Truncated cuboctahedron

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Truncated cuboctahedron
Truncatedcuboctahedron.jpg
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 26, E = 72, V = 48 (χ = 2)
Faces by sides 12{4}+8{6}+6{8}
Conway notation bC or taC
Schläfli symbols tr{4,3} or t\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}
t0,1,2{4,3}
Wythoff symbol 2 3 4 |
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group Oh, BC3, [4,3], (*432), order 48
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 4-6:arccos(-sqrt(6)/3)=144°44'08"
4-8:arccos(-sqrt(2)/3)=135°
6-8:arccos(-sqrt(3)/3)=125°15'51"
References U11, C23, W15
Properties Semiregular convex zonohedron
Great rhombicuboctahedron.png
Colored faces		 Great rhombicuboctahedron vertfig.png
4.6.8
(Vertex figure)
Disdyakisdodecahedron.jpg
Disdyakis dodecahedron
(dual polyhedron)		 Truncated cuboctahedron flat.svg
Net

In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron.

Other names

Alternate interchangeable names are:

The name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles. However, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular.

The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the same planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. Compare to small rhombicuboctahedron.

One unfortunate point of confusion: There is a nonconvex uniform polyhedron by the same name. See nonconvex great rhombicuboctahedron.

Cartesian coordinates

The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all permutations of:

(±1, ±(1+√2), ±(1+2√2))

Area and volume

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

A = 12\left(2+\sqrt{2}+\sqrt{3}\right) a^2 \approx 61.7551724a^2
V = \left(22+14\sqrt{2}\right) a^3 \approx 41.7989899a^3.

Dissection

The truncated cuboctahedron can be dissected into a central rhombicuboctahedron, with 6 square cupolas above each primary square face, 8 triangular cupola above each triangular face, and 12 cubes above the secondary square faces.

A dissected truncated cuboctahedron can create a genus 5, 7 or 11 Stewart toroid by removing the central rhombicuboctahedron and either the square cupola, the triangular cupola or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing a subset of these dissected components. For example, removing half of the triangular cupola creates a genus 3 torus, which (if they are chosen appropriately) has tetrahedral symmetry. [4][5]

Stewart toroids
Genus 3 Genus 5 Genus 7 Genus 11
Excavated truncated cuboctahedron4.png Excavated truncated cuboctahedron2.png Excavated truncated cuboctahedron3.png Excavated truncated cuboctahedron.png

Uniform colorings

There is only one uniform coloring of the faces of this polyhedron, one color for each face type.

A 2-uniform coloring, with tetrahedral symmetry, exists with alternately colored hexagons.

Orthogonal projections

The truncated cuboctahedron has two special orthogonal projections in the A2 and B2 Coxeter planes with [6] and [8] projective symmetry, and numerous [2] symmetries can be constructed from various projected planes relative to the polyhedron elements.

Orthogonal projections
Centered by Vertex Edge
4-6		 Edge
4-8		 Edge
6-8		 Face normal
4-6
Image Cube t012 v.png Cube t012 e46.png Cube t012 e48.png Cube t012 e68.png Cube t012 f46.png
Projective
symmetry		 [2]+ [2] [2] [2] [2]
Centered by Face normal
Square		 Face normal
Octagon		 Face
Square		 Face
Hexagon		 Face
Octagon
Image Cube t012 af4.png Cube t012 af8.png Cube t012 f4.png 3-cube t012.svg 3-cube t012 B2.svg
Projective
symmetry		 [2] [2] [2] [6] [8]

Spherical tiling

The truncated 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-t012.png Truncated cuboctahedron stereographic projection square.png
square-centered		 Truncated cuboctahedron stereographic projection hexagon.png
hexagon-centered		 Truncated cuboctahedron stereographic projection octagon.png
octagon-centered
Orthogonal projection Stereographic projections

Related polyhedra

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

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

Truncated cuboctahedral graph

Truncated cuboctahedral graph
Truncated cuboctahedral graph.png
4-fold symmetry
Vertices 48
Edges 72
Automorphisms 48
Chromatic number 2
Properties Cubic, Hamiltonian, regular, zero-symmetric

In the mathematical field of graph theory, a truncated cuboctahedral graph (or great rhombcuboctahedral graph) is the graph of vertices and edges of the truncated cuboctahedron, one of the Archimedean solids. It has 48 vertices and 72 edges, and is a zero-symmetric and cubic Archimedean graph.[6]

See also

References

  1. ↑ Lua error in package.lua at line 80: module 'strict' not found. (Model 15, p. 29)
  2. ↑ Lua error in package.lua at line 80: module 'strict' not found. (Section 3-9, p. 82)
  3. ↑ Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999). (p. 82)
  4. ↑ B. M. Stewart, Adventures Among the Toroids (1970) ISBN 978-0-686-11936-4
  5. ↑ http://www.doskey.com/polyhedra/Stewart05.html
  6. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  • Lua error in package.lua at line 80: module 'strict' not found.

External links

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