In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a

3-ball Three-ball (or "3-ball", colloquially) is a folk game of pool played with any three standard pool and . The game is frequently gambled upon. The goal is to () the three object balls in as few shots as possible.Manhattan () metric.

Regular octahedron

Dimensions

If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :

r_u = \frac a \approx 0.707 \cdot a

and the radius of an inscribed sphere ( tangent to each of the octahedron's faces) is :

r_i = \frac a \approx 0.408\cdot a

while the midradius, which touches the middle of each edge, is :

r_m = \tfrac a = 0.5\cdot a

Orthogonal projections

The ''octahedron'' has four special orthogonal projections, centered, on an edge, vertex, face, and normal to a face. The second and third correspond to the B₂ and A₂ Coxeter planes.

Spherical tiling

The octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Cartesian coordinates

An octahedron with edge length can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then : ( ±1, 0, 0 ); : ( 0, ±1, 0 ); : ( 0, 0, ±1 ). In an ''x''–''y''–''z'' Cartesian coordinate system, the octahedron with center coordinates (''a'', ''b'', ''c'') and radius ''r'' is the set of all points (''x'', ''y'', ''z'') such that :

\left, x - a\ + \left, y - b\ + \left, z - c\ = r.

Area and volume

The surface area ''A'' and the volume ''V'' of a regular octahedron of edge length ''a'' are: :

A=2\sqrta^2 \approx 3.464a^2

V=\frac \sqrta^3 \approx 0.471a^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 rather than 4 triangles). If an octahedron has been stretched so that it obeys the equation :

\left, \frac\+\left, \frac\+\left, \frac\ = 1,

the formulas for the surface area and volume expand to become :

A=4 \, x_m \, y_m \, z_m \times \sqrt,

V=\frac\,x_m\,y_m\,z_m.

Additionally the inertia tensor of the stretched octahedron is :

I =
\begin
  \frac m (y_m^2+z_m^2) & 0 & 0 \\
  0 & \frac m (x_m^2+z_m^2) & 0 \\
  0 & 0 & \frac m (x_m^2-y_m^2)
\end.

These reduce to the equations for the regular octahedron when :

x_m=y_m=z_m=a\,\frac.

Geometric relations

Using the standard nomenclature for Johnson solids, an octahedron would be called a ''

square bipyramid In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

''.

Dual

The octahedron is the dual polyhedron of the cube. : Dual Cube-Octahedron

If an octahedron of edge length

= a

is inscribed in a cube, then the length of an edge of the cube

= \sqrt a

Stellation

The interior of the compound of two dual

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

is an octahedron, and this compound, called the stella octangula, is its first and only

stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specif ...

. 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 A rectifier is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in only one direction. The reverse operation (converting DC to AC) is performed by an inver ...

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 In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...

relate to the other Platonic solids.

Snub octahedron

One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

. 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''. An icosahedron produced this way is called a snub octahedron.

Tessellations

Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space. This and the regular tessellation of cubes are the only such

uniform honeycomb In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of fac ...

s in 3-dimensional space.

Characteristic orthoscheme

Like all regular convex polytopes, the octahedron can be dissected into an integral number of disjoint orthoschemes, all of the same shape characteristic of the polytope. A polytope's characteristic orthoscheme is a fundamental property because the polytope is generated by reflections in the

facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

of its orthoscheme. The orthoscheme occurs in two

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...

forms which are mirror images of each other. The characteristic orthoscheme of a regular polyhedron is a quadrirectangular irregular tetrahedron. The faces of the octahedron's characteristic tetrahedron lie in the octahedron's mirror planes of

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

. 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, among its mirror planes, some that do not pass through any of its faces. The octahedron's

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

is denoted B₃. The octahedron and its

dual polytope 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 ...

