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Encyclopedia > Octahedron
Regular Octahedron

Type Platonic solid
Elements F = 8, E = 12, V = 6 (χ = 2)
Faces by sides 8{3}
Schläfli symbol {3,4} and $begin{Bmatrix} 3 3 end{Bmatrix}$
Wythoff symbol 4 | 2 3
2 | 3 3
Coxeter-Dynkin
Symmetry Oh
References U05, C17, W2
Properties Regular convex deltahedron
Dihedral angle 109.47122° = arccos(-1/3)

3.3.3.3
(Vertex figure)

Cube
(dual polyhedron)

Net

The octahedron's symmetry group is Oh, of order 48. This group's subgroups include D3d (order 12), the symmetry group of a triangular antiprism; D4h (order 16), the symmetry group of a square bipyramid; and Td (order 24), the symmetry group of a rectified tetrahedron. These symmetries can be emphasized by different decorations of the faces. The symmetry group of an object (e. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

It is a three-dimensional cross polytope. In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ...

An octahedron can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

( ±1, 0, 0 );
( 0, ±1, 0 );
( 0, 0, ±1 ).

## Area and volume

The area A and the volume V of a regular octahedron of edge length a are: The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...

$A=2sqrt{3}a^2 approx 3.46410162a^2$
$V=frac{1}{3} sqrt{2}a^3 approx 0.471404521a^3$

Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles). A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

## Geometric relations

The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids. One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound. A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Stella octangula The stella octangula (eight-pointed star), also known as the stellated octahedron, is the polyhedral compound of two tetrahedra. ... Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ... In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... In philosophy (especially that of Aristotle), the golden mean is the felicitous middle between two extremes, one of excess and the other of deficiency; for this meaning, see golden mean (philosophy). ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, and is one of the 28 convex uniform honeycombs. Another is a tessellation of octahedra and cuboctahedra. The tetrahedral-octahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of alternating tetrahedra and octahedra. ... A tessellation of space fills space with solids, e. ... Simplified space frame roof with the nearest unit polygon hightlighted in blue A space frame is a truss-like, light weight rigid structure constructed from interlocking struts in a geometric pattern. ... Richard Buckminster â€œBuckyâ€ Fuller (July 12, 1895 â€“ July 1, 1983)[1] was an American visionary, designer, architect, poet, author, and inventor. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...

The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces.

Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid. The elongated square gyrobicupola (J37), a Johnson solid This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles. ...

## Related polyhedra

The octahedron can also be considered a rectified tetrahedron. This can be shown by a 2-color face model. With this coloring, the octahedron has tetrahedral symmetry. In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ...

Compare this truncation sequence between a tetrahedron and its dual:

 Tetrahedron Truncated tetrahedron octahedron Truncated tetrahedron Tetrahedron

## Octahedra in the physical world

• If each edge of an octahedron is replaced by a one ohm resistor, the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.[1]
• Natural crystals of diamond or alum are commonly octahedral.

A roleplaying game (RPG) is a type of game in which players assume the roles of characters and collaboratively create stories. ... Dice (the plural of die, from Old French de, from Latin datum something given or played [1]) are small polyhedral objects, usually cubical, used for generating random numbers or other symbols. ... Rolling dice Dice (the plural of the word die, probably from the Latin dare: to give) are, in general, small polyhedral objects with the faces marked with numbers or other symbols, thrown in order to choose one of the faces randomly. ... The ohm (symbol: Î©) is the SI unit of electric resistance. ... Resistor symbols (non-European) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ... This article is about the gemstone. ... A crystal of alum Alum, Allom [aluminium potassium sulphate], in chemistry, is a term given to the crystallized double sulfates of the typical formula M+2SO4Â·M3+2(SO4)3Â·24H2O, where M+ is the sign of an alkali metal (lithium, sodium, potassium, rubidium, or caesium), and M3+ denotes one...

## Other octahedra

The regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; nonregular octahedra may have as many as 12 vertices and 18 edges.[1]

In geometry, the hexagonal prism is a prism with hexagonal base. ... This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... The truncated tetrahedron is an Archimedean solid. ... The tetragonal trapezohedron or deltohedron is the second in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. ...

Spinning octahedron, made by me using POV-Ray, see image:poly. ... Stella octangula The stella octangula (eight-pointed star), also known as the stellated octahedron, is the polyhedral compound of two tetrahedra. ... A triakis octahedron is a catalan solid which looks a bit like an overinflated octahedron. ... A disdyakis dodecahedron, or hexakis octahedron, is the Catalan solid whose Archimedean dual is the truncated cuboctahedron. ... The truncated octahedron is an Archimedean solid. ... A generic octahedral molecule. ...

## References

1. ^ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. Retrieved on 2006-09-30.

Year 2006 (MMVI) was a common year starting on Sunday (link displays full 2006 calendar) of the Gregorian calendar. ... is the 273rd day of the year (274th in leap years) in the Gregorian calendar. ...

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