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| {{For|the album by The Mars Volta|Octahedron (album)}}
| | I am Ernesto from Safnern. I love to play Euphonium. Other hobbies are Drawing. |
| {{Reg polyhedra db|Reg polyhedron stat table|O}}
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| In [[geometry]], an '''octahedron''' (plural: octahedra) is a [[polyhedron]] with eight faces. A '''regular''' octahedron is a [[Platonic solid]] composed of eight [[equilateral triangle]]s, four of which meet at each [[wikt:vertex|vertex]].
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| A regular octahedron is the [[dual polyhedron]] of a [[cube]]. It is a [[Rectification (geometry)|rectified]] [[tetrahedron]]. It is a square [[bipyramid]] in any of three [[orthogonal]] orientations. It is also a triangular [[antiprism]] in any of four orientations.
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| An octahedron is the three-dimensional case of the more general concept of a [[cross polytope]].
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| ==Regular octahedron==
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| ===Dimensions===
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| 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
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| :<math>r_u = \frac{a}{2} \sqrt{2} \approx 0.7071067 \cdot a</math>
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| and the radius of an inscribed sphere ([[tangent]] to each of the octahedron's faces) is
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| :<math>r_i = \frac{a}{6} \sqrt{6} \approx 0.4082482\cdot a</math>
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| while the midradius, which touches the middle of each edge, is
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| :<math>r_m = \frac{a}{2} = 0.5\cdot a</math>
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| ===Orthogonal projections===
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| The ''octahedron'' has four special [[orthogonal projection]]s, centered, on an edge, vertex, face, and normal to a face. The second and third correspond to the B<sub>2</sub> and A<sub>2</sub> [[Coxeter plane]]s.
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| {|class=wikitable width=480
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| |+ Orthogonal projections
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| |-
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| !Centered by
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| !Edge
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| !Face<BR>Normal
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| !Vertex
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| !Face
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| |-
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| !Image
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| |[[File:Cube t2 e.png|100px]]
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| |[[File:Cube t2 fb.png|100px]]
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| |[[File:3-cube t2 B2.svg|100px]]
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| |[[File:3-cube t2.svg|100px]]
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| |- align=center
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| !Projective<BR>symmetry
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| |[2]
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| |[2]
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| |[4]
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| |[6]
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| |}
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| ===Cartesian coordinates===
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| An octahedron with edge length sqrt(2) can be placed with its center at the origin and its vertices on the coordinate axes; the [[Cartesian coordinates]] of the vertices are then
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| : ( ±1, 0, 0 );
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| : ( 0, ±1, 0 );
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| : ( 0, 0, ±1 ).
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| In an ''x''–''y''–''z'' [[Cartesian coordinate system]], the octahedron with center [[Coordinate system|coordinates]] (''a'', ''b'', ''c'') and radius ''r'' is the set of all points (''x'', ''y'', ''z'') such that
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| :<math>\left|x - a\right| + \left|y - b\right| + \left|z - c\right| = r.</math>
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| ===Area and volume===
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| The surface area ''A'' and the [[volume]] ''V'' of a regular octahedron of edge length ''a'' are:
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| :<math>A=2\sqrt{3}a^2 \approx 3.46410162a^2</math>
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| :<math>V=\frac{1}{3} \sqrt{2}a^3 \approx 0.471404521a^3</math>
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| 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).
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| If an octahedron has been stretched so that it obeys the equation:
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| :<math>\left|\frac{x}{x_m}\right|+\left|\frac{y}{y_m}\right|+\left|\frac{z}{z_m}\right| = 1</math>
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| The formula for the surface area and volume expand to become:
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| :<math>A=4 \, x_m \, y_m \, z_m \times \sqrt{\frac{1}{x_m^2}+\frac{1}{y_m^2}+\frac{1}{z_m^2}}</math>
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| :<math>V=\frac{4}{3}\,x_m\,y_m\,z_m</math>
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| Additionally the inertia tensor of the stretched octahedron is:
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| :<math>
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| I = | |
| \begin{bmatrix}
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| \frac{1}{10} m (y_m^2+z_m^2) & 0 & 0 \\
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| 0 & \frac{1}{10} m (x_m^2+z_m^2) & 0 \\
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| 0 & 0 & \frac{1}{10} m (x_m^2+y_m^2)
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| \end{bmatrix}
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| </math>
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| These reduce to the equations for the regular octahedron when:
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| :<math>x_m=y_m=z_m=a\,\frac{\sqrt{2}}{2}</math>
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| ===Geometric relations===
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| [[File:Compound of two tetrahedra.png|left|thumb|The octahedron represents the central intersection of two tetrahedra]]
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| The interior of the [[polyhedral compound|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. [[Rectification (geometry)|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 ratio|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''.
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| [[Tetrahedral-octahedral honeycomb|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 [[cube]]s, and is one of the 28 [[convex uniform honeycomb]]s. Another is a tessellation of octahedra and [[cuboctahedra]].
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| 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.
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| Using the standard nomenclature for [[Johnson solid]]s, an octahedron would be called a ''square bipyramid''. Truncation of two opposite vertices results in a [[square bifrustum]].
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| The octahedron is [[k-vertex-connected graph|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 polytope|simplicial]] [[well-covered graph|well-covered]] polyhedra, meaning that all of the [[maximal independent set]]s of its vertices have the same size. The other three polyhedra with this property are the [[pentagonal dipyramid]], the [[snub disphenoid]], and an irregular polyhedron with 12 vertices and 20 triangular faces.<ref>{{Cite journal
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| |last1=Finbow |first1=Arthur S.
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| |last2=Hartnell |first2=Bert L.
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| |last3=Nowakowski |first3=Richard J.
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| |last4=Plummer |first4=Michael D. | author4-link = Michael D. Plummer
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| |doi=10.1016/j.dam.2009.08.002
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| |issue=8
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| |journal=Discrete Applied Mathematics
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| |mr=2602814
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| |pages=894–912
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| |title=On well-covered triangulations. III
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| |volume=158
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| |year=2010}}</ref>
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| ===Uniform colorings and symmetry===
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| There are 3 [[uniform coloring]]s of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111.
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| The octahedron's [[symmetry group]] is O<sub>h</sub>, of order 48, the three dimensional [[hyperoctahedral group]]. This group's [[subgroup]]s include D<sub>3d</sub> (order 12), the symmetry group of a triangular [[antiprism]]; '''D<sub>4h</sub>''' (order 16), the symmetry group of a square [[bipyramid]]; and T<sub>d</sub> (order 24), the symmetry group of a [[Octahedron#Tetratetrahedron|rectified tetrahedron]]. These symmetries can be emphasized by different colorings of the faces.
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| {| class=wikitable
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| !Name
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| !Octahedron
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| ![[Rectification (geometry)|Rectified]] [[tetrahedron]]<BR>(Tetratetrahedron)
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| !Triangular [[antiprism]]
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| !Square [[bipyramid]]
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| !Rhombic fusil
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| |- align=center
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| !Image<BR>(Face coloring)
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| |[[File:Uniform polyhedron-43-t2.png|100px]]<BR>(1111)
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| |[[File:Uniform polyhedron-33-t1.png|100px]]<BR>(1212)
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| |[[File:Trigonal antiprism.png|100px]]<BR>(1112)
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| |[[File:Square bipyramid.png|100px]]<BR>(1111)
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| |- align=center
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| ![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
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| |{{CDD|node_1|3|node|4|node}}
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| |{{CDD|node|3|node_1|3|node}}
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| |{{CDD|node_h|2x|node_h|6|node}}<BR>{{CDD|node_h|2x|node_h|3|node_h}}
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| |{{CDD|node_f1|2|node_f1|4|node}}
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| |{{CDD|node_f1|2|node_f1|2|node_f1}}
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| |- align=center
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| ![[Schläfli symbol]]
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| |{3,4}
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| |t<sub>1</sub>{3,3}
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| |s{2,6}<BR>sr{2,3}
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| |fs{2,4}<BR>{ } + {4}
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| |fsr{2,2}<BR>{ } + { } + { }
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| |- align=center
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| ![[Wythoff symbol]]
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| | 4 | 3 2
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| | 2 | 4 3
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| | 2 | 6 2 <BR> | 2 3 2
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| | ||
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| |- align=center
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| ![[List of spherical symmetry groups|Symmetry]]
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| |O<sub>h</sub>, [4,3], (*432)
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| |T<sub>d</sub>, [3,3], (*332)
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| |D<sub>3d</sub>, [2<sup>+</sup>,6], (2*3)<BR>D<sub>3</sub>, [2,3]<sup>+</sup>, (322)
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| |D<sub>4h</sub>, [2,4], (*422)
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| |D<sub>2h</sub>, [2,2], (*222)
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| |- align=center
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| !Symmetry order
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| |48
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| |24
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| |12<BR>6
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| |16
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| |8
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| |}
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| ===Dual===
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| The octahedron is the [[dual polyhedron]] to the [[cube]].
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| :[[File:Dual Cube-Octahedron.svg|240px]]
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| ===Nets===
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| It has eleven arrangements of [[net (polyhedron)|nets]].
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| ==Irregular octahedra==
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| 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.
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| *''Triangular [[antiprism]]s'': Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles.
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| *Tetragonal [[bipyramid]]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.
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| *[[Schönhardt polyhedron]], a nonconvex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.
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| === Other convex octahedra ===
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| 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.<ref>[http://www.uwgb.edu/dutchs/symmetry/polynum0.htm]</ref>
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| 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.<ref>[http://www.numericana.com/data/polycount.htm Counting polyhedra]</ref><ref>http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm</ref> (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.)
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| Some better known irregular octahedra include the following:
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| *[[Hexagonal prism]]: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges.
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| *Heptagonal [[Pyramid (geometry)|pyramid]]: 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.
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| *[[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.
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| *[[Tetragonal trapezohedron]]: The eight faces are congruent [[kite (geometry)|kites]].
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| ==Related polyhedra==
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| 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]].
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| {| class=wikitable
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| |[[File:Triangulated tetrahedron.png|120px]]
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| |[[File:Compound_of_two_tetrahedra.png|120px]]
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| |-
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| ![[tetrahedron]]
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| ![[stellated octahedron]]
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| |}
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| The octahedron is one of a family of uniform polyhedra related to the cube.
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| {{Octahedral truncations}}
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| The octahedron is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,''n''}, continuing into the [[Hyperbolic space|hyperbolic plane]].
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| {{Triangular regular tiling}}
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| ===Tetratetrahedron===
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| The regular octahedron can also be considered a ''[[rectification (geometry)|rectified]] tetrahedron'' – and can be called a ''tetratetrahedron''. This can be shown by a 2-color face model. With this coloring, the octahedron has [[tetrahedral symmetry]].
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| Compare this truncation sequence between a tetrahedron and its dual:
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| {{Tetrahedron family}} <!-- This template shows too many figures. It needs replacing with the simple set described in the text -->
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| The above shapes may also be realized as slices orthogonal to the long diagonal of a [[tesseract]]. If this diagonal is oriented vertically with a height of 1, then the first five slices above occur at heights ''r'', 3/8, 1/2, 5/8, and ''s'', where ''r'' is any number in the range (0,1/4], and ''s'' is any number in the range [3/4,1).
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| The tetratetrahedron can be seen in a sequence of [[quasiregular polyhedron]]s and tilings:
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| {{Quasiregular figure table}}
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| ===Trigonal antiprism===
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| As a [[antiprism|trigonal antiprism]], the octahedron is related to the hexagonal dihedral symmetry family.
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| {{Hexagonal dihedral truncations}}
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| {{UniformAntiprisms}}
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| ===Square bipyramid===
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| {{Bipyramids}}
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| ===Tetrahemihexahedron===
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| The regular octahedron shares its edges and vertex arrangement with one [[nonconvex uniform polyhedron]]: the [[tetrahemihexahedron]], with which it shares four of the triangular faces.
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| {| class="wikitable" width="400" style="vertical-align:top;text-align:center"
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| | [[Image:octahedron.png|100px]]<br>Octahedron
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| | [[Image:tetrahemihexahedron.png|100px]]<br>[[Tetrahemihexahedron]]
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| |}
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| ===Tetrahedral Truss===
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| A framework of repeating tetrahedrons and octahedrons was invented by [[Buckminster Fuller]] in the 1950s, known as a [[space frame]], commonly regarded as the strongest structure for resisting [[cantilever]] stresses.
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| ==Octahedra in the physical world==
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| ===Octahedra in nature===
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| [[Image:Fluorite octahedron.jpg|thumb|[[Fluorite]] octahedron.]]
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| *Natural crystals of [[diamond]], [[alum]] or [[fluorite]] are commonly octahedral, as the space-filling [[tetrahedral-octahedral honeycomb]].
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| *The plates of [[kamacite]] alloy in [[octahedrite]] [[meteorites]] are arranged paralleling the eight faces of an octahedron.
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| *Many metal ions [[Coordination chemistry|coordinate]] six ligands in an octahedral or [[Jahn-Teller effect|distorted]] octahedral configuration.
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| * [[Widmanstätten pattern]]s in [[nickel]]-[[iron]] [[crystal]]s
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| ===Octahedra in art and culture===
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| [[Image:Rubiks snake octahedron.jpg|thumb|Two identically formed [[rubik's snake]]s can approximate an octahedron.]]
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| *Especially in [[roleplaying game]]s, this solid is known as a "d8", one of the more common [[dice#Non-cubical dice|non-cubical dice]].
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| *If each edge of an octahedron is replaced by a one [[ohm (unit)|ohm]] [[resistor]], the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.<ref>{{cite journal |last= Klein |first=Douglas J. |year=2002 |title=Resistance-Distance Sum Rules |journal=Croatica Chemica Acta |volume=75 |issue=2 |pages=633–649 |url=http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf |format=PDF |accessdate=2006-09-30}}</ref>
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| *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]].
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| ==See also==
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| *[[:Image:Octahedron.gif|Spinning octahedron]]
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| *[[Stella octangula]]
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| *[[Triakis octahedron]]
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| *[[Disdyakis dodecahedron|Hexakis octahedron]]
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| *[[Truncated octahedron]]
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| *[[Octahedral molecular geometry]]
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| *[[Octahedral symmetry]]
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| *[[Octahedral graph]]
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| ==References==
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| <!--See [[Wikipedia:Footnotes]] for instructions.-->
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| <references/>
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| ==External links==
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| {{Wikisource1911Enc}}
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| *{{mathworld |urlname=Octahedron |title=Octahedron}}
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| *{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3o4o - oct}}
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| *[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1S2GJRuqXD7blH9ixbn1mPoTSPo6vkjWddm7xoNxFj&name=Octahedron#applet Editable printable net of an octahedron with interactive 3D view]
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| *[http://www.software3d.com/Octahedron.php Paper model of the octahedron]
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| *[http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra]
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| *[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
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| {{Polyhedra}}
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| {{Polyhedron navigator}}
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| {{Polytopes}}
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| {{Use dmy dates|date=December 2010}}
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| [[Category:Deltahedra]]
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| [[Category:Platonic solids]]
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| [[Category:Prismatoid polyhedra]]
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| [[Category:Pyramids and bipyramids]]
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