, the cube, have the same symmetry group but different characteristic tetrahedra. The characteristic tetrahedron of the regular octahedron can be found by a canonical dissection of the regular octahedron which subdivides it into 48 of these characteristic orthoschemes surrounding the octahedron's center. Three left-handed orthoschemes and three right-handed orthoschemes meet in each of the octahedron's eight faces, the six orthoschemes collectively forming a trirectangular tetrahedron: a triangular pyramid with the octahedron face as its equilateral base, and its cube-cornered apex at the center of the octahedron. If the octahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths

\sqrt

1

\sqrt

(the exterior right triangle face, the ''characteristic triangle'' 𝟀, 𝝓, 𝟁 of the octahedron), plus

\sqrt

1

\sqrt

(edges that are the ''characteristic radii'' of the octahedron). The 3-edge path along orthogonal edges of the orthoscheme is

1

\sqrt

\sqrt

, first from an octahedron vertex to an octahedron edge center, then turning 90° to an octahedron face center, then turning 90° to the octahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 90-60-30 triangle which is one-sixth of an octahedron face. The three faces interior to the octahedron are: a 45-90-45 triangle with edges

1

\sqrt

1

, a right triangle with edges

\sqrt

1

\sqrt

, and a right triangle with edges

\sqrt

\sqrt

\sqrt

Topology

The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the

maximal independent set In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maxi ...

s of its vertices have the same size. The other three polyhedra with this property are the

pentagonal dipyramid In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid (). Each bipyramid is the dual of a uniform prism. Although it is face-transitive, it is not a Platonic ...

, the

snub disphenoid In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some ve ...

, and an irregular polyhedron with 12 vertices and 20 triangular faces.

Nets

The regular octahedron has eleven arrangements of nets.

Faceting

The uniform

tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...

is a

tetrahedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection ...

faceting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. New edges of a faceted polyhedron may be ...

of the regular octahedron, sharing

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...

and

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...

. It has four of the triangular faces, and 3 central squares.

Uniform colorings and symmetry

There are 3 uniform colorings of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111. The octahedron's

is O_h, of order 48, the three dimensional

hyperoctahedral group In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a paramete ...

. This group's

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s include D_3d (order 12), the symmetry group of a triangular antiprism; D_4h (order 16), the symmetry group of a square bipyramid; and T_d (order 24), the symmetry group of a

rectified tetrahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...

. These symmetries can be emphasized by different colorings of the faces.

Irregular octahedra

The following polyhedra are combinatorially equivalent to the regular polyhedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of a regular octahedron. * ''Triangular antiprisms'': Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. * Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilaterals are planar squares. * Schönhardt polyhedron, a non-convex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices. * Bricard octahedron, a non-convex self-crossing flexible polyhedron More generally, an octahedron can be any polyhedron with eight faces. 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. There are 257 topologically distinct ''convex'' octahedra, excluding mirror images. More specifically there are 2, 11, 42, 74, 76, 38, 14 for octahedra with 6 to 12 vertices respectively. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) Some better known irregular octahedra include the following: * Hexagonal prism: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges. * Heptagonal

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...

: One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles). It is not possible for all triangular faces to be equilateral. * Truncated tetrahedron: The four faces from the tetrahedron are truncated to become regular hexagons, and there are four more equilateral triangle faces where each tetrahedron vertex was truncated. *

Tetragonal trapezohedron In geometry, a tetragonal trapezohedron, or deltohedron, is the second in an infinite series of trapezohedra, which are dual to the antiprisms. It has eight faces, which are congruent kites, and is dual to the square antiprism. In mesh gene ...

: The eight faces are congruent

kites A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the face ...

. * Octagonal

hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune ha ...

: degenerate in Euclidean space, but can be realized spherically.

Octahedra in the physical world

Octahedra in nature

* Natural crystals of

diamond Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, b ...

alum An alum () is a type of chemical compound, usually a hydrated double sulfate salt of aluminium with the general formula , where is a monovalent cation such as potassium or ammonium. By itself, "alum" often refers to potassium alum, with the ...

fluorite Fluorite (also called fluorspar) is the mineral form of calcium fluoride, CaF2. It belongs to the halide minerals. It crystallizes in isometric cubic habit, although octahedral and more complex isometric forms are not uncommon. The Mohs sca ...

are commonly octahedral, as the space-filling

tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names i ...

