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In geometry, an octahedron (pl.: octahedra or octahedrons) is a polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures.
For compounds with the formula MX 6, the chief alternative to octahedral geometry is a trigonal prismatic geometry, which has symmetry D 3h. In this geometry, the six ligands are also equivalent. There are also distorted trigonal prisms, with C 3v symmetry; a prominent example is W(CH 3) 6.
146 magnetic balls, packed in the form of an octahedron. In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The n th octahedral number can be obtained by the formula: [1] = (+).
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.
This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] It corresponds to the Euler characteristic of the sphere (i.e. χ = 2 {\displaystyle \ \chi =2\ } ), and applies identically to spherical polyhedra .
Example: an octahedron is a birectification of a cube: {3,4} = 2r{4,3}. Another type of truncation, cantellation , cuts edges and vertices, removing the original edges, replacing them with rectangles, removing the original vertices, and replacing them with the faces of the dual of the original regular polyhedra or tiling.
Xenon hexafluoride, which has a distorted octahedral geometry. Some AX 6 E 1 molecules, e.g. xenon hexafluoride (XeF 6) and the Te(IV) and Bi(III) anions, TeCl 2− 6, TeBr 2− 6, BiCl 3− 6, BiBr 3− 6 and BiI 3− 6, are octahedral, rather than pentagonal pyramids, and the lone pair does not affect the geometry to the degree predicted by ...
These include the hemi-cube, hemi-octahedron, hemi-dodecahedron, and hemi-icosahedron. They are (globally) projective polyhedra , and are the projective counterparts of the Platonic solids . The tetrahedron does not have a projective counterpart as it does not have pairs of parallel faces which can be identified, as the other four Platonic ...