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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 ...
However, Eberhard's theorem states that it should be possible to form a simple polyhedron by adding some number of hexagons, and in this case one hexagon suffices: bisecting a cube on a regular hexagon passing through six of its faces produces two copies of a simple roofless polyhedron with three triangle faces, three pentagon faces, and one ...
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas.
For every convex polyhedron, there exists a dual polyhedron having faces in place of the original's vertices and vice versa, and; the same number of edges. The dual of a convex polyhedron can be obtained by the process of polar reciprocation. [34] Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:
Its (two-dimensional) faces are regular p-gons ({p}), the cells are regular polyhedra of type {p,q}, the vertex figures are regular polyhedra of type {q,r}, and the edge figures are regular r-gons (type {r}). See the six convex regular and 10 regular star 4-polytopes. For example, the 120-cell is represented by {5,3,3}.
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. It has 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, 60 vertices , and 120 edges .
where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. Any convex polyhedron's surface has Euler characteristic = + = . This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3]