A (symmetric) n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices. The "n-gonal" in the name of a bipyramid does not refer to a face but to the internal polygon base, lying in the mirror plane that connects the two pyramid halves. (If it were a face, then each of its edges would connect three faces instead of two.) A "regular" bipyramid has a regular polygon base. It is usually implied to be also a right bipyramid. A right bipyramid has its two apices right above and right below the center or the centroid of its polygon base. A "regular" right (symmetric) n-gonal bipyramid has Schläfli symbol { } + {n}. A right (symmetric) bipyramid has Schläfli symbol { } + P, for polygon base P. The "regular" right (thus face-transitive) n-gonal bipyramid with regular vertices is the dual of the n-gonal uniform (thus right) prism, and has congruent isosceles triangle faces. A "regular" right (symmetric) n-gonal bipyramid can be projected on a sphere or globe as a "regular" right (symmetric) n-gonal spherical bipyramid: n equally spaced lines of longitude going from pole to pole, and an equator line bisecting them. Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are equilateral triangles, and thus the bipyramid is a deltahedron): the "regular" right (symmetric) triangular, tetragonal, and pentagonal bipyramids. The tetragonal or square bipyramid with same length edges, or regular octahedron, counts among the Platonic solids; the triangular and pentagonal bipyramids with same length edges count among the Johnson solids (J12 and J13). A "regular" right (symmetric) n-gonal bipyramid has dihedral symmetry group Dnh, of order 4n, except in the case of a regular octahedron, which has the larger octahedral symmetry group Oh, of order 48, which has three versions of D4h as subgroups.

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Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is a polyhedron that bounds a convex set. Every convex polyhedron can be constructed as the convex hull of its vertices, and for every finite set of points, not all on the same plane, the convex hull is a convex polyhedron. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions.
Uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also face- and edge-transitive), quasi-regular (if also edge-transitive but not face-transitive), or semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions. The Schläfli symbol is a recursive description, starting with {p} for a p-sided regular polygon that is convex.
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