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Half Line The notion of LINE or STRAIGHT LINE was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature ) with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined in this manner: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which will leave from its imaginary moving some vestige in length, exempt of any width [...More...]  "Half Line" on: Wikipedia Yahoo 

Symmetry SYMMETRY (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection , rotation or scaling . Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together. Mathematical symmetry may be observed with respect to the passage of time ; as a spatial relationship ; through geometric transformations ; through other kinds of functional transformations; and as an aspect of abstract objects , theoretic models , language , music and even knowledge itself [...More...]  "Symmetry" on: Wikipedia Yahoo 

Similarity (geometry) Two geometrical objects are called SIMILAR if they both have the same shape , or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation , rotation and reflection . This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other [...More...]  "Similarity (geometry)" on: Wikipedia Yahoo 

Onedimensional Space In physics and mathematics , a sequence of n numbers can specify a location in ndimensional space. When n = 1, the set of all such locations is called a ONEDIMENSIONAL SPACE . An example of a onedimensional space is the number line , where the position of each point on it can be described by a single number. In algebraic geometry there are several structures that are technically onedimensional spaces but referred to in other terms. A field k is a onedimensional vector space over itself. Similarly, the projective line over k is a onedimensional space. In particular, if k = ℂ, the complex numbers , then the complex projective line P1(ℂ) is onedimensional with respect to ℂ, even though it is also known as the Riemann sphere . More generally, a ring is a lengthone module over itself. Similarly, the projective line over a ring is a onedimensional space over the ring [...More...]  "Onedimensional Space" on: Wikipedia Yahoo 

Point (geometry) In modern mathematics , a POINT refers usually to an element of some set called a space . More specifically, in Euclidean geometry Euclidean geometry , a point is a primitive notion upon which the geometry is built, meaning that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms , that it must satisfy. In particular, the geometric points do not have any length , area , volume or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space Euclidean space [...More...]  "Point (geometry)" on: Wikipedia Yahoo 

Ray (geometry) The notion of LINE or STRAIGHT LINE was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature ) with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined in this manner: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which will leave from its imaginary moving some vestige in length, exempt of any width [...More...]  "Ray (geometry)" on: Wikipedia Yahoo 

Congruence (geometry) In geometry , two figures or objects are CONGRUENT if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called CONGRUENT if, and only if, one can be transformed into the other by an isometry , i.e., a combination of RIGID MOTIONS, namely a translation , a rotation , and a reflection . This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted. In elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects [...More...]  "Congruence (geometry)" on: Wikipedia Yahoo 

Vertex (geometry) In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place [...More...]  "Vertex (geometry)" on: Wikipedia Yahoo 

Diagonal In geometry , a DIAGONAL is a line segment joining two vertices of a polygon or polyhedron , when those vertices are not on the same edge . Informally, any sloping line is called diagonal. The word "diagonal" derives from the ancient Greek διαγώνιος diagonios, "from angle to angle" (from διά dia, "through", "across" and γωνία gonia, "angle", related to gony "knee"); it was used by both Strabo Strabo and Euclid Euclid to refer to a line connecting two vertices of a rhombus or cuboid , and later adopted into Latin as diagonus ("slanting line"). In matrix algebra , a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are also other, nonmathematical uses [...More...]  "Diagonal" on: Wikipedia Yahoo 

Orthogonal In mathematics , ORTHOGONALITY is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms . Two elements u and v of a vector space with bilinear form B are ORTHOGONAL when B(u,v) = 0. When the bilinear form corresponds to a pseudo Euclidean space Euclidean space , there are nonperpendicular vectors that are hyperbolicorthogonal . In the case of function spaces , families of orthogonal functions are used to form a basis [...More...]  "Orthogonal" on: Wikipedia Yahoo 

Perpendicular In elementary geometry , the property of being PERPENDICULAR (PERPENDICULARITY) is the relationship between two lines which meet at a right angle (90 degrees ). The property extends to other related geometric objects . A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles . Perpendicularity can be shown to be symmetric , meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order. Perpendicularity easily extends to segments and rays [...More...]  "Perpendicular" on: Wikipedia Yahoo 

Parallel (geometry) In geometry , PARALLEL lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in threedimensional Euclidean space Euclidean space that do not share a point are said to be parallel. However, two lines in threedimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines . Parallel planes are planes in the same threedimensional space that never meet. Parallel lines are the subject of Euclid Euclid 's parallel postulate . Parallelism is primarily a property of affine geometries and Euclidean space is a special instance of this type of geometry. Some other spaces, such as hyperbolic space , have analogous properties that are sometimes referred to as parallelism [...More...]  "Parallel (geometry)" on: Wikipedia Yahoo 

Length In geometric measurements, LENGTH is the most extended dimension of an object. In the International System of Quantities , length is any quantity with dimension distance. In other contexts "length" is the measured dimension of an object. For example, it is possible to cut a length of a wire which is shorter than wire thickness. Length Length may be distinguished from height , which is vertical extent, and width or breadth, which are the distance from side to side, measuring across the object at right angles to the length. Length Length is a measure of one dimension, whereas area is a measure of two dimensions (length squared) and volume is a measure of three dimensions (length cubed). In most systems of measurement , the unit of length is a base unit , from which other units are derived. The metric length of one kilometre is equivalent to the imperial measurement of 0.62137 miles [...More...]  "Length" on: Wikipedia Yahoo 

Twodimensional Space In physics and mathematics , TWODIMENSIONAL SPACE or BIDIMENSIONAL SPACE is a geometric model of the planar projection of the physical universe . The two dimensions are commonly called length and width. Both directions lie in the same plane . A sequence of n real numbers can be understood as a location in ndimensional space. When n = 2, the set of all such locations is called twodimensional space or bidimensional space, and usually is thought of as a Euclidean space Euclidean space [...More...]  "Twodimensional Space" on: Wikipedia Yahoo 

Square In geometry , a SQUARE is a regular quadrilateral , which means that it has four equal sides and four equal angles (90degree angles, or right 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 {displaystyle square } ABCD [...More...]  "Square" on: Wikipedia Yahoo 

Parallelogram In Euclidean geometry , a PARALLELOGRAM is a simple (nonselfintersecting ) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The threedimensional counterpart of a parallelogram is a parallelepiped . The etymology (in Greek παραλληλόγραμμον, a shape "of parallel lines") reflects the definition [...More...]  "Parallelogram" on: Wikipedia Yahoo 