1.
Vierkant (meetkunde)
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
2.
Regelmatige vijfhoek
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In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting, a self-intersecting regular pentagon is called a pentagram. A regular pentagon has Schläfli symbol and interior angles are 108°, a regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5. The diagonals of a regular pentagon are in the golden ratio to its sides. The area of a regular convex pentagon with side length t is given by A = t 225 +1054 =5 t 2 tan 4 ≈1.720 t 2. A pentagram or pentangle is a regular star pentagon and its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio. The area of any polygon is, A =12 P r where P is the perimeter of the polygon. Substituting the regular pentagons values for P and r gives the formula A =12 ×5 t × t tan 2 =5 t 2 tan 4 with side length t, like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the r of the inscribed circle. Like every regular polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, the regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon, one method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwells Polyhedra. The top panel shows the construction used in Richmonds method to create the side of the inscribed pentagon, the circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius and this point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the axis at point Q. A horizontal line through Q intersects the circle at point P, to determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras theorem and two sides, the hypotenuse of the triangle is found as 5 /2
3.
Euclidische meetkunde
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
4.
Passer (gereedschap)
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A pair of compasses, also known simply as a compass, is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can also be used as tools to measure distances, Compasses can be used for mathematics, drafting, navigation and other purposes. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. Prior to computerization, compasses and other tools for manual drafting were often packaged as a bow set with interchangeable parts, today these facilities are more often provided by computer-aided design programs, so the physical tools serve mainly a didactic purpose in teaching geometry, technical drawing, etc. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. The handle is usually half a inch long. Users can grip it between their pointer finger and thumb, there are two types of legs in a pair of compasses, the straight or the steady leg and the adjustable one. Each has a purpose, the steady leg serves as the basis or support for the needle point. The screw on your hinge holds the two legs in its position, the hinge can be adjusted depending on desired stiffness, the tighter the screw the better the compass’ performance. The needle point is located on the leg, and serves as the center point of circles that are drawn. The pencil lead draws the circle on a paper or material. This holds the lead or pen in place. Circles can be made by fastening one leg of the compasses into the paper with the spike, putting the pencil on the paper, the radius of the circle can be adjusted by changing the angle of the hinge. Distances can be measured on a map using compasses with two spikes, also called a dividing compass, to use a pair of compasses, place the points on a ruler and open it to the measurement of ½ of the measurement of the circle that is desired. For instance, if one desires to draw a 3 inch circle, next, place the point on the spot that you wish the center of your circle to be, and then rotate the section that has the pencil lead around the point, using the handle. Compasses-and-straightedge constructions are used to illustrate principles of plane geometry, although a real pair of compasses is used to draft visible illustrations, the ideal compass used in proofs is an abstract creator of perfect circles
5.
Liniaal
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A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing, engineering and building to measure distances or to rule straight lines. The ruler is a straightedge which may also contain calibrated lines to measure distances, rulers have long been made from different materials and in a wide range of sizes. Plastics have also used since they were invented, they can be molded with length markings instead of being scribed. Metal is used for more durable rulers for use in the workshop,12 inches or 30 cm in length is useful for a ruler to be kept on a desk to help in drawing. Shorter rulers are convenient for keeping in a pocket, longer rulers, e. g.18 inches are necessary in some cases. Rigid wooden or plastic yardsticks,1 yard long, and meter sticks,1 meter long, are also used, classically, long measuring rods were used for larger projects, now superseded by tape measure or laser rangefinders. Desk rulers are used for three purposes, to measure, to aid in drawing straight lines and as a straight guide for cutting and scoring with a blade. Practical rulers have distance markings along their edges, a line gauge is a type of ruler used in the printing industry. These may be made from a variety of materials, typically metal or clear plastic, units of measurement on a basic line gauge usually include inches, agate, picas, and points. More detailed line gauges may contain sample widths of lines, samples of common type in several point sizes, measuring instruments similar in function to rulers are made portable by folding or retracting into a coil when not in use. When extended for use, they are straight, like a ruler, the illustrations on this page show a 2-meter carpenters rule, which folds down to a length of 24 cm to easily fit in a pocket, and a 5-meter-long tape, which retracts into a small housing. A flexible length measuring instrument which is not necessarily straight in use is the tailors fabric tape measure and it is used to measure around a solid body, e. g. a persons waist measurement, as well as linear measurement, e. g. inside leg. It is rolled up when not in use, taking up little space, a contraction rule is made having larger divisions than standard measures to allow for shrinkage of a metal casting. They may also be known as a shrinkage or shrink rule, a ruler software program can be used to measure pixels on a computer screen or mobile phone. These programs are known as screen rulers. In geometry, a ruler without any marks on it may be used only for drawing lines between points. A straightedge is used to help draw accurate graphs and tables. A ruler and compass construction refers to using an unmarked ruler
6.
