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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.[1] 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
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Face (geometry)
In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object;[1] a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).[2]Contents1 Polygonal face1.1 Number of polygonal faces of a polyhedron2 k-face2.1 Cell or 3-face 2.2 Facet or (n-1)-face 2.3 Ridge or (n-2)-face 2.4 Peak or (n-3)-face3 See also 4 References 5 External linksPolygonal face[edit] In elementary geometry, a face is a polygon on the boundary of a polyhedron.[2][3] Other names for a polygonal face include side of a polyhedron, and tile of a Euclidean plane tessellation. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope
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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.[1] 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[2] (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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Geometry
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
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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.[1] 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.[2]Contents1 Existence and uniqueness 2 Shortest path 3 Higher-dimensional polytope nets 4 See also 5 References 6 External linksExistence and uniqueness[edit] Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated
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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
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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
Desargues' theorem
is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings[1] although it is "a very pervasive and important concept in (modern) mathematics"[2] and "an important general theme that has manifestations in almost every area of mathematics".[3] 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 Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.Contents1 Definitions1.1 As a flat slice 1.2 As a spherical polygon 1.3 As the set of connected vertices 1.4 Abstract definition2 General properties 3 Constructions3.1 From the adjacent vertices 3.2 Dorman Luke
Dorman Luke
construction 3.3 Regular polytopes4 An example vertex figure of a honeycomb 5 Edge figure 6 See also 7 References7.1 Notes 7.2 Bibliography8 External linksDefinitions[edit]"Whole-edge" vertex figure of the cubeSpherical vertex figure of the cubePoint-set vertex figure of the cubeTake some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex
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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
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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.[1] 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[2] (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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Convex Polyhedron
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.[1] 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[2] (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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Euclidean Geometry
Euclidean geometry
Euclidean geometry
is a mathematical system attributed to the Alexandrian Greek mathematician 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,[1] Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.[2] 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
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Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, CC (February 9, 1907 – March 31, 2003)[2] was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London
London
but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald.[3] He was most noted for his work on regular polytopes and higher-dimensional geometries
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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). 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, there is only one plane and all triangles are contained in it; however, in higher dimensional Euclidean spaces this is no longer true
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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.[1] The word circumsphere is sometimes used to mean the same thing.[2] 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,[3] and the center point of this sphere is called the circumcenter of P.[4]Contents1 Existence and optimality 2 Related concepts 3 References 4 External linksExistence and optimality[edit] When 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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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.Contents1 Interpretations 2 See also 3 References 4 External linksInterpretations[edit] All 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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