Cuboctahedron

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Cuboctahedron
Cuboctahedron
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 14, E = 24, V = 12 (χ = 2)
Faces by sides 8{3}+6{4}
Conway notation aC
aaT
Schläfli symbols r{4,3} or \begin{Bmatrix} 4 \\ 3 \end{Bmatrix}
rr{3,3} or r\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}
t1{4,3} or t0,2{3,3}
Wythoff symbol 2 | 3 4
3 3 | 2
Coxeter diagram CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry group Oh, BC3, [4,3], (*432), order 48
Td, [3,3], (*332), order 24
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 125.26°
 \sec^{-1} \left(-\sqrt{3}\right)
References U07, C19, W11
Properties Semiregular convex quasiregular
Cuboctahedron.png
Colored faces
Cuboctahedron
3.4.3.4
(Vertex figure)
Rhombicdodecahedron.jpg
Rhombic dodecahedron
(dual polyhedron)
Cuboctahedron flat.svg
Net

In 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, i.e. an Archimedean solid, being vertex-transitive and edge-transitive.

Its dual polyhedron is the rhombic dodecahedron.

Other names[edit]

Area and volume[edit]

The area A and the volume V of the cuboctahedron of edge length a are:

A = \left(6+2\sqrt{3}\right)a^2 \approx 9.4641016a^2
V = \frac{5}{3} \sqrt{2}a^3 \approx 2.3570226a^3.

Orthogonal projections[edit]

The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes. The skew projections show a square and hexagon passing through the center of the cuboctahedron.

Cuboctahedron (orthogonal projections)
Square
Face
Triangular
Face
Vertex Edge Skew
3-cube t1 B2.svg 3-cube t1.svg Cube t1 v.png Cube t1 e.png Cuboctahedron B2 planes.png Cuboctahedron 3 planes.png
[4] [6] [2] [2]
Rhombic dodecahedron (Dual polyhedron)
Dual cube t1 B2.png Dual cube t1.png Dual cube t1 v.png Dual cube t1 e.png Dual cube t1 skew1.png Dual cube t1 skew2.png

Spherical tiling[edit]

The cuboctahedron 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-t1.png Cuboctahedron stereographic projection square.png
square-centered
Cuboctahedron stereographic projection triangle.png
triangle-centered
orthographic projection Stereographic projections

Cartesian coordinates[edit]

The Cartesian coordinates for the vertices of a cuboctahedron (of edge length √2) centered at the origin are:

(±1,±1,0)
(±1,0,±1)
(0,±1,±1)

An alternate set of coordinates can be made in 4-space, as 12 permutations of:

(0,1,1,2)

This construction exists as one of 16 orthant facets of the cantellated 16-cell.

Root vectors[edit]

The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B3.

Dissection[edit]

The cuboctahedron can be dissected into two triangular cupola by a common hexagon passing through the center of the cuboctahedron. If these two triangular cupola are twisted so triangles and squares line up, Johnson solid J27, triangular orthobicupola is created.

Cuboctahedron 3 planes.pngTriangular cupola.pngTriangular orthobicupola.png

The cuboctahedron can also be dissected into 6 square pyramids, and 8 tetrahedra meeting at a central point. This dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra.

TetraOctaHoneycomb-VertexConfig.svg

Geometric relations[edit]

A cuboctahedron can be obtained by taking an appropriate cross section of a four-dimensional 16-cell.

A cuboctahedron has octahedral symmetry. Its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either.

The cuboctahedron is a rectified cube and also a rectified octahedron.

It is also a cantellated tetrahedron. With this construction it is given the Wythoff symbol: 3 3 | 2. Cantellated tetrahedron.png

A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, and six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmetry group and its vertices are equivalent under that group.

The edges of a cuboctahedron form four regular hexagons. If the cuboctahedron is cut in the plane of one of these hexagons, each half is a triangular cupola, one of the Johnson solids; the cuboctahedron itself thus can also be called a triangular gyrobicupola, the simplest of a series (other than the gyrobifastigium or "digonal gyrobicupola"). If the halves are put back together with a twist, so that triangles meet triangles and squares meet squares, the result is another Johnson solid, the triangular orthobicupola, also called an anticuboctahedron.

Both triangular bicupolae are important in sphere packing. The distance from the solid's center to its vertices is equal to its edge length. Each central sphere can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal close-packed lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's center.

Cuboctahedra appear as cells in three of the convex uniform honeycombs and in nine of the convex uniform polychora.

The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron.

Vertex arrangement[edit]

The cuboctahedron shares its edges and vertex arrangement with two nonconvex uniform polyhedra: the cubohemioctahedron (having the square faces in common) and the octahemioctahedron (having the triangular faces in common). It also serves as a cantellated tetrahedron, as being a rectified tetratetrahedron.

Cuboctahedron.png
Cuboctahedron
Cubohemioctahedron.png
Cubohemioctahedron
Octahemioctahedron.png
Octahemioctahedron

The cuboctahedron 2-covers the tetrahemihexahedron,[1] which accordingly has the same abstract vertex figure (two triangles and two squares: 3.4.3.4) and half the vertices, edges, and faces. (The actual vertex figure of the tetrahemihexahedron is 3.4.3/2.4, with the a/2 factor due to the cross.)