. * The plates of kamacite alloy in octahedrite

meteorites A meteorite is a solid piece of debris from an object, such as a comet, asteroid, or meteoroid, that originates in outer space and survives its passage through the atmosphere to reach the surface of a planet or moon. When the original object ...

are arranged paralleling the eight faces of an octahedron. * Many metal ions

coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...

six ligands in an octahedral or distorted octahedral configuration. *

Widmanstätten pattern Widmanstätten patterns, also known as Thomson structures, are figures of long nickel–iron crystals, found in the octahedrite iron meteorites and some pallasites. They consist of a fine interleaving of kamacite and taenite bands or ribbons ...

s in

nickel Nickel is a chemical element with symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and ductile transition metal. Pure nickel is chemically reactive but large pieces are slow t ...

iron Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in ...

crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...

Octahedra in art and culture

* Especially in roleplaying games, this solid is known as a "d8", one of the more common

polyhedral dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing ga ...

. * If each edge of an octahedron is replaced by a one-

ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm (bor ...

resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...

, the resistance between opposite vertices is ohm, and that between adjacent vertices ohm. * Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad; see

hexany In musical tuning systems, the hexany, invented by Erv Wilson, represents one of the simplest structures found in his combination product sets. It is referred to as an uncentered structure, meaning that it implies no tonic. It achieves this b ...

Tetrahedral octet truss

space frame In architecture and structural engineering, a space frame or space structure ( 3D truss) is a rigid, lightweight, truss-like structure constructed from interlocking struts in a geometric pattern. Space frames can be used to span large areas w ...

of alternating tetrahedra and half-octahedra derived from the

Tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names i ...

was invented by

Buckminster Fuller Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing ...

in the 1950s. It is commonly regarded as the strongest building structure for resisting

cantilever A cantilever is a rigid structural element that extends horizontally and is supported at only one end. Typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. Like other structural elements, a cant ...

stresses.

Related polyhedra

A regular octahedron can be augmented into a tetrahedron by adding 4 tetrahedra on alternated faces. Adding tetrahedra to all 8 faces creates the

stellated octahedron The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depic ...

. The octahedron is one of a family of uniform polyhedra related to the cube. It is also one of the simplest examples of a

hypersimplex In polyhedral combinatorics, the hypersimplex \Delta_ is a convex polytope that generalizes the simplex. It is determined by two integers d and k, and is defined as the convex hull of the d-dimensional vectors whose coefficients consist of k ones ...

, a polytope formed by certain intersections of a

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...

with a

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

. The octahedron is topologically related as a part of sequence of regular polyhedra with

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...

s , continuing into the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...

Tetratetrahedron

The regular octahedron can also be considered a '' rectified tetrahedron'' – and can be called a ''tetratetrahedron''. This can be shown by a 2-color face model. With this coloring, the octahedron has

. Compare this truncation sequence between a tetrahedron and its dual: The above shapes may also be realized as slices orthogonal to the long diagonal of a

tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of e ...

. If this diagonal is oriented vertically with a height of 1, then the first five slices above occur at heights ''r'', , , , and ''s'', where ''r'' is any number in the range , and ''s'' is any number in the range . The octahedron as a ''tetratetrahedron'' exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.''n'')², progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With

orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The adva ...

symmetry of *''n''32 all of these tilings are Wythoff constructions within a

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...

of symmetry, with generator points at the right angle corner of the domain.

Trigonal antiprism

As a trigonal antiprism, the octahedron is related to the hexagonal dihedral symmetry family.

Square bipyramid

Other related polyhedra

Truncation of two opposite vertices results in a

square bifrustum The square bifrustum or ''square truncated bipyramid'' is the second in an infinite series of bifrustum polyhedra. It has 4 trapezoidal and 2 square faces. This polyhedron can be constructed by taking a square bipyramid (octahedron) and trunca ...

. The octahedron can be generated as the case of a 3D superellipsoid with all exponent values set to 1.

References

External links

* * *
Editable printable net of an octahedron with interactive 3D view

Paper model of the octahedron

* ttp://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra
Virtual Reality Polyhedra
The Encyclopedia of Polyhedra *

Try: dP4 {{Authority control Deltahedra Individual graphs Platonic solids Prismatoid polyhedra Pyramids and bipyramids