Octahedron

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Regular octahedron
Octahedron.jpg
(Click here for rotating model)
Type Platonic solid
Elements F = 8, E = 12
V = 6 (χ = 2)
Faces by sides8{3}
Conway notation O
aT
Schläfli symbols {3,4}
r{3,3} or
Face configuration V4.4.4
Wythoff symbol 4 | 2 3
Coxeter diagram CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry Oh, BC3, [4,3], (*432)
Rotation group O, [4,3]+, (432)
References U 05, C 17, W 2
Properties regular, convex deltahedron
Dihedral angle 109.47122° = arccos(−13)
Octahedron vertfig.png
3.3.3.3
(Vertex figure)
Hexahedron.png
Cube
(dual polyhedron)
Octahedron flat.svg
Net
3D model of regular octahedron. Octahedron.stl
3D model of regular octahedron.

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.

Contents

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 in the Manhattan (1) 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

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

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

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.

Orthogonal projections
Centered byEdgeFace
Normal
VertexFace
Image Cube t2 e.png Cube t2 fb.png 3-cube t2 B2.svg 3-cube t2.svg
Projective
symmetry
[2][2][4][6]

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.

Uniform tiling 432-t2.png Octahedron stereographic projection.svg
Orthographic projection Stereographic projection

Cartesian coordinates

An octahedron with edge length 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

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

In an xyz 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

Area and volume

The surface 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 (because we have 8 rather than 4 triangles).

If an octahedron has been stretched so that it obeys the equation

the formulas for the surface area and volume expand to become

Additionally the inertia tensor of the stretched octahedron is

These reduce to the equations for the regular octahedron when

Geometric relations

Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid .

Dual

The octahedron is the dual polyhedron of the cube.

Dual Cube-Octahedron.svg

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

Stellation

The octahedron represents the central intersection of two tetrahedra Compound of two tetrahedra.png
The octahedron represents the central intersection of two tetrahedra

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.

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. 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 honeycombs 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 of its orthoscheme. The orthoscheme occurs in two chiral 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. 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 is denoted B3. The octahedron and its dual polytope, 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 [1] of the regular octahedron CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png which subdivides it into 48 of these characteristic orthoschemes CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 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. [2]

Characteristics of the regular octahedron [3]
edgearcdihedral
𝒍90°109°28′
𝟀54°44′8″90°
𝝓45°60°
𝟁35°15′52″45°

If the octahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths , , (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁 of the octahedron), plus , , (edges that are the characteristic radii of the octahedron). The 3-edge path along orthogonal edges of the orthoscheme is , , , 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 , , , a right triangle with edges , , , and a right triangle with edges , , .

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 sets 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. [4]

Nets

The regular octahedron has eleven arrangements of nets.

Faceting

The uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement. It has four of the triangular faces, and 3 central squares.

Uniform polyhedron-33-t1.png
Octahedron
Tetrahemihexahedron.png
Tetrahemihexahedron

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 symmetry group is Oh, of order 48, the three dimensional hyperoctahedral group. 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 colorings of the faces.

NameOctahedron Rectified tetrahedron
(Tetratetrahedron)
Triangular antiprism Square bipyramid Rhombic fusil
Image
(Face coloring)
Uniform polyhedron-43-t2.png
(1111)
Uniform polyhedron-33-t1.png
(1212)
Trigonal antiprism.png
(1112)
Square bipyramid.png
(1111)
Rhombic bipyramid.png
(1111)
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1.pngCDel nodes.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 2x.pngCDel node f1.png
Schläfli symbol {3,4}r{3,3}s{2,6}
sr{2,3}
ft{2,4}
{ } + {4}
ftr{2,2}
{ } + { } + { }
Wythoff symbol 4 | 3 22 | 4 32 | 6 2
| 2 3 2
Symmetry Oh, [4,3], (*432)Td, [3,3], (*332)D3d, [2+,6], (2*3)
D3, [2,3]+, (322)
D4h, [2,4], (*422)D2h, [2,2], (*222)
Order 482412
6
168

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.

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. [5] 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. [6] [7] (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:

Octahedra in the physical world

Octahedra in nature

Fluorite octahedron. Fluorite octahedron.jpg
Fluorite octahedron.

Octahedra in art and culture

Two identically formed Rubik's Snakes can approximate an octahedron. Rubiks snake octahedron.jpg
Two identically formed Rubik's Snakes can approximate an octahedron.

Tetrahedral octet truss

A space frame of alternating tetrahedra and half-octahedra derived from the Tetrahedral-octahedral honeycomb was invented by Buckminster Fuller in the 1950s. It is commonly regarded as the strongest building structure for resisting cantilever stresses.

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.

Triangulated tetrahedron.png Compound of two tetrahedra.png
tetrahedron stellated octahedron

The octahedron is one of a family of uniform polyhedra related to the cube.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.png Uniform polyhedron-33-t2.png Uniform polyhedron-33-t01.png Uniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.svg
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.jpg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.jpg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.jpg Triakistetrahedron.jpg Dodecahedron.jpg

It is also one of the simplest examples of a hypersimplex, a polytope formed by certain intersections of a hypercube with a hyperplane.

The octahedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.

*n32 symmetry mutation of regular tilings: {3,n}
SphericalEuclid.Compact hyper.Paraco.Noncompact hyperbolic
Trigonal dihedron.svg Uniform tiling 332-t2.png Uniform tiling 432-t2.png Uniform tiling 532-t2.png Uniform polyhedron-63-t2.png Order-7 triangular tiling.svg H2-8-3-primal.svg H2 tiling 23i-4.png H2 tiling 23j12-4.png H2 tiling 23j9-4.png H2 tiling 23j6-4.png H2 tiling 23j3-4.png
3.3 33 34 35 36 37 38 3 312i39i36i33i

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 tetrahedral symmetry.

Compare this truncation sequence between a tetrahedron and its dual:

Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332)[3,3]+, (332)
Uniform polyhedron-33-t0.png Uniform polyhedron-33-t01.png Uniform polyhedron-33-t1.png Uniform polyhedron-33-t12.png Uniform polyhedron-33-t2.png Uniform polyhedron-33-t02.png Uniform polyhedron-33-t012.png Uniform polyhedron-33-s012.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
{3,3} t{3,3} r{3,3} t{3,3} {3,3} rr{3,3} tr{3,3} sr{3,3}
Duals to uniform polyhedra
Tetrahedron.svg Triakistetrahedron.jpg Hexahedron.svg Triakistetrahedron.jpg Tetrahedron.svg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Dodecahedron.svg
V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.3.3.3.3

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 < r1/4, and s is any number in the range 3/4s < 1.

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 symmetry of *n32 all of these tilings are Wythoff constructions within a fundamental domain of symmetry, with generator points at the right angle corner of the domain. [9] [10]

*n32 orbifold symmetries of quasiregular tilings: (3.n)2
Quasiregular fundamental domain.png
Construction
Spherical EuclideanHyperbolic
*332*432*532*632*732*832...*32
Quasiregular
figures
Uniform tiling 332-t1-1-.png Uniform tiling 432-t1.png Uniform tiling 532-t1.png Uniform tiling 63-t1.svg Triheptagonal tiling.svg H2-8-3-rectified.svg H2 tiling 23i-2.png
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.)2

Trigonal antiprism

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

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622)[6,2]+, (622)[6,2+], (2*3)
Hexagonal dihedron.png Dodecagonal dihedron.png Hexagonal dihedron.png Spherical hexagonal prism.png Spherical hexagonal hosohedron.png Spherical truncated trigonal prism.png Spherical dodecagonal prism2.png Spherical hexagonal antiprism.png Spherical trigonal antiprism.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
{6,2} t{6,2} r{6,2} t{2,6} {2,6} rr{6,2} tr{6,2} sr{6,2} s{2,6}
Duals to uniforms
Spherical hexagonal hosohedron.png Spherical dodecagonal hosohedron.png Spherical hexagonal hosohedron.png Spherical hexagonal bipyramid.png Hexagonal dihedron.png Spherical hexagonal bipyramid.png Spherical dodecagonal bipyramid.png Spherical hexagonal trapezohedron.png Spherical trigonal trapezohedron.png
V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V3.3.3.3
Family of uniform n-gonal antiprisms
Antiprism name Digonal antiprism (Trigonal)
Triangular antiprism
(Tetragonal)
Square antiprism
Pentagonal antiprism Hexagonal antiprism Heptagonal antiprism Octagonal antiprism Enneagonal antiprism Decagonal antiprism Hendecagonal antiprismDodecagonal antiprism... Apeirogonal antiprism
Polyhedron image Digonal antiprism.png Trigonal antiprism.png Square antiprism.png Pentagonal antiprism.png Hexagonal antiprism.png Antiprism 7.png Octagonal antiprism.png Enneagonal antiprism.png Decagonal antiprism.png Hendecagonal antiprism.png Dodecagonal antiprism.png ...
Spherical tiling image Spherical digonal antiprism.png Spherical trigonal antiprism.png Spherical square antiprism.png Spherical pentagonal antiprism.png Spherical hexagonal antiprism.png Spherical heptagonal antiprism.png Spherical octagonal antiprism.png Plane tiling image Infinite antiprism.svg
Vertex config. 2.3.3.33.3.3.34.3.3.35.3.3.36.3.3.37.3.3.38.3.3.39.3.3.310.3.3.311.3.3.312.3.3.3...∞.3.3.3

Square bipyramid

"Regular" right (symmetric) n-gonal bipyramids:
Bipyramid nameDigonal bipyramid Triangular bipyramid
(See: J12)
Square bipyramid
(See: O)
Pentagonal bipyramid
(See: J13)
Hexagonal bipyramid Heptagonal bipyramid Octagonal bipyramid Enneagonal bipyramid Decagonal bipyramid ... Apeirogonal bipyramid
Polyhedron image Triangular bipyramid.png Square bipyramid.png Pentagonale bipiramide.png Hexagonale bipiramide.png Heptagonal bipyramid.png Octagonal bipyramid.png Enneagonal bipyramid.png Decagonal bipyramid.png ...
Spherical tiling image Spherical digonal bipyramid.svg Spherical trigonal bipyramid.png Spherical square bipyramid.svg Spherical pentagonal bipyramid.svg Spherical hexagonal bipyramid.png Spherical heptagonal bipyramid.png Spherical octagonal bipyramid.png Spherical enneagonal bipyramid.png Spherical decagonal bipyramid.png Plane tiling image Infinite bipyramid.svg
Face config. V2.4.4V3.4.4V4.4.4V5.4.4V6.4.4V7.4.4V8.4.4V9.4.4V10.4.4...V∞.4.4
Coxeter diagram CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2x.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 5.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 7.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 8.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 9.pngCDel node.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 10.pngCDel node.png...CDel node f1.pngCDel 2.pngCDel node f1.pngCDel infin.pngCDel node.png

Truncation of two opposite vertices results in a square bifrustum.

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

See also

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A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

<span class="mw-page-title-main">Cube</span> Solid object with six equal square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

<span class="mw-page-title-main">Regular icosahedron</span> One of the five Platonic solids

In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

<span class="mw-page-title-main">Tetrahedron</span> Polyhedron with 4 faces

In geometry, a tetrahedron, 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 the ordinary convex polyhedra and the only one that has fewer than 5 faces.

<span class="mw-page-title-main">Truncated tetrahedron</span> Archimedean solid with 8 faces

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges. It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.

<span class="mw-page-title-main">Truncated octahedron</span> Archimedean solid

In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces, 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron.

<span class="mw-page-title-main">Truncated cube</span>

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.

<span class="mw-page-title-main">24-cell</span> Regular object in four dimensional geometry

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

<span class="mw-page-title-main">5-cell</span> Four-dimensional analogue of the tetrahedron

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.

<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

<span class="mw-page-title-main">16-cell</span> Four-dimensional analog of the octahedron

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid [sic?].

<span class="mw-page-title-main">Disdyakis dodecahedron</span> Geometric shape with 48 faces

In geometry, a disdyakis dodecahedron,, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron. The net of the rhombic dodecahedral pyramid also shares the same topology.

<span class="mw-page-title-main">Pyramid (geometry)</span> Conic solid with a polygonal base

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, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.

<span class="mw-page-title-main">Tetrahedral-octahedral honeycomb</span> Quasiregular space-filling tesselation

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

In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.

<span class="mw-page-title-main">Alternation (geometry)</span> Removal of alternate vertices

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

<span class="mw-page-title-main">Regular dodecahedron</span> Platonic solid

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.

References

  1. Coxeter 1973, p. 130, §7.6 The symmetry group of the general regular polytope; "simplicial subdivision".
  2. Coxeter 1973, pp. 70–71, Characteristic tetrahedra; Fig. 4.7A.
  3. Coxeter 1973, pp. 292–293, Table I(i); "Octahedron, 𝛽3".
  4. Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.; Plummer, Michael D. (2010). "On well-covered triangulations. III". Discrete Applied Mathematics. 158 (8): 894–912. doi: 10.1016/j.dam.2009.08.002 . MR   2602814.
  5. "Enumeration of Polyhedra". Archived from the original on 10 October 2011. Retrieved 2 May 2006.
  6. "Counting polyhedra".
  7. "Polyhedra with 8 Faces and 6-8 Vertices". Archived from the original on 17 November 2014. Retrieved 14 August 2016.
  8. Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta. 75 (2): 633–649. Archived from the original (PDF) on 10 June 2007. Retrieved 30 September 2006.
  9. Coxeter Regular Polytopes , Third edition, (1973), Dover edition, ISBN   0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
  10. "Two Dimensional symmetry Mutations by Daniel Huson".
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds