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A regular octahedron is an octahedron that is a regular polyhedron. All the faces of a regular octahedron are equilateral triangles of the same size, and exactly four triangles meet at each vertex. A regular octahedron is convex, meaning that for any two points within it, the line segment connecting them lies entirely within it.
Faces are 8-by-8 combined to larger faces for a = b = 0 (cube) and 6-by-6 for a = b = c (octahedron). The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6 (drawn in purple and red), representing in two orthogonal subsymmetries: D 2h , and T d .
The convex forms are listed in order of degree of vertex configurations from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be shown. There are infinitely many prisms and antiprisms, one for each regular polygon; the ones up to the 12-gonal cases are listed.
The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition: The vertices of a convex regular polyhedron all lie on a sphere. All the dihedral angles of the polyhedron are equal; All the vertex figures of the polyhedron are regular polygons.
Some fields of study allow polyhedra to have curved faces and edges. Curved faces can allow digonal faces to exist with a positive area. When the surface of a sphere is divided by finitely many great arcs (equivalently, by planes passing through the center of the sphere), the result is called a spherical polyhedron. Many convex polytopes having ...
The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4 π steradians) divided by the number of faces. This is equal to the angular deficiency of its dual. The various angles associated with the Platonic solids are tabulated below.
However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces. Because of the ambiguity of the term octahedron and tilarity of the various eight-sided figures, the term is rarely used without clarification. Before sharpening, many pencils take the shape of a long hexagonal prism. [2]
The dual of a cube is an octahedron.Vertices of one correspond to faces of the other, and edges correspond to each other. In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. [1]