Polyhedron

Platonic solidKepler-Poinsot solidArchimedean solidUniform star-polyhedronCatalan solidIn geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- (stem of πολύς, 'many') + -hedron (form of ἕδρα, 'base' or 'seat').'The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ... the writers failed to define what are the polyhedra'. In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- (stem of πολύς, 'many') + -hedron (form of ἕδρα, 'base' or 'seat'). A convex polyhedron is the convex hull of finitely many points, not all on the same plane.Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic.Many definitions of 'polyhedron' have been given within particular contexts, some more rigorous than others, and there is not universal agreement over which of these to choose.Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or includeshapes that are often not considered as valid polyhedra (such as solids whose boundaries are not manifolds). As Branko Grünbaum observed, Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its vertices (corner points), edges (line segments connecting certain pairs of vertices),faces (two-dimensional polygons), and sometimes by its three-dimensional interior volume.One can distinguish among these different definitions according to whether they describe the polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its incidence geometry. In all of these definitions, a polyhedron is typically understood as a three-dimensional example of the more general polytope in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional 'cells'.However, some of the literature on higher-dimensional geometry uses the term 'polyhedron' to mean something else: not a three-dimensional polytope, but a shape that is different from a polytope in some way. For instance, some sources define a convex polyhedron to be the intersection of finitely many half-spaces, and a polytope to be a bounded polyhedron. The remainder of this article considers only three-dimensional polyhedra. Polyhedra may be classified and are often named according to the number of faces. The naming system is based on Classical Greek, for example tetrahedron (4), pentahedron (5), hexahedron (6), triacontahedron (30), and so on. For a complete list of the Greek numeral prefixes see Numeral prefixes>Table of number prefixes in English>Greek>Quantitative