Elementen van Euclides
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Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
7.
Lijn (meetkunde)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
8.
Cirkel
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
9.
Middelpunt (meetkunde)
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In geometry, a centre of an object is a point in some sense in the middle of the object. According to the definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of groups then a centre is a fixed point of all the isometries which move the object onto itself. The centre of a circle is the point equidistant from the points on the edge, similarly the centre of a sphere is the point equidistant from the points on the surface, and the centre of a line segment is the midpoint of the two ends. For objects with several symmetries, the centre of symmetry is the point left unchanged by the symmetric actions, so the centre of a square, rectangle, rhombus or parallelogram is where the diagonals intersect, this being the fixed point of rotational symmetries. Similarly the centre of an ellipse or a hyperbola is where the axes intersect. For an equilateral triangle, these are the point, which lies at the intersection of the three axes of symmetry of the triangle, one third of the distance from its base to its apex. e. F=thf for some real power h, thus the position of a centre is independent of scale. F is symmetric in its last two arguments, i. e. f= f, thus position of a centre in a triangle is the mirror-image of its position in the original triangle. This strict definition excludes pairs of points such as the Brocard points. The Encyclopedia of Triangle Centers lists over 9,000 different triangle centres, a tangential polygon has each of its sides tangent to a particular circle, called the incircle or inscribed circle. The centre of the incircle, called the incentre, can be considered a centre of the polygon, a cyclic polygon has each of its vertices on a particular circle, called the circumcircle or circumscribed circle. The centre of the circumcircle, called the circumcentre, can be considered a centre of the polygon, if a polygon is both tangential and cyclic, it is called bicentric. The incentre and circumcentre of a polygon are not in general the same point. The centre of a general polygon can be defined in different ways. The vertex centroid comes from considering the polygon as being empty, the side centroid comes from considering the sides to have constant mass per unit length. The usual centre, called just the centroid comes from considering the surface of the polygon as having constant density and these three points are in general not all the same point. Centrepoint Centre of mass Chebyshev centre Fixed points of isometry groups in Euclidean space
10.
Getallenlijn
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In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by R. Every point of a line is assumed to correspond to a real number. Often the integers are shown as specially-marked points evenly spaced on the line, although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, a number line is usually represented as being horizontal, but in a Cartesian coordinate plane the vertical axis is also a number line. Another convention uses only one arrowhead which indicates the direction in which numbers grow, if a particular number is farther to the right on the number line than is another number, then the first number is greater than the second. The distance between them is the magnitude of their difference—that is, it measures the first number minus the second one, taking this difference is the process of subtraction. Thus, for example, the length of a segment between 0 and some other number represents the magnitude of the latter number. Two numbers can be added by picking up the length from 0 to one of the numbers and this gives a result that is 3 combined lengths of 5 each, since the process ends at 15, we find that 5 ×3 =15. This puts the right end of the length 2 at the end of the length from 0 to 6. Since three lengths of 2 filled the length 6,2 goes into 6 three times, the section of the number line between two numbers is called an interval. If the section includes both numbers it is said to be an interval, while if it excludes both numbers it is called an open interval. If it includes one of the numbers but not the other one, all the points extending forever in one direction from a particular point are together known as a ray. If the ray includes the point, it is a closed ray. Sometimes it is convenient to scale the numbers on the line with a logarithmic scale. This approach is useful, for example, in illustrating a sequence of events in the history of the universe or of evolution, a line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called line, extends the number line to a complex number plane. Together these lines form what is known as the Cartesian coordinate system, further, the Cartesian coordinate system can itself be extended by visualizing a third number line coming out of the screen, measuring a third variable called z. Positive numbers are closer to the eyes than the screen is, while negative numbers are behind the screen