = -y \,,
but
: $y$ = -k^2y \,;
similarly for cosine.

This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.

Other definitions

Theorem: There exists exactly one pair of real functions
s, c with the following properties:

For any $x, y \in\mathbb$ :

: $s(x)^2 + c(x)^2 = 1,\,$

: $s(x+y) = s(x)c(y) + c(x)s(y),\,$

: $c(x+y) = c(x)c(y) - s(x)s(y),\,$

: $0 < xc(x) < s(x) < x\ \mathrm\ 0 < x < 1.$

Computation

The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. In this section, however, we describe more details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found. (Below, it suffices to consider a small range of angles, say 0 to π/2, since all other angles can be reduced to this range by the periodicity and symmetries of the trigonometric functions.)

Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of their values, calculated to many significant figures. Such tables have been available for as long as trigonometric functions have been described (see History, above), and were typically generated by repeated application of the half-angle and angle-addition}