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POLYHEDRON a Polyhedron (Plural Polyhedra Or Polyhedrons) Is A

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POLYHEDRON a Polyhedron (Plural Polyhedra Or Polyhedrons) Is A POLYHEDRON A polyhedron (plural polyhedra or polyhedrons) is a geometric solid in three dimensions with flat faces and straight edges. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- (stem of πολύς, "many") + -edron (form of έδρα, "base", "seat", or "face"). Defining a polyhedron as a solid bounded by flat faces and straight edges is not very precise and, to a modern mathematician, quite unsatisfactory. Grünbaum (1994, p. 43) observed, "The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... [in that] at each stage ... the writers failed to define what are the 'polyhedra' ...." Since then rigorous definitions of "polyhedron" have been given within particular contexts. However such definitions are seldom compatible in other contexts. a.) Basis for definition Any polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions: 3 dimensions: The body is bounded by the faces, and is usually the volume enclosed by them. 2 dimensions: A face is a polygon bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral surface. 1 dimension: An edge joins one vertex to another and one face to another, and is usually a line segment. The edges together make up the polyhedral skeleton. 0 dimensions: A vertex (plural vertices) is a corner point. -1 dimension: The null polytope is a kind of non-entity required by abstract theories. More generally in mathematics and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract. A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. b.) Characteristics i. Names of polyhedral - Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for example tetrahedron (4), pentahedron (5), hexahedron (6), heptahedron (7), triacontahedron (30), and so on. - Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. the Pentagonal dodecahedron. - Other common names indicate that some operation has been performed on a simpler polyhedron, for example the truncated cube looks like a cube with its corners cut off, and has 14 faces (so it is also an example of a tetrakaidecahedron). - Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron. ii. Edges - Edges have two important characteristics (unless the polyhedron is complex): An edge joins just two vertices. An edge joins just two faces. - These two characteristics are dual to each other. iii. Euler characteristic - The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron: - For a simply connected polyhedron, χ = 2. For a detailed discussion, see Proofs and Refutations by Imre Lakatos. iv. Orientability - Some polyhedra, such as all convex polyhedra, have two distinct sides to their surface, for example one side can consistently be coloured black and the other white. We say that the figure is orientable. - But for some polyhedra this is not possible, and the figure is said to be non- orientable. All polyhedra with odd-numbered Euler characteristic are non- orientable. A given figure with even χ < 2 may or may not be orientable. v. Vertex figure - For every vertex one can define a vertex figure, which describes the local structure of the figure around the vertex. If the vertex figure is a regular polygon, then the vertex itself is said to be regular. vi. Duality - For every polyhedron there exists a dual polyhedron having: faces in place of the original's vertices and vice versa, the same number of edges the same Euler characteristic and orientability The dual of a convex polyhedron can be obtained by the process of polar reciprocation. vii. Volume - The volume of an orientable polyhedron having an identifiable centroid can be calculated using Green's theorem: by choosing the function where (x,y,z) is the centroid of the surface enclosing the volume under consideration. As we have, Hence the volume can be calculated as: where the normal of the surface pointing outwards is given by: The final expression can be written as where S is the surface area of the polyhedron. SOME POLYHEDRAL Dodecahedron Small stellated dodecahedron (Regular polyhedron) (Regular star) Icosidodecahedron Great cubicuboctahedron (Uniform) (Uniform star) Rhombic triacontahedron Elongated pentagonal cupola (Uniform dual) (Convex regular-faced) Octagonal prism Square antiprism (Uniform prism) (Uniform antiprism) .
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