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Polyhedron
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{{short description|Three-dimensional shape with flat faces, straight edges, and sharp corners}} {{other uses}} {{redirect-distinguish|Polyhedra|Polyhedra (software)}} {{infobox | name = Polyhedron | title = Examples of polyhedra | image = {{multiple image|border=infobox|perrow=2/2/2|total_width=350 | image1 = Tetrahedron.jpg | alt1 = | caption1 = [[Regular tetrahedron]]<br>([[Platonic solid]]) | image2 = Small stellated dodecahedron.png | alt2 = | caption2 = [[Small stellated dodecahedron]]<br>([[Kepler–Poinsot polyhedron]]) | image3 = Icosidodecahedron.png | alt3 = | caption3 = [[Icosidodecahedron]]<br>([[Archimedean solid]]) | image4 = Great cubicuboctahedron.png | alt4 = | caption4 = [[Great cubicuboctahedron]]<br>([[Uniform star-polyhedron]]) | image5 = Rhombic triacontahedron.png | alt5 = | caption5 = [[Rhombic triacontahedron]]<br>([[Catalan solid]]) | image6 = Hexagonal torus.svg | alt6 = | caption6 = A [[toroidal polyhedron]] }} | label1 = Definition | data1 = A three-dimensional example of the more general [[polytope]] in any number of dimensions. | label2 = Characteristics | data2 = number of faces,<br> topological classification and [[Euler characteristic]],<br>[[Dual polyhedron|duality]],<br>[[vertex figure]]s,<br>[[surface area]] and [[volume]],<br>lines as in [[geodesic]]s and [[diagonal]]s,<br>[[Dehn invariant]],<br>[[Symmetry group|highly symmetrical]]. }} In [[geometry]], a '''polyhedron''' ({{plural form}}: '''polyhedra''' or '''polyhedrons'''; {{ety|el|''[[wikt:πολύς|πολύ]]'' {{nowrap|(poly-)}} |many||''[[wikt:ἕδρα|ἕδρον]]'' {{nowrap|(-hedron)}} |base, seat}}) is a [[three-dimensional figure]] with flat [[polygon]]al [[Face (geometry)|faces]], straight [[Edge (geometry)|edges]] and sharp corners or [[Vertex (geometry)|vertices]]. The term "polyhedron" may refer either to a [[solid figure]] or to its boundary [[surface (mathematics)|surface]]. The terms '''solid polyhedron''' and '''polyhedral surface''' are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole [[structure (mathematics)|structure]] formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional [[polygon]] and a three-dimensional specialization of a [[polytope]], a more general concept in any number of [[dimension]]s. Polyhedra have several general characteristics that include the number of faces, topological classification by [[Euler characteristic]], [[Dual polyhedron|duality]], [[vertex figure]]s, [[surface area]], [[volume]], interior lines, [[Dehn invariant]], and [[symmetry]]. The symmetry of a polyhedron means that the polyhedron's appearance is unchanged by the transformation such as rotating and reflecting. The ''convex polyhedron'' is well-defined with several equivalent standard definitions, one of which is a polyhedron that is a [[convex set]], or the polyhedral surface that bounds it. Every convex polyhedron is the [[convex hull]] of its vertices, and the convex hull of a finite set of points is a polyhedron. There are many families of convex polyhedra, and the most common examples are [[cube]] and the family of [[Pyramid (geometry)|pyramids]]. {{TOC limit|3}}
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