Octahedron
An octahedron is a
polyhedron with eight faces. A regular octahedron is a
Platonic solid composed of eight
equilateral triangles, four of which meet at each vertex.
The regular octahedron is a special kind of triangular
antiprism and of square
bipyramid, and is dual to the
cube. The regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges. A regular octahedron is a three-dimensional
cross polytope.
Encyclopedia
An
octahedron is a
polyhedron with eight faces. A
regular octahedron is a
Platonic solid composed of eight
equilateral triangles, four of which meet at each vertex.
The regular octahedron is a special kind of triangular
antiprism and of square
bipyramid, and is dual to the
cube. The regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges. A regular octahedron is a three-dimensional
cross polytope.
The octahedron in general
There are four important topological types of octahedra with
dihedral symmetry:
The term
octahedron is rarely used in this general sense, because these have little in common other than the same number of faces.
Cartesian coordinates
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
- ;
- ;
- .
Area and volume
The area
A and the volume
V of a regular octahedron of edge length
a are:
Thus the volume is four times that of a regular
tetrahedron with the same edge length, while the surface area is twice .
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 . 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.
Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tiling 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 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.
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.
Using the standard nomenclature for
Johnson solids, an octahedron would be called a
square bipyramid.
Uses
Especially in roleplaying games, this solid is known as a
d8, one of the more common
Polyhedral dice.
If each edge of an octahedron is replaced by a one ohm
resistor, the resistance between opposite vertices is 0.5 ohms, and that between adjacent vertices 5/12 ohms.
Many
diamond crystals are naturally octahedral.
See also
...
External links
- - Mathworld.com
- - MathsIsFun.com
- The Encyclopedia of Polyhedra
- Many links
