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Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is also possible to use abstract polyhedra as the basis of a definition of geometric polyhedra. A realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the ...
A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In ...
These require a more general definition of polyhedra. Grünbaum (1994) gave a rather complicated definition of a polyhedron, while McMullen & Schulte (2002) gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization. Here an ...
The regular dodecahedron is a polyhedron with twelve pentagonal faces, thirty edges, and twenty vertices. [1] It is one of the Platonic solids, a set of polyhedrons in which the faces are regular polygons that are congruent and the same number of faces meet at a vertex. [2] This set of polyhedrons is named after Plato.
In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on its vertices; today, this is more commonly referred to as a uniform polyhedron (this follows from Thorold Gosset's 1900 definition of the more general semiregular polytope). [1] [2] These polyhedra include:
A convex polyhedron whose faces are regular polygons is known as a Johnson solid, or sometimes as a Johnson–Zalgaller solid. Some authors exclude uniform polyhedra from the definition. A uniform polyhedron is a polyhedron in which the faces are regular and they are isogonal ; examples include Platonic and Archimedean solids as well as prisms ...
(dual polyhedron) Net In geometry , the rhombicosidodecahedron is an Archimedean solid , one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces .
Its dual polyhedron is the great stellated dodecahedron { 5 / 2 , 3}, having three regular star pentagonal faces around each vertex. Stellated icosahedra Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron.