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Tetrahedron
In geometry , a TETRAHEDRON (plural: TETRAHEDRA or TETRAHEDRONS), also known as a TRIANGULAR PYRAMID, is a polyhedron composed of four triangular faces , six straight edges , and four vertex corners . The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3-SIMPLEX. The tetrahedron is one kind of pyramid , which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra , a tetrahedron can be folded from a single sheet of paper. It has two such nets
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Triangular
A TRIANGLE is a polygon with three edges and three vertices . It is one of the basic shapes in geometry . A triangle with vertices A, B, and C is denoted A B C {displaystyle triangle ABC} . In Euclidean geometry
Euclidean geometry
any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space
Euclidean space
), in other words, there is only one plane that contains that triangle and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane
Euclidean plane
, there is only one plane and all triangles are contained in it, however, in higher dimensional Euclidean spaces this is no longer true. This article is about triangles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted
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Edge (geometry)
In geometry , an EDGE is a particular type of line segment joining two vertices in a polygon , polyhedron , or higher-dimensional polytope . In a polygon, an edge is a line segment on the boundary, and is often called a SIDE. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal . CONTENTS * 1 Relation to edges in graphs * 2 Number of edges in a polyhedron * 3 Incidences with other faces * 4 Alternative terminology * 5 See also * 6 References * 7 External links RELATION TO EDGES IN GRAPHSIn graph theory , an edge is an abstract object connecting two graph vertices , unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment
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Geometry
GEOMETRY (from the Ancient Greek : γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer . Geometry
Geometry
arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . Geometry
Geometry
began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid
Euclid
, whose treatment, Euclid\'s Elements , set a standard for many centuries to follow. Geometry
Geometry
arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC
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Net (polyhedron)
In geometry a NET of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard. An early instance of polyhedral nets appears in the works of Albrecht Dürer , whose 1525 book Unterweysung der Messung mit dem Zyrkel und Rychtscheyd included nets for the Platonic solids and several of the Archimedean solids
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Vertex Figure
In geometry , a VERTEX FIGURE, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. CONTENTS* 1 Definitions – theme and variations * 1.1 As a flat slice * 1.2 As a spherical polygon * 1.3 As the set of connected vertices * 1.4 Abstract definition * 2 General properties * 3 Dorman Luke construction * 4 Regular polytopes * 5 An example vertex figure of a honeycomb * 6 Edge figure * 7 See also * 8 References * 8.1 Notes * 8.2 Bibliography * 9 External links DEFINITIONS – THEME AND VARIATIONSTake some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance
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Self-dual
In mathematics , a DUALITY, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points , so that the dual of A is A itself. For example, Desargues\' theorem is self-dual in this sense under the standard duality in projective geometry . In mathematical contexts, duality has numerous meanings although it is "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings , bilinear functions from an object of one type and another object of the second type to some family of scalars
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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
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Convex Polytope
A CONVEX POLYTOPE is a special case of a polytope , having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms "convex polytope" and "CONVEX POLYHEDRON" interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope. In addition, some texts require a polytope to be a bounded set , while others (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or (n-1)-manifold. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming . A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum
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Tangent
In geometry , the TANGENT LINE (or simply TANGENT) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passes through the point (c, f (c)) on the curve and has slope f '(c) where f ' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space
Euclidean space
. As it passes through the point where the tangent line and the curve meet, called the POINT OF TANGENCY, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. Similarly, the TANGENT PLANE to a surface at a given point is the plane that "just touches" the surface at that point
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Area (mathematics)
AREA is the quantity that expresses the extent of a two-dimensional figure or shape , or planar lamina , in the plane . Surface area is its analog on the two-dimensional surface of a three-dimensional object . Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares
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Insphere
In geometry , the INSCRIBED SPHERE or INSPHERE of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual to the dual polyhedron 's circumsphere . The radius of the sphere inscribed in a polyhedron P is called the INRADIUS of P. CONTENTS * 1 Interpretations * 2 See also * 3 References * 4 External links INTERPRETATIONSAll regular polyhedra have inscribed spheres, but most irregular polyhedra do not have all facets tangent to a common sphere, although it is still possible to define the largest contained sphere for such shapes
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Circumsphere
In geometry , a CIRCUMSCRIBED SPHERE of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word CIRCUMSPHERE is sometimes used to mean the same thing. As in the case of two-dimensional circumscribed circles , the radius of a sphere circumscribed around a polyhedron P is called the circumradius of P, and the center point of this sphere is called the circumcenter of P. CONTENTS * 1 Existence and optimality * 2 Related concepts * 3 References * 4 External links EXISTENCE AND OPTIMALITYWhen it exists, a circumscribed sphere need not be the smallest sphere containing the polyhedron ; for instance, the tetrahedron formed by a vertex of a cube and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator
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Euclidean Geometry
EUCLIDEAN GEOMETRY is a mathematical system attributed to the Alexandrian Greek mathematician Euclid
Euclid
, which he described in his textbook on geometry : the Elements . Euclid's method consists in assuming a small set of intuitively appealing axioms , and deducing many other propositions (theorems ) from these. Although many of Euclid's 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 secondary school as the first axiomatic system and the first examples of formal proof . It goes on to the solid geometry of three dimensions . Much of the Elements states results of what are now called algebra and number theory , explained in geometrical language
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Convex Polyhedra
A CONVEX POLYTOPE is a special case of a polytope , having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms "convex polytope" and "CONVEX POLYHEDRON" interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope. In addition, some texts require a polytope to be a bounded set , while others (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or (n-1)-manifold. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming . A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum
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Dihedral Angle
A DIHEDRAL ANGLE is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge . In higher dimension, a dihedral angle represents the angle between two hyperplanes . CONTENTS * 1 Definitions * 2 Dihedral angles in stereochemistry * 3 Dihedral angles of proteins * 3.1 Converting from dihedral angles to Cartesian coordinates in chains * 4 Calculation of a dihedral angle * 5 Dihedral angles in polyhedra * 6 See also * 7 References * 8 External links DEFINITIONSA dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection
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