In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, an
octahedron (plural: octahedra) is a
polyhedronIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
with eight faces. A
regular octahedron is a
Platonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
composed of eight
equilateral triangles, four of which meet at each vertex.
An octahedron is the three-dimensional case of the more general concept of a cross polytope.
Dimensions
If the edge length of a regular octahedron is
, the
radiusIn classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of a circumscribed
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
(one that touches the octahedron at all vertices) is
-
and the radius of an inscribed sphere (
tangentIn geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
to each of the octahedron's faces) is
-
while the midradius, which touches the middle of each edge, is
-
Cartesian coordinates
Orthographic projectionOrthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...
s
|
|
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
- ( ±1, 0, 0 );
- ( 0, ±1, 0 );
- ( 0, 0, ±1 ).
Area and volume
The surface area
A and the
volumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
V of a regular octahedron of edge length
a are:
Thus the volume is four times that of a regular
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).
Geometric relations
The interior of the
compoundA polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram....
of two dual
tetrahedraIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
is an octahedron, and this compound, called the
stella octangulaThe stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers...
, is its first and only
stellationStellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...
. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e.
rectifyingIn 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 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
cuboctahedronIn geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
and
icosidodecahedronIn geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon...
relate to the other Platonic solids. One can also divide the edges of an octahedron in the ratio of the
golden meanGolden mean may refer to:*Doctrine of the Golden Mean , a chapter in Li Ji, one of the Four Books of Confucianism*Golden mean , the felicitous middle between the extremes of excess and deficiency...
to define the vertices of an
icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
. 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 tetrahedraThe tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1:2....
can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by
Buckminster FullerRichard Buckminster “Bucky” Fuller was an American systems theorist, author, designer, inventor, futurist and second president of Mensa International, the high IQ society....
. This is the only such tiling save the regular tessellation of
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
s, and is one of the 28
convex uniform honeycombIn geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...
s. Another is a tessellation of octahedra and
cuboctahedraIn geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
.
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 solidIn geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
s, an octahedron would be called a
square bipyramid. Truncation of two opposite vertices results in a
square bifrustumThe square bifrustum or square truncated bipyramid is the second in an infinite series of bifrustum polyhedra. It has 4 trapezoid and 2 square faces....
.
The octahedron is
4-connectedIn graph theory, a graph G with vertex set V is said to be k-vertex-connected if the graph remains connected when you delete fewer than k vertices from the graph...
, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected
simplicialIn geometry, a simplicial polytope is a d-polytope whose facets are all simplices.For example, a simplicial polyhedron contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph....
well-coveredIn graph theory, a well-covered graph is an undirected graph in which every minimal vertex cover has the same size as every other minimal vertex cover. Well-covered graphs were defined and first studied by .-Definitions:...
polyhedra, meaning that all of the
maximal independent setIn graph theory, a maximal independent set or maximal stable set is an independent set that is not a subset of any other independent set. That is, it is a set S such that every edge of the graph has at least one endpoint not in S and every vertex not in S has at least one neighbor in S...
s of its vertices have the same size. The other three polyhedra with this property are the
pentagonal dipyramidIn geometry, the pentagonal bipyramid is third of the infinite set of face-transitive bipyramids.Each bipyramid is the dual of a uniform prism.If the faces are equilateral triangles, it is a deltahedron and a Johnson solid...
, the
snub disphenoidIn geometry, the snub disphenoid is one of the Johnson solids . It is a three-dimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron. It is not a regular polyhedron because some vertices have four faces and others have five...
, and an irregular polyhedron with 12 vertices and 20 triangular faces.
Uniform colorings and symmetry
There are 3
uniform coloringIn geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive...
s of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111.
The octahedron's
symmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
is
Oh, of order 48, the three dimensional
hyperoctahedral groupIn 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. Groups of this type are identified by a parameter n, the dimension of the hypercube....
. This group's
subgroupIn group theory, 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 of G if the restriction of * to H x H is a group operation on H...
s include
D3d (order 12), the symmetry group of a triangular
antiprismIn geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
;
D4h (order 16), the symmetry group of a square
bipyramidAn n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...
; and
Td (order 24), the symmetry group of a rectified tetrahedron. These symmetries can be emphasized by different colorings of the faces.
| Name |
Octahedron |
Rectified 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...
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
(Tetratetrahedron) |
Triangular antiprismIn geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
|
Square bipyramidAn n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...
|
| Coxeter-Dynkin In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
|
|
|
|
|
| Schläfli symbol |
{3,4} |
t1{3,3} |
s{3,2} |
|
| Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles....
|
4 | 3 2 |
2 | 4 3 |
2 | 6 2 |
|
| Symmetry |
Oh
[4,3]
(*432) |
Td
[3,3]
(*332) |
D3d
[2+,6]
(2*3) |
D4h
[2,3]
(*322) |
| Symmetry order |
48 |
24 |
12 |
16 |
Image
(uniform coloring) |
(1111) |
(1212) |
(1112) |
 |
Dual
The octahedron is the
dual polyhedronIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
to the
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
.
Nets
It has eleven arrangements of
netsIn geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron...
.
This example depicts it as both a dipyramid and an antiprism:
Tetratetrahedron
The regular octahedron can also be considered a
rectifiedIn 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...
tetrahedron - and can be called a
tetratetrahedron. This can be shown by a 2-color face model. With this coloring, the octahedron has
tetrahedral symmetry150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...
.
Compare this truncation sequence between a tetrahedron and its dual:
TetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
Truncated tetrahedronIn geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
|
Octahedron |
Truncated tetrahedronIn geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
|
TetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
The above shapes may also be realized as slices orthogonal to the long diagonal of a
tesseractIn geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog 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 6 square faces, the hypersurface of the tesseract consists of 8...
. If this diagonal is oriented vertically with a height of 1, then the five slices above occur at heights
, 3/8, 1/2, 5/8, and
, where
is any number in the range (0,1/4], and
is any number in the range [3/4,1).
This polyhedron is topologically related as a part of sequence of regular polyhedra with
Schläfli symbols {3,n}, continuing into the
hyperbolic planeIn mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
.
{3,3}In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
{3,4}In geometry, 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....
|
{3,5}In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
|
{3,6} |
{3,7} |
{3,8} |
{3,9} |
Tetrahemihexahedron
The regular octahedron shares its edges and vertex arrangement with one
nonconvex uniform polyhedronIn geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting...
: the
tetrahemihexahedronIn geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....
, with which it shares four of the triangular faces.
Octahedron |
TetrahemihexahedronIn geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....
|
Octahedra in the physical world
- Especially in roleplaying games, this solid is known as a "d8", one of the more common non-cubical dice.
- If each edge of an octahedron is replaced by a one ohm resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
, the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.
- Natural crystals of diamond
In mineralogy, diamond is an allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the conversion rate from diamond to graphite is negligible at ambient conditions...
, alumAlum is both a specific chemical compound and a class of chemical compounds. The specific compound is the hydrated potassium aluminium sulfate with the formula KAl2.12H2O. The wider class of compounds known as alums have the related empirical formula, AB2.12H2O.-Chemical properties:Alums are...
or fluoriteFluorite is a halide mineral composed of calcium fluoride, CaF2. It is an isometric mineral with a cubic habit, though octahedral and more complex isometric forms are not uncommon...
are commonly octahedral, as the space=filling tetrahedral-octahedral honeycombThe tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1:2....
.
- The plates of kamacite
Kamacite is a mineral. It is an alloy of iron and nickel, usually in the proportions of 90:10 to 95:5 although impurities such as cobalt or carbon may be present. On the surface of Earth, it occurs naturally only in meteorites. It has a metallic luster, is gray and has no clear cleavage although...
alloy in octahedriteOctahedrites are a class of iron meteorites within the structural classification. They are the most common class of iron meteorites.They are composed primarily of the nickel-iron alloys: taenite - high nickel content, and kamacite - low nickel content....
meteorites are arranged paralleling the eight faces of an octahedron.
- Many metal ions coordinate six ligands in an octahedral or distorted
The Jahn–Teller effect, sometimes also known as Jahn–Teller distortion, or the Jahn–Teller theorem, describes the geometrical distortion of non-linear molecules under certain situations. This electronic effect is named after Hermann Arthur Jahn and Edward Teller, who proved, using group theory,...
octahedral configuration.
Octahedra in music
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
hexanyIn music theory, the hexany is a six-note just intonation scale, with the notes placed on the vertices of an octahedron, equivalently the faces of a cube...
.
Other 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 antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
s: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles.
- Tetragonal bipyramid
An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...
s, 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
In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices...
, a nonconvex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.
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; nonregular octahedra may have as many as 12 vertices and 18 edges.
http://www.uwgb.edu/dutchs/symmetry/polynum0.htm
Other nonregular octahedra include the following.
- Hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...
: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges.
- Heptagonal pyramid
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
: 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
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
. 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
The tetragonal trapezohedron or deltohedron is the second in an infinite series of face-uniform polyhedra which are dual to the antiprisms. It has eight faces which are congruent kites.- External links :*...
. The eight faces are congruent kitesIn Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...
.
See also
- Spinning octahedron
- Stella octangula
The stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers...
- Triakis octahedron
In geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more...
- Hexakis octahedron
In geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons...
- Truncated octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
- Octahedral molecular geometry
In chemistry, octahedral molecular geometry describes the shape of compounds where in six atoms or groups of atoms or ligands are symmetrically arranged around a central atom, defining the vertices of an octahedron...
- Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...
- Octahedral graph
External links