Cuboctahedron.png
Cuboctahedron
Tetrahemihexahedron.png
Tetrahemihexahedron

Related polyhedra[edit]

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

Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332) [3,3]+, (332)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
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.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
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Tetrahedron.svg Triakistetrahedron.jpg Hexahedron.svg Triakistetrahedron.jpg Tetrahedron.svg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg POV-Ray-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
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.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel 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.png CDel 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.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
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.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg POV-Ray-Dodecahedron.svg

The cuboctahedron can be seen in a sequence of quasiregular polyhedrons and tilings:

Dimensional family of quasiregular polyhedra and tilings: 3.n.3.n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
p6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
 
[iπ/λ,3]
Quasiregular
figures
configuration
Uniform tiling 332-t1-1-.png
3.3.3.3
Uniform tiling 432-t1.png
3.4.3.4
Uniform tiling 532-t1.png
3.5.3.5
Uniform tiling 63-t1.png
3.6.3.6
Uniform tiling 73-t1.png
3.7.3.7
Uniform tiling 83-t1.png
3.8.3.8
H2 tiling 23i-2.png
3.∞.3.∞
3.∞.3.∞
Coxeter diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 3.pngCDel node.png
Dual
(rhombic)
figures
configuration
Hexahedron.svg
V3.3.3.3
Rhombicdodecahedron.jpg
V3.4.3.4
Rhombictriacontahedron.svg
V3.5.3.5
Rhombic star tiling.png
V3.6.3.6
Order73 qreg rhombic til.png
V3.7.3.7
Uniform dual tiling 433-t01-yellow.png
V3.8.3.8
Ord3infin qreg rhombic til.png
V3.∞.3.∞
Coxeter diagram CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel ultra.pngCDel node f1.pngCDel 3.pngCDel node.png
Dimensional family of quasiregular polyhedra and tilings: 4.n.4.n
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
 
[iπ/λ,4]
Coxeter CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 4.pngCDel node.png
Quasiregular
figures
configuration
Uniform tiling 432-t1.png
4.3.4.3
Uniform tiling 44-t1.png
4.4.4.4
Uniform tiling 54-t1.png
4.5.4.5
Uniform tiling 64-t1.png
4.6.4.6
Uniform tiling 74-t1.png
4.7.4.7
Uniform tiling 84-t1.png
4.8.4.8
H2 tiling 24i-2.png
4.∞.4.∞
4.∞.4.∞
Dual figures
Coxeter CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel ultra.pngCDel node f1.pngCDel 4.pngCDel node.png
Dual
(rhombic)
figures
configuration
Rhombicdodecahedron.jpg
V4.3.4.3
Uniform tiling 44-t0.png
V4.4.4.4
Order-5-4 quasiregular rhombic tiling.png
V4.5.4.5
Ord64 qreg rhombic til.png
V4.6.4.6
Ord74 qreg rhombic til.png
V4.7.4.7
Ord84 qreg rhombic til.png
V4.8.4.8
Ord4infin qreg rhombic til.png
V4.∞.4.∞
V4.∞.4.∞

This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

Dimensional family of expanded polyhedra and tilings: 3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
 
*832
[8,3]...
 
*∞32
[∞,3]
 
Expanded
figure
Spherical triangular prism.png
3.4.2.4
Uniform tiling 332-t02.png
3.4.3.4
Uniform tiling 432-t02.png
3.4.4.4
Uniform tiling 532-t02.png
3.4.5.4
Uniform polyhedron-63-t02.png
3.4.6.4
Uniform tiling 73-t02.png
3.4.7.4
Uniform tiling 83-t02.png
3.4.8.4
H2 tiling 23i-5.png
3.4.∞.4
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{2,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{4,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{5,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{6,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{7,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{8,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{∞,3}
Deltoidal figure Triangular dipyramid.png
V3.4.2.4
Rhombicdodecahedron.jpg
V3.4.3.4
Deltoidalicositetrahedron.jpg
V3.4.4.4
Deltoidalhexecontahedron.jpg
V3.4.5.4
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V3.4.6.4
Deltoidal triheptagonal til.png
V3.4.7.4
Deltoidal trioctagonal til.png
V3.4.8.4
Deltoidal triapeirogonal til.png
V3.4.∞.4
Coxeter CDel node f1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png

Related polytopes[edit]

The cuboctahedron can be decomposed into a regular octahedron and eight irregular but equal octahedra in the shape of the convex hull of a cube with two opposite vertices removed. This decomposition of the cuboctahedron corresponds with the cell-first parallel projection of the 24-cell into three dimensions. Under this projection, the cuboctahedron forms the projection envelope, which can be decomposed into six square faces, a regular octahedron, and eight irregular octahedra. These elements correspond with the images of six of the octahedral cells in the 24-cell, the nearest and farthest cells from the 4D viewpoint, and the remaining eight pairs of cells, respectively.

Cultural occurrences[edit]

Two cuboctahedra on a chimney. Israel.
  • In the Star Trek episode "By Any Other Name", aliens seize the Enterprise by transforming crew members into inanimate cuboctahedra.
  • The "Geo Twister" fidget toy [1] is a flexible cuboctahedron.
  • The Coriolis space stations in the computer game Elite are cuboctahedron-shaped.

See also[edit]

References[edit]

  • Ghyka, Matila (1977). The geometry of art and life. ([Nachdr.] ed.). New York: Dover Publications. pp. 51–56, 81–84. ISBN 9780486235424. 
  • Richter, David A., Two Models of the Real Projective Plane 
  • Weisstein, Eric W. (2002). "Cuboctahedron". CRC Concise Encyclopedia of Mathematics. (2nd ed.). Hoboken: CRC Press. pp. 620–621. ISBN 9781420035223. 
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
  • Cromwell, P. Polyhedra, CUP hbk (1997), pbk. (1999). Ch.2 p.79-86 Archimedean solids

External links[